# Probability Seminar at King's

The Probability Seminar at King's is dedicated to presenting the latest advances in Pure and Applied Probability. If only a room number is given for a talk, then it is taking place in King's College London, Strand Campus.

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## Talks

Forthcoming Past### October 2022

03/10/2022 - Ilya Pavlyukevich (Friedrich Schiller University Jena)

15:30-16:30 STRAND BLDG S4.29

We revisit a one-dimensional selection problem for a stochastic differential equation $dX=|X|^\beta\sign(X) dt + \epsilon dL$ driven by an alpha-stable Lévy process L, $\epsilon \to 0$, $\beta\in (0,1)$, considered in by Pilipenko and Proske (Statist.
Probab. Lett., 132:62--73, 2018) and show that the selection phenomenon pertains in the multiplicative noise setting and is robust with respect to certain perturbations of the irregular drift and of the small jumps of the noise.
This is the joint work with A. Pilipenko (ECP, 25:1--14, 2020).

### June 2022

27/06/2022 - Jordan Stoyanov (Bulgarian Academy of Sciences, Sofia)

15:30-16:30 STRAND BLDG S3.41

We start with a couple of examples and refreshing comments on the
moment determinacy of probability distributions (M-det or M-indet). This will
illustrate the direction of our discussion.
(1) We suggest a new geometric approach when presenting old classical results
of the type "iff" for M-determinacy in terms of the smallest eigenvalues of Hankel
matrices. New proofs of known results will be given. These are beautiful results,
however they rely on "non-checkable" conditions.
(2) We turn to other recent results based on "checkable" conditions which are only
sufficient or only necessary for either M-determinacy, or M-indeterminacy. We treat
both absolutely continuous and discrete distributions.
(3) We discuss a few related and challenging questions by involving the cumulants,
multivariate distributions and describing new methods to construct Stieltjes classes
for M-indet distributions.
(4) The new results, ideas and techniques will be explained and well referenced.
A series of open questions will be briefly outlined.

### March 2022

28/03/2022 - Boris Khoruzhenko

Draw a matrix at random from one of the classical compact groups of large dimension and remove an equal number of rows and columns from it. What can be said about the distribution of eigenvalues of the truncated matrix? This question for the complex unitary group was addressed by Sommers and Zyczkowski in their pioneering paper in 2000. The real orthogonal case was addressed by a number of research groups and is, nowadays, well understood too. Surprisingly, little was known about the remaining case of the unitary symplectic matrices. In the first part of this talk I will explain why truncations of unitary symplectic matrices are interesting too, and in the second part I will describe the distribution of their complex eigenvalues in the regimes of weak and strong non-unitarity and explain some analytic challenges of getting asymptotics of the eigenvalue correlation functions in planar symplectic ensembles.

14/03/2022 - Alexander Drewitz

15:30-16:30

Percolation models have been playing a fundamental role in statistical physics for several decades by now. They had initially been investigated in the gelation of polymers during the 1940s by chemistry Nobel laureate Flory and Stockmayer. From a mathematical point of view, the birth of percolation theory was the introduction of Bernoulli percolation by Broadbent and Hammersley in 1957, motivated by research on gas masks for coal miners. One of the key features of this model is the inherent stochastic independence which simplifies its investigation, and which has lead to very deep mathematical results. During recent years, the investigation of the more realistic and at the same time more complex situation of percolation models with long-range correlations has attracted significant attention.
We will exhibit some recent progress for two emblematic examples of such models, the Gaussian free field and the model of Random Interlacements. In this context, a particular focus is put on the understanding of the critical parameters in the associated percolation models. What is more, we also survey recent progress in the understanding of the model at criticality via its critical exponents.

07/03/2022 - Christoph Reinsinger (University of Oxford)

15:30-15:30

### February 2022

28/02/2022 - Mohammud Foondun (University of Strathclyde)

15:30-16:30

We will review some recent results about non-existence of global solutions to some non-linear stochastic heat equations. We will then look at corresponding results for the wave equations and the stochastic wave equations. We will see how noise can affect blow-up or non-existence properties of the solution.

21/02/2022 - Daniel Lacker (Columbia University)

15:30-16:30

14/02/2022 - Michael Högele (Universidad de los Andes)

15:30-16:30

Many stochastic systems -think of strongly irreducible Markov chains on finite state space- have the tendency to converge (in some metric) as a function of time to a unique dynamical equilibrium. In this talk we give a very short introduction to the so-called cutoff phenomenon for systems which depend on a parameter (for instance, the growing size of the state space or the decreasing noise intensity). It states that in terms of this parameter the system converges sharply along a precise deterministic time scale. That is, morally speaking, any lag behind this time scale implies large distances to the equilibrium, while any advance ahead of this time scale leads to small distances. We explain recent results on the topic for stochastic linear equations with a stable state for small noise in the Wasserstein distance and extensions. These results are part of an ongoing research project with G. Barrera (U. Helsinki) and J.C. Pardo (CIMAT, Mexico).

07/02/2022 - Ting-Kam Leonard Wong (University of Toronto)

15:30-16:30

In stochastic portfolio theory, functionally generated portfolio provides a robust method to exploit the long-term stability of large equity markets. An important practical question is the optimal choice of the "generating function". After reviewing the main ideas of this theory, we propose a concrete and fully implementable approach to the optimization of functionally generated portfolio. The main idea is to optimize over a family of rank-based portfolios parameterized by an exponentially concave function on the unit interval. The resulting optimization problem, which is convex, allows for various regularizations and constraints to be imposed on the generating function. We discuss the theoretical properties of this optimization problem and present empirical examples using CRSP data from the US stock market, including the performance of the portfolios allowing for dividends, defaults, and transaction costs. This is joint work with my PhD student Steven Campbell.

### January 2022

31/01/2022 - Kirstin Strokorb (Cardiff University)

15:30-16:30

Statistical modelling of complex dependencies in extreme events requires meaningful sparsity structures in multivariate extremes. In this context two perspectives on conditional independence and graphical models have recently emerged: One that focuses on threshold exceedances and multivariate pareto distributions, and another that focuses on max-linear models and directed acyclic graphs. What connects these notions is the exponent measure that lies at the heart of each approach. In this work we develop a notion of conditional independence defined directly on the exponent measure (and even more generally on measures that explode at the origin) that extends recent work of Engelke and Hitz (2019), who had been confined to homogeneous measures with density. We prove easier checkable equivalent conditions to verify this new conditional independence in terms of a reduction to simple test classes, probability kernels and density factorizations. This provides a pathsway to graphical modelling among general multivariate (max-)infinitely distributions. Structural max-linear models turn out to form a Bayesian network with respect to our new form of conditional independence.
Joint work (in progress) with Sebastian Engelke and Jevgenijs Ivanovs.

24/01/2022 - Olivier Guéant (University Paris 1)

15:30-16:30

In recent years, interest has grown for portfolio construction methods that do not rely on expected returns. Among risk-based methods, the most popular ones are minimum variance, maximum diversification, and risk budgeting (especially equal risk contribution, aka. ERC). Risk budgeting is particularly attracting because of its versatility: based on the Euler decomposition of positively homogenous functions, it can be used with a large range of risk « measures », from volatility to expected shortfall/CVaR and beyond. The goal of this talk is to present new mathematical results about the construction of risk budgeting portfolios for a very wide spectrum of risk « measures » and to show that, in many cases, stochastic optimization techniques can be used to build risk budgeting portfolios.

17/01/2022 - Raluca Balan (University of Ottawa)

15:30-16:30

In this talk, we introduce some tools which are needed for the stochastic analysis with respect to a Lévy noise, using the random field approach introduced by Walsh (1986). We consider the case of a Lévy white noise, with possibly infinite variance (such as the α-stable Lévy noise). The focus will be on the stochastic wave equation with this type of noise, on the entire space R^d, in dimension is d = 1 or d = 2. In this equation, the noise is multiplied
by a Lipschitz function σ(u) of the solution. We will show that the solution exists and has a càdlàg modification in the local fractional Sobolev space of order r < 1/4 if d = 1, respectively r < −1 if d = 2.

### December 2021

06/12/2021 - Delia Coculescu

### November 2021

29/11/2021 - Dmitry Zaporozhets

One of the classical Sylvester questions asks for the probability that four points $X_1, X_2, X_3, X_4$ independently and uniformly distributed in some plane convex figure $K\subset\mathbb R^2$ form a triangle. Blaschke answered it showing that for any convex figure $K$,
\begin{align}\label{2328}
\frac{35}{12\pi^2}\leq\mathbf P[\mathrm{conv}(X_1,X_2,X_3, X_4) \text{ is triangle}]\leq\frac{1}{3}.
\end{align}
The lower bound is achieved if and only if $K$ is an ellipse, and the
upper one -- if and only if $K$ is a triangle.
It is straightforward that
\begin{align*}
\mathbf P[\mathrm{conv}(X_1,X_2,X_3, X_4) \text{ is triangle}]=4\frac{\mathbf E\, \mathrm S(\mathrm{conv}(X_1,X_2,X_3))}{\mathrm S(K)},
\end{align*}
where $\mathrm S(\cdot)$ denotes the area of a plane figure.
Therefore~\eqref{2328} is equivalent to
\begin{align*}
\frac{35}{48\pi^2}\leq\frac{\mathbf E\, \mathrm S(\mathrm{conv}(X_1,X_2,X_3))}{\mathrm S(K)}\leq\frac{1}{12},
\end{align*}
which gives the optimal lower and upper bounds for the normalized average area of the random triangle inside a convex figure.
If we have only two random points inside $K$, they form a random segment with some random length. Thus it is natural to ask for the optimal bounds of the normalized average length of this segment. While the area of the random triangle is normalized by the area of $K$, the length of the random segment should be normalized by the perimeter of $K$ denoted by $P(K)$.
To answer this question, we will show that for any convex figure
$K\subset\mathbb R^2$ with non-empty interior,
\begin{align}\label{1316}
\frac{7}{60}<\frac{\mathbf E \|X_1-X_2\|)}{\mathrm P(K)}<\frac{1}{6}.
\end{align}
The both lower and upper bounds are optimal. We will also generalize~\eqref{1316} to any dimension.
Based on the joint paper:
G. Bonnet, A. Gusakova, Ch. Thäle, and D. Zaporozhets, ``Sharp
inequalities for the mean distance of random points in convex
bodies'', Adv. Math., 386 (2021)

22/11/2021 - Martin Schweizer

15:30-16:30

In recent years, there have been several criticisms concerning classical arbitrage theory in the spirit of Delbaen and Schachermayer, and some new approaches with different goals and ideas have emerged. We re-examine the classical theory with a particular focus on making it as robust to changes of numeraire as possible in as general a setting as possible. This requires to develop a new (and more general) concept of absence of arbitrage, which we then characterise in a dual manner by martingale properties of the given financial market. As a teaser, we invite you to think about the following question: When exactly is the Black-Scholes model, consisting of a stock given by a geometric Brownian motion and a bank account with a deterministic interest rate, arbitrage-free on an infinite horizon? The talk is based on joint work with Daniel Balint.

15/11/2021 - Maria Eulália Vares

My basic goal in this talk is to discuss one further example of the interplay between SPDEs and percolation methods. We consider a highly anisotropic finite-range bond percolation on $\mathbb{Z}^2$: on horizontal lines we have edges connecting two vertices within distance $N$ and vertical edges are only between nearest neighbor vertices. On this graph we consider the following independent percolation model: horizontal edges are open with probability $1/(2N)$, while vertical edges are open with probability $\epsilon$ to be suitably tuned as $N$ grows to infinity. The main result tells that if $\epsilon = \kappa N^{-\frac25}$, then we see a phase transition in $\kappa$: there exist positive and finite constants $C_1, C_2$ so that there is no percolation if $\kappa <C_1$ while percolation occurs for $\kappa >C_2$. The question is motivated by a result on the analogously layered ferromagnetic Ising model at mean field critical temperature, treated in [2]. The proof relies on the analysis of the scaling limit of the growth process restricted to each horizontal layer. This is inspired by works of Mueller and Tribe on the long range contact process (PTRF, 1995). A renormalization scheme is used for the percolative regime.
The talk is based on a joint work with Thomas Mountford and Hao Xue [1].

08/11/2021 - Gregoire Loeper

15:30-16:30

We provide a survey of recent results on model calibration by Optimal Transport. We present the general framework and then discuss the calibration of local, and local-stochastic, volatility models to European options, the joint VIX/SPX calibration problem as well as calibration to some path-dependent options. We explain the numerical algorithms and present examples both on synthetic and market data.

01/11/2021 - Alon Nishry

15:30-16:30

We consider weakly confined particle systems (i.e. point processes) in the plane, characterized by a large number of outliers away from a droplet, where the bulk of the particles accumulate in the many-particle limit. For Coulomb gases at determinantal inverse temperature, and zeros of random polynomials, we observe that the limiting outlier process only depends on the shape of the uncharged region containing them (and the global net excess charge).
In particular, for a determinantal Coulomb gas confined by a sufficiently regular background measure, the outliers in a simply connected uncharged region converge to the corresponding Bergman point process. Moreover, the outliers in different uncharged regions are asymptotically independent, even if the regions have common boundary points.
Based on a joint work with R. Butez, D. García-Zelada, and A. Wennman (arXiv:2104.03959).

### October 2021

25/10/2021 - Boualem Djehiche

15:30-16:30

I will review recent results on a class of one-dimensional continuous reflected backward stochastic Volterra integral equations (RBSVIE) driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). Moreover, I will show how the solution to the RBSVIE is related to a time-inconsistent optimal stopping problem and derive an optimal strategy.

18/10/2021 - Jurgen Angst

15:30-16:30

We investigate the large degree asymptotics of the expected number of real zeros of random trigonometric polynomials with Gaussian dependent coefficients. We show that, quite surprisingly, the latter is universal or not, depending on the fact that the associated spectral density vanishes on a set of positive Lebesgue measure. Under some negative moment condition on the same spectral density, we also establish the almost sure asymptotics of the number of zeros. Our approach combines the classical use of Kac--Rice formula, generalizations of Fejér--Lebesgue estimates and extensions of Salem--Zygmund almost sure CLT to the dependent framework.

04/10/2021 - Guillaume Poly

We shall study second order asymptotic in the almost sure central limit Theorem of Salem and Zygmund. The method is based on a novel approach of Hermite expansions for random fields which are not Gaussian. Relying on some universality principles for random homogeneous sums we shall explain how "Hermite expansion methodology" for establishing CLTs may be extended beyond the usual requirement of Gaussianity of the random fields.

### April 2020

06/04/2020 - Alessandro Gnoatto TBC

16:00-17:00

### March 2020

30/03/2020 - Martin Herdegen (University of Warwick)

16:00-17:00

We revisit portfolio selection in a one-period financial market under a coherent risk measure constraint, the most prominent example being Expected Shortfall (ES). Unlike in the case of classical mean-variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation regulatory arbitrage. We show that the presence or absence of regulatory arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual characterisation of the risk measure. In the special case of ES, our results show that the market does not admit regulatory arbitrage for ES at confidence level $\alpha$ if and only if there exists an EMM $Q \approx P$ such that $\Vert \frac{dQ}{dP} \Vert_\infty < \frac{1}{\alpha}$. The talk is based on joint work with my PhD student Nazem Khan.

16/03/2020 - Eveliina Peltola

16:00-17:00 S4.29

For a number of lattice models in 2D statistical physics, it has been proven that the scaling limit of an interface at criticality (with suitable boundary conditions) is a conformally invariant random curve, Schramm-Loewner evolution (SLE). Similarly, collections of several interfaces converge to collections of interacting SLEs. Connection probabilities of these interfaces encode crossing probabilities in the lattice models, which should also be related to correlation functions of appropriate fields in the corresponding conformal field theory (CFT); the latter, however, being mathematically ill-defined. I discuss results pertaining to make sense of this relationship.

09/03/2020 - Pierre-François Rodriguez

16:00-17:00 S4.29

We consider the Gaussian free field on a large class of transient weighted graphs G, and prove that its sign clusters contain an infinite connected component. In fact, we show that the sign clusters fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs G belonging to this class include e.g. Cayley graphs of suitable volume growth, a variety of fractal graphs and cover cases in which the random walk on G exhibits anomalous diffusive behavior. Based on joint work with A. Prévost and A. Drewitz.

### February 2020

24/02/2020 - Pierre Le Doussal

16:00-17:00 S4.29

We consider the KPZ equation for the stochastic growth of a one dimensional interface. While the typical fluctuations of the height field have been much studied, until recently there were few results for the large fluctuations. We present the recent results in physics and mathematics for the tails of the distribution of height in the large deviation regime, both in the small time limit (crossover to the Edward-Wilkinson equation) and the large time limit. Our approach is based on the exact solution of the KPZ equation at all time, and its connection to random matrices. The large time left tail turns out to have a rich structure, which can be investigated using Coulomb gas methods. The results are compared with high precision numerical simulations.

17/02/2020 - Marie Kratz, ESSEC Business School, CREAR (risk research center)

16:00-17:00 S4.29

First we review the notions of regular variation both first order and second order as well as univariate and multivariate, with some possible extensions. Then we discuss the relations between multivariate second order regular variation for a vector and the second order regular variation property for the sum of its components. We illustrate the main results with examples, before turning to applications, studying the risk concentration of a portfolio of heavy-tailed risk factors.

10/02/2020 - Gechun Liang (Warwick University)

17:00-18:00

We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations.
Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space.
The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation.
We establish the convergence of the scheme and determine the convergence rate, using the comparison principles for both the scheme and the equation together with a mollification procedure.
The main application is to obtain the convergence rate of Peng's robust central limit theorem for the general situation.
Joint work with Shuo Huang.

10/02/2020 - Zorana Grbac (Université Paris Diderot, France)

16:00-17:00

In this talk a term structure modelling framework taking stochastic
discontinuities explicitly into account will be presented.
Stochastic discontinuities are a key feature in interest rate markets,
as for example the jumps of the term structures in correspondence to
monetary policy meetings of the European Central Bank show. However, this
feature has been mostly neglected in the existing term structure
models. We provide a general analysis of multiple curve markets allowing for
stochastic discontinuities under minimal assumptions in an
extended HJM framework and give a fundamental theorem of asset pricing
based on no arbitrage theory for large markets. The approach
with stochastic discontinuities permits to embed market models
directly, unifying seemingly different modeling philosophies.
A tractable class of models in this framework can be obtained using
affine semimartingales beyond stochastic continuity. These processes have been
introduced in a recent paper by Keller-Ressel et al. (2019) and
possesses several appealing properties for this purpose. Their jumps can be both inaccessible and predictable and their corresponding semimartingale characteristics still have an affine form. The conditional characteristic function can be represented using solutions to measure differential equations of Riccati type. Some examples of such tractable models will given.
This is joint work with C. Fontana, S. Guembel and T. Schmidt.

03/02/2020 - Alexander Gnedin (Queen Mary University of London)

16:00-17:00

In 1981 Samuels and Steele showed that the maximum expected length of subsequence which can be
selected from $n$ iid observations by a nonanticipating online strategy is asymptotic to $\sqrt{2n}$.
This can be compared with the classic $2\sqrt{n}$ asymptotics for the (offline) longest increasing subsequence in the Ulam-Hammersley problem.
The talk will survey history of the online selection problem and then focus on recent results on its poissonised version.
The latter include a renewal-style approximation
leading to an asymptotic expansion of the moments, as well as functional limits for the transversal and longitudinal fluctuations of the shape of the increasing subsequence under the optimal or near-optimal selection strategies.
The talk is based on a joint work with Amirlan Seksenbayev.

### January 2020

20/01/2020 - Stefan Geiss ( University of Jyväskylä)

16:00-17:00 S4.29

We study functional fractional smoothness properties of stochastic differential equations. Our techniques are based on the decoupling method introduced in [1]. The approach in this talk uses potentials to describe the regularity of the coefficients and allows for a unified treatment of standard SDEs, stopped diffusions, local times, Kusuoka-Stroock processes, and more. As a special case we obtain embedding theorems for real interpolation spaces on the Wiener space for McKean-Vlasov SDEs.
[1] S. Geiss, J. Ylinen: Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs. arXiv:1409.5322v3. To appear in Mem. AMS.

### December 2019

02/12/2019 - Mira Shamis (QMUL)

15:00-16:00 S4.29

We shall discuss the Ky Fan inequality for the eigenvalues of the sum of two Hermitian matrices. As an application, we shall derive a sharp version of a recent result of Hislop and Marx pertaining to the dependence of the integrated density of states of random Schroedinger operators on the distribution of the potential. Time permitting, we shall also discuss an application to quasiperiodic operators.

### November 2019

18/11/2019 - Martin Hairer (Imperial College)

15:00-16:00 S4.29

A "rubber band" constrained to remain on a manifold evolves by trying to shorten its length, eventually settling on a closed geodesic, or collapsing entirely. It is natural to try to consider a noisy version of such a model where each segment of the band gets pulled in random directions. Trying to build such a model turns out to be surprisingly difficult and generates a number of nice geometric insights, as well as some beautiful algebraic and analytical objects. We will survey some of the main results obtained on the way to this construction.

04/11/2019 - Raluca Balan (University of Ottawa)

15:00-16:00 S4.29

In this talk, we introduce some tools which are needed for the stochastic analysis with respect to a Lévy noise L, using the random ﬁeld approach introduced by Walsh (1986). We consider two cases: (i) ﬁnite variance Lévy noise; (ii) α-stable Lévy noise. The stochastic integral with respect to this noise is constructed using diﬀerent methods in the two cases, using Ito theory in case (i), respectively a maximal inequality for the tail of the integral process in case (ii). Then, we will show how to use these tools for the study of the non-linear SPDE Lu(t,x) = σ(u(t,x)) ˙ L(t,x),t > 0,x ∈ Rd , where L is a second-order diﬀerential operatorand σ is a Lipschitz function. Examples include the case when L is the heat operator or the wave operator. This study is motivated by the work of Peszat and Zabczyk (2007) using the approach based on integration with respect to Hilbert-space valued processes, as well as the work of Mueller (1998) and Mytnik (2002) who examined the stochastic heat equation with α-stable Lévy noise and non-Lipschitz function σ.

### October 2019

28/10/2019 - Sander Willems - RBS

15:00-16:00 Kings College London, Strand Building, S4.29

This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the instantaneous volatility level. The steady-state distribution of the instantaneous volatility belongs to the class of Generalized Inverse Gaussian distributions. We show that the quadratic term in the drift is crucial to avoid moment explosions and to preserve the martingale property of the stock price process. Using a conveniently chosen change of measure, we relate the model to the class of polynomial diffusions. This remarkable relation allows us to develop a highly accurate option price approximation technique based on orthogonal polynomial expansions.

21/10/2019 - Zdzislaw Brzeźniak (University of York)

15:00-16:00 S4.29

18/10/2019 - Alex Waston (UCL)

15:00-16:00 S5.20

Growth-fragmentation processes describe the evolution of systems of cells which grow continuously and fragment suddenly; they are used in models of cell division and protein polymerisation. In the long term, the concentrations of cells with given masses typically increase at some exponential rate and, after compensating for this, they arrive at an asymptotic profile. Up to now, this has mainly been studied for the average behaviour of the system, often by means of a natural partial integro-differential equation and the associated spectral theory. However, the behaviour of the system as a whole, rather than only its average, is more delicate. We obtain a criterion for the convergence of the entire collection of cells to a certain asymptotic profile, and we find some improved explicit conditions for this to occur. Joint work with Jean Bertoin.

07/10/2019 - Olga Iziumtseva

15:00-16:00 S4.29

The present talk is devoted to evolution of Gaussian field in the flow of interacting particles. We present completely new approach for the description of evolution based on the equation with interaction introduced by A.A. Dorogovtsev in 2003. It allows to describe the motion of field taking into account its shape. For defined random field we prove the existence of self-intersection local times and describe its asymptotics.
References:
1. A.A. Dorogovtsev, O.L. Izyumtseva, Hilbert-valued self-intersection local times for planar Brownian motion, Stochastics, 91, no.1, 2019, 143-154
2. A.A. Dorogovtsev, Stochastic flows with interaction and measure-valued processes, International Journal of Mathematics and Mathematical Sciences 63 (2003), 3963-3977
3. A.A. Dorogovtsev, O.L. Izyumtseva, Local times of self-intersection, Ukrainian Mathematical Journal, 68, no. 3, 2016, 325-379

### September 2019

23/09/2019 - Cécile Mailler

15:00-16:00 S4.29

Measure-valued Pólya processes (MVPPs) are the generalization of Pólya urns to infnitely-many colours. In this joint work with Denis Villemonais (Nancy), we use stochastic approximation techniques to prove strong convergence of MVPPs to the quasi-stationnary distribution of a Markov process with absorption (that depends on the MVPP considered). The main difficulty is that we allow our set of colours to be non-compact and thus need to consider a stochastic approximation taking values in the set of measures on a non-compact space; we use Lyapunov functions to treat this non-compact case.

### April 2019

01/04/2019 - Robert Dalang (École Polytechnique Fédérale de Lausanne)

15:00-16:00 S4.36

Mild solutions and generalized solutions are two natural
notions of solution to a linear stochastic PDE driven by a pure jump
symmetric Lévy white noise. We identify necessary and sufficient
conditions for existence for these two kinds of solutions, and we apply
this result to the linear stochastic heat, wave and Poisson equations
driven by a symmetric $\alpha$-stable white noise. This talk, intended
for a general probability audience, is based on joint work with Thomas
Humeau.

### March 2019

25/03/2019 - Bruno Bouchard (Université Paris Dauphine)

15:00-16:00 S4.36

We consider a generalized version of the model of linear market impact of Bouchard et al. 2017. In a Brownian Markovian setting, we provide a viscosity solution of the super-hedging price and show that it actually coincides with a perfect hedging price under additional regularity assumptions. The PDE formulation suggests that it admits a dual formulation in the form of a control problem in standard form, which paves the way to the study of a class of second order BSDEs with feedback effect. This link is further discussed in a non-Markovian framework. We will also provide a small non-linearity asymptotic.

18/03/2019 - Diane Holcomb

15:00-16:00 S4.36

In 1984 Trotter described a tridiagonal random matrix model that has the same eigenvalues as the Gaussian Orthogonal Ensemble. This model and a later generalization share many structural properties with discrete Schrödinger operators. This led to the conjecture that the largest eigenvalues of the ensemble converged to the the eigenvalues a certain random differential operator. We will give an overview of where the conjecture comes from and a bit of the proof. We will then look at a process that appears when looking at eigenvalues of submatrices of the tridiagonal model. This process is notably different than the one that appears when considering submatrices of the full matrix model. This talk will survey work by Dumitriu-Edelman, Edelman-Sutton, and Ramirez-Rider-Virag. It finishes up with work that is joint A. Gonzalez

04/03/2019 - Christian Andrés Fonseca-Mora (University of Costa Rica)

15:00-16:00 S4.36

The objective of this talk is the study of semimartingales and cylindrical semimartingales in infinite dimensional spaces. Our main motivation is the introduction of a theory of stochastic integration on duals of nuclear spaces, in particular on spaces of distributions. We start by considering conditions for a cylindrical semimartingale in the strong dual E' of a nuclear space E to have a E'-valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples will be provided. Later, we will show that under some general conditions every E'-valued semimartingale has a canonical decomposition. In the last part of our talk we explain how our aforementioned results on duals of nuclear spaces can be used to establish sufficient conditions for a cylindrical semimartingale in a Hilbert space H to have a H-valued semimartingale version whose paths are right-continuous with left limits. As an application of our result we will obtain the theorem of radonification by a single Hilbert-Schmidt operator for cylindrical semimartingales in H.

### February 2019

25/02/2019 - Eugene Bogomolny

15:00-15:00 S4.36

In the talk I will discuss the construction of high-excited wave functions in spin-chains models, chaotic and pseudo-integrable quantum billiards. Special attention is given to the calculation of fractal properties of their wave functions.

18/02/2019 - Jan van Neerven (Delft)

15:00-16:00 S4.36

Extending results of Pardoux and Peng and Hu and Peng, we prove well-posedness results for backward stochastic evolution equations in UMD Banach spaces. This is joint work with Qi Lü.

11/02/2019 - Elena Issoglio (Leeds)

This talk focuses on a multidimensional SDE where the drift is an element of a fractional Sobolev space with negative order, hence a distribution. This SDE admits a unique weak solution in a suitable sense - this was proven in [Flandoli, Issoglio, Russo (2017)]. The aim here is to construct a numerical scheme to approximate this solution. One of the key problems is that the drift cannot be evaluated pointwise, hence we approximate it with suitable functions using Haar wavelets, and then apply (an extended version of) Euler-Maruyama scheme. We then show that the algorithm converges in law, and in the special 1-dimensional case we also get a rate of convergence (and in fact convergence in L^1).
This talk is based on a joint work with T. De Angelis and M. Germain.

04/02/2019 - Alejandro Rivera

15:00-16:00 S4.36

The canonical white noise W on a compact manifold M can be written as
a random sum over the Laplace eigenfunctions on M with iid standard
normal coefficients. By truncating this sum, one obtains a smooth
Gaussian field on M: the cut-off white noise. In the past decade, much
has been said about the geometry and topology of this random field. In
particular, its average statistics are in a sense universal.
On the other hand, one can generalize the above definition and study
fields defined as sums over Laplace eigenfunctions with independent
normal coefficients whose variance decays at a certain rate. These are
known as fractional gaussian fields. This wide family contains for
instance, the periodic Brownian motion and the Gaussian Free Field. We
will discuss the geometry of the level set of these fields.

### January 2019

21/01/2019 - Mathew Penrose (Bath)

15:00-16:00 S4.36

Imagine it is Autumn in the forest, and randomly shaped leaves fall sequentially at random onto the ground until it is completely covered. The visible parts of leaves on the ground at a given instant then form a random tessellation of the plane. Mathematically, the leaves and their times of arrival are modelled as an independently marked Poisson process in space-time with the marks determining their shapes. This dead leaves (or confetti) model was proposed by Matheron in 1968 and has applications in modelling natural images and in materials science; see [1]. The one-dimensional version of the model (leaves on the line) can be obtained by simply taking a linear section through the two-dimensional tessellation but is also open to other interpretations, for example the tree-trunks visible from the edge of the forest.
We discuss new and old results on some or all of the following in both one and two dimensions:
* Exact formula for the number of cells of the tessellation per unit volume.
* Asymptotic variance, CLT and time evolution for the total number of cells visible in a large window.
* Analogous results in one dimension.
* In two dimensions, similar results for the total length of cell boundaries within a window.
* In one dimension, the distribution of the size of a `typical' cell.
We make heavy use of the Mecke formula from the theory of Poisson processes; see e.g. [3]. Also relevant is a variant of the classical `Buffon's needle' problem. We also build on earlier work on items 1 and 3 by Cowan and Tsang [2].

14/01/2019 - Enrico Scalas (Sussex)

15:00-16:00 Room S.4.36 Strand Building

The fractional non-homogeneous Poisson process was introduced by a time-change of the non-homogeneous Poisson process with the inverse alpha-stable subordinator. We propose a similar definition for the (non-homogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional non-homogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe's theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

### December 2018

10/12/2018 - Claudio Bellani (Imperial College London)

15:30-16:00 S.4.36

The understanding of continuous trajectories on a Banach space greatly benefited from the introduction of the concept of signature (K-T. Chen 1954, T. Lyons 1998), both in deterministic and probabilistic settings. In the context of mathematical models for option trading, the first two levels of the underlying path’s signature are the essential features on which the PDE pricing and hedging technology rests. The talk will show such essential dependence by presenting how, in Armstrong et al. (2018), the authors rewrote the classical formulas of Mathematical Finance exclusively relying on signature-lifted price paths, refraining in particular from the deployment of probabilistic tools (Ito calculus). This was possible thanks to the fact that the classical Gamma sensitivity is the Gubinelli’s derivative of the Delta hedging, and yielded a signature-inspired understanding of the fundamental theorem of derivative trading.

10/12/2018 - Imanol Pérez Arribas (Oxford University and JP Morgan)

15:00-15:30 S.4.36

We introduce signature payoffs, a family of path-dependent derivatives that are given in terms of the signature of the price path of the underlying asset. We show that these derivatives are dense in the space of continuous payoffs, a result that is exploited to quickly price arbitrary continuous payoffs. This approach to pricing derivatives is then tested with European options, Asian options, lookback options and variance swaps.

03/12/2018 - Nadia Sidorova

15:00-16:00

The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It describes the mean-field behaviour of a continuous-time branching random walk. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials it is localised at just one point. However, in a partially symmetric parabolic Anderson model, the one-point localisation breaks down for heavy-tailed potentials and remains unchanged for light-tailed potentials, exhibiting a range of phase transitions.

### November 2018

26/11/2018 - Andrey Dorogovtsev

15:00-16:00

The purpose of the talk is to present different characteristics of the compact sets
in Hilbert space, relationships between them and Kolmogorov entropy, and to show applications to the random processes theory. We study the conditions under which the compact set can be covered by a Hilbert-Schmidt brick. The notion of geometric entropy of the sets in Hilbert space is introduced. It is closely connected with the time of free motion in stochastic flows.
The lecture is partially based on the joint work with Mihail Popov

19/11/2018 - Rene Schilling

S2.28

We consider SDEs of the form
$dX_t=b(t,X_{t−}) dt + dL_t$
with initial condition $X_0=x$ driven by a $d$-dimensional Levy process.
We establish conditions on the Lévy process and the drift coefficient $b$ such that the Euler--Maruyama approximation converges strongly to a solution of the SDE with an explicitly given rate. The convergence rate depends on the regularity of $b$ and the behaviour of the Lévy measure at the origin. As a by-product of the proof, we obtain that the SDE has a pathwise unique solution. Our result covers many important examples of Lévy processes, e.g. isotropic stable, relativistic stable, tempered stable and layered stable.

12/11/2018 - Daniel Hug

15:00-16:00 S2.28

In Euclidean space, the asymptotic shape of large cells in various types of Poisson driven random tessellations has been the subject of a famous conjecture due to David Kendall. Since shape is a geometric concept and large cells are identified by means of geometric size functionals, the resolution of the conjecture is connected with geometric inequalities of isoperimetric type and related stability results. We start by reviewing briefly some of the results
that have been obtained in this context within the last two decades.
While random tessellations have been studied extensively in Euclidean space, random tessellations and, more generally, stochastic geometry in spherical space have come into focus recently. The current work [1] explores specific and typical cells of random tessellations in spherical space. We obtain probabilistic deviation inequalities and asymptotic distributions for quite
general size functionals. In contrast to the Euclidean setting, where the asymptotic regime concerns large size, in the spherical framework the asymptotic analysis is concerned with high intensities.
In addition to results for Poisson great hypersphere and Poisson Voronoi tessellations in spherical space, we also report on the recent work [2] on splitting tessellation processes in spherical space, which correspond to STIT-tessellation models stable iteration in Euclidean space and have been studied intensively in recent years. Expectations and variances of spherical curvature measures induced by splitting tessellation processes are studied by means of auxiliary martingales and tools from spherical integral geometry. The spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Various other cell characteristics can be treated as well and related to distributions of Poisson great hypersphere tessellations.

05/11/2018 - Christina Goldschmidt

15:00-16:00 S2.28

Consider a graph with label set \{1,2, \ldots,n\} chosen uniformly at random from those such that vertex i has degree D_i, where D_1, D_2, \ldots, D_n are i.i.d. strictly positive random variables. The condition for criticality (i.e. the threshold for the emergence of a giant component) in this setting is E[D^2] = 2 E[D], and we assume additionally that P(D = k) \sim c k^{-(\alpha + 2)} as k tends to infinity, for some \alpha \in (1,2). In this situation, it turns out that the largest components have sizes on the order of n^{\alpha/(\alpha+1)}. Building on earlier work of Adrien Joseph, we show that the components have scaling limits which can be related to a forest of stable trees a la Duquesne-Le Gall-Le Jan) via an absolute continuity relation. This gives a natural generalisation of the scaling limit for the Erdos-Renyi random graph which I obtained in collaboration with Louigi Addario-Berry and Nicolas Broutin a few years ago (extending results of Aldous), and complements recent work on random graph scaling limits of various authors including Bhamidi, Broutin, Duquesne, van der Hofstad, van Leeuwaarden, Riordan, Sen, M. Wang and X. Wang.
This is joint work with Guillaume Conchon-Kerjan (Paris 7).

### October 2018

29/10/2018 - Noga Alon

15:00-16:00 S2.28

A graph is called k universal if it contains every undirected graph
on k vertices as an induced subgraph. What is the largest k=k(n) so
that the binomial random graph G(n,0.5) is k-universal with high
probability ? I will describe briefly the motivation and history of
the problem, and sketch a proof of a result that determines k(n)
precisely. The argument suggests the analysis of a random process
that can be called the vertex exposure process, and combines
probabilistic tools with group theoretic methods.

01/10/2018 - Naomi Feldheim

15:00-16:00 S2.28

Let X and Y be two unbounded positive independent random variables.
Write Min_m for the probability of the event {min(X,Y) > m} and Mean_m
for that of the event {(X+Y)/2 > m}. We show that the limit inferior of Min_m / Mean_m is always 0 (as m approaches infinity), regardless of the distributions of X and Y. This may be viewed as a universal anti-concentration result. We will present an elementary proof of analytic nature, and discuss related
open problems. We will also discuss how this result is used to analyze a
probabilistic model for evolving social groups, suggested by Alon,
Feldhman, Mansour, Oren and Tennenholz. Based on joint work with Ohad Feldheim, arXiv:1609.03004 and arXiv:1807.05550.

### June 2018

18/06/2018

Oxford University

### May 2018

24/05/2018 - Panos Vassiliou

14:00-15:00 Room 1102, Department of Statistical Science, UCL, 1-19 Torrington Place (1st floor)

In the present talk we establish Laws of Large Numbers for Non-Homogeneous Markov Systems and Cyclic Non-homogeneous Markov systems. We start with a theorem, where we establish, that for a NHMS under certain conditions, the fraction of time that a membership is in a certain state, asymptotically converges in mean square to the limit of the relative population structure of memberships in that state. We continue by proving a theorem which provides the conditions under which the mode of convergence is almost surely. We continue by proving under which conditions a Cyclic NHMS is Cesaro strongly ergodic. We then proceed to prove, that for a Cyclic NHMS under certain conditions the fraction of time that a membership is in a certain state, asymptotically converges in mean square to the limit of the relative population structure in the strongly Cesaro sense of memberships in that state. We then proceed to establish a founding Theorem, which provides the conditions under which, the relative population structure asymptotically converges in the strongly Cesaro sense with geometrical rate. This theorem is the basic instrument missing to prove, under what conditions the Law of Large Numbers for a Cyclic NHMS is with almost surely mode of convergence. Finally, we present two applications. Firstly for geriatric and stroke patients in a hospital and secondly for the population of students in a University system. In the second Part of the talk we study Laws of Large Numbers for Non-Homogeneous Markov Systems when the sequence of transition matrices consist of arbitrary stochastic matrices all with the same incidence matrix. With this step forward we reach full possible generality in terms of assumptions involved on the type of stochastic transition matrices.

10/05/2018 - Tyler Helmuth

14:00-15:00 Room 102, Department of Statistical Science, UCL, 1-19 Torrington Place (1st floor)

The vertex-reinforced jump process (VRJP) is a linearly reinforced random walk in continuous time. Reinforcing means that the VRJP prefers to jump to vertices it has previously visited; the strength of the reinforcement is a parameter of the model.
Sabot and Tarres have shown that the VRJP is related to a model known as the H22 supersymmetric hyperbolic spin model, which originated in the study of random band matrices. By making use of results for the H22 model they proved the VRJP is recurrent for sufficiently strong reinforcement. I will present a new and direct connection between the VRJP and hyperbolic spin models (both supersymmetric and classical), and show how this connection can be used to prove that the VRJP is recurrent in two dimensions for all reinforcement strengths.
This talk is about joint work with R. Bauerschmidt and A. Swan.

### March 2018

26/03/2018 - Guenter Last

15:00-16:00 KCL Strand, Room S4.29

We consider a Poisson process (Poisson random measure) on a general phase space. Using an explicit Fock space isometry based on expected difference operators, we shall derive covariance identities for square-integrable functions
of the Poisson process. As a first application we give a short proof of the Poincare and the FKG inequalities. Our second application are explicit bounds for the normal approximation of Poisson functionals derived via Stein's method. Finally we shall discuss concentration inequalities.

19/03/2018 - Andreas Kyprianou (Bath)

16:00-17:00 KCL Bush House NE 1.04

We look at at a coupled system of stochastic differential equations that describe an infinite parametric family of genealogical skeletal decompositions of a general continuous state branching process (CSBP), supercritical, critical and subcritical. This puts into a common framework a number of known and new path decompositions of CSBPs, including those which involve continuum random trees, and allow us to connect the notion of Evans-O'Connell immortal particle decomposition to that of the skeletal decomposition.
This is joint work with Dorka Fekete (Bath) and Joaquin Fontbona (U. de Chile).

19/03/2018 - Mo Dick Wong

15:00-16:00 KCL Bush House NE 1.04

We first consider Boltzmann-Gibbs measures associated with log-correlated Gaussian fields, the multifractal properties of which are captured by the multifractal exponents. While both the quenched and the annealed multifractal exponents exhibit a phase transition at low temperature known as freezing, we show that the leading-order term of the annealed exponent exhibits another phase transition at some intermediate temperature which may be explained by a partial localisation mechanism similar to that when freezing occurs, and
verify/extend Fyodorov's prefreezing conjecture in 2009. We then discuss
the connection between prefreezing and the Seiberg bound in Liovuille quantum field theory (LQFT), and present a different approach to the problem which allows us to establish a further phase transition in the second-order term at the Seiberg bound.

19/03/2018 - Juan Carlos Pardo (CIMAT)

14:00-15:00 KCL Bush House NE 1.04

In this talk, we are interested on the extinction time of continuous state branching
processes with competition in a Lévy random environment. In particular we deduce, under the so-called Grey’s condition together with the assumption that the Levy random environment does not drift towards infinity, that for any starting point the process gets extinct in finite time a.s. Moreover if we replace the condition on the Levy random environment by a technical integrability condition on the competition mechanism, then the process also gets extinct in finite time a.s. and it comes down from infinity under the condition that the negative jumps associated to the environment are driven by a compound Poisson process.
The case when the random environment is driven by a Brownian motion and the competition is logistic is also treated. Our arguments are base on a Lamperti-type
representation where the driven process turns out to be a perturbed Feller diffusion by an independent spectrally positive Lévy process. If the independent random perturbation is a subordinator then the process converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case, we provide the expectation and the Laplace transform of the absorption time, as a functional of the solution to a Riccati differential equation.

12/03/2018 - Zeev Rudnick

15:00-16:00 KCL Strand, Room S4.29

Given a tuple of independent random permutations on N letters, what is the probability that they have the same cycle structure? I will use this question and its answer, to model some of Erdos' favorite problems, which deal with the frequency of occurrence of "locally repeated" values of arithmetic functions, such as the Erdos-Mirsky problem on coincidences of the divisor function at consecutive integers, the analogous problem for the Euler totient function, and
the quantitative conjectures of Erdos with Pomerance and Sarkozy and of Graham, Holt and Pomerance on the frequency of occurrences. I will introduce the corresponding problems in the setting of polynomials over a finite field, and completely solve them in the large finite field limit, using the results on cycle structures of random permutations.

05/03/2018 - Andreas Eberle (Bonn)

17:00-18:00 KCL Strand, Room K6.63

Carefully constructed Markovian couplings and specifically designed Kantorovich metrics can be used to derive relatively precise bounds on the distance between the laws of two Langevin processes. In the case of two overdamped Langevin diffusions with the same drift, the processes are coupled by reflection, and the metric is an L1 Wasserstein distance based on an appropriately chosen concave distance function. If the processes have different drifts, then the reflection coupling can be replaced by a „sticky coupling“ where the distance between the two copies is bounded from above by a one-dimensional diffusion process with a sticky boundary at 0. This new type of coupling leads to long-time stable bounds on the total variation distance between the two laws. Similarly, two kinetic Langevin processes can be coupled using a particular combination of a reflection and a synchronous coupling that is sticky on a hyperplane. Again, the coupling distance is contractive on average w.r.t. an appropriately designed Wasserstein distance. This can be applied to derive new bounds of kinetic order for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. Similar approaches are also useful when studying mean-field interacting particle systems, McKean-Vlasov diffusions, or diffusions on infinite dimensional state spaces.
(joint work with Arnaud Guillin and Raphael Zimmer).

05/03/2018 - Matthias Winkel (Oxford)

16:00-17:00 KCL Strand, Room K6.63

We construct diffusions on a space of interval partitions of [0,1] that
are stationary with Poisson-Dirichlet laws. The processes of ranked
interval lengths of our partitions are diffusions introduced by Ethier and
Kurtz (1981) and Petrov (2009). Specifically, we decorate the jumps of a
spectrally positive stable process with independent squared Bessel
excursions. In the spirit of Ray-Knight theorems, we form a process
indexed by level, in our case by extracting intervals from jumps crossing
the level. We show that the fluctuating total interval lengths form
another squared Bessel process of different dimension parameter. By
interweaving two such constructions, we can match dimension parameters to
equal -1. We normalise interval partitions and change time by total
interval length in a de-Poissonisation procedure. These interval partition
diffusions are a key ingredient to construct a diffusion on the space of
real trees whose existence has been conjectured by Aldous. This is joint
work, partly in progress, with Noah Forman, Soumik Pal and Douglas
Rizzolo.

05/03/2018 - Domenico Marinucci (Rome)

15:00-16:00 KCL Strand, Room K6.63

We shall first review some recent results on the geometry of
Lipschitz-Killing curvatures for the excursion sets of random
2-dimensional spherical harmonics f_{\ell} of high degree. We shall then
focus on the asymptotic behaviour of the nodal length, i.e. the length
of their zero set. It is found that the nodal lengths are asymptotically
equivalent, in the L^2 -sense, to the "sample trispectrum", i.e., the
integral of the fourth-order Hermite polynomial of the values of
f_{\ell}. A particular by-product of this is a Quantitative Central
Limit Theorem (in Wasserstein distance) for the nodal length, in the
high energy limit.
Based on joint work with Maurizia Rossi and Igor Wigman.

### February 2018

26/02/2018 - Guillaume Rémy

15:00-16:00

Starting from the restriction of a 2d Gaussian free field (GFF) to the
unit circle one can define a Gaussian multiplicative chaos (GMC) measure
whose density is formally given by the exponential of the GFF. In 2008
Fyodorov and Bouchaud conjectured an exact formula for the density of the
total mass of this GMC. In this talk we will give a rigorous proof of this
formula. Our method is inspired by the technology developed by Kupiainen,
Rhodes and Vargas to derive the DOZZ formula in the context of Liouville
conformal field theory on the Riemann sphere. In our case the key
observation is that the negative moments of the total mass of GMC on the
circle determine its law and are equal to one-point correlation functions
of Liouville theory in the unit disk. Finally we will discuss applications
in random matrix theory, asymptotics of the maximum of the GFF, and tail
expansions of GMC.

19/02/2018 - Manjunath Krishnapur

15:00-16:00 S4.29

What is the probability that a linear combination of i.i.d. random variables with small coefficients dominates another linear combination of the same random variables with large coefficients? In joint work with Sourav Sarkar, we raised this question and conjectured that the answer is the ratio of the L^2 norms of the two coefficient vectors, up to an additive term that describes the arithmetic structure of the larger coefficients. We present the motivation and partial proofs.

12/02/2018 - Valentina Cammarota

15:00-16:00 S4.29

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemannian manifolds. This is known to be true on average. We will discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.

05/02/2018 - Sasha Sodin

15:00-16:00 S4.29

Non-Hermitian random Schroedinger operators were put forth in the
mid 1990-s by Hatano and Nelson, and mathematically studied by Goldsheid
and Khoruzhenko. We shall describe the model and discuss some results
obtained in a recent joint work with I. Goldsheid.

### January 2018

22/01/2018 - Ilya Molchanov

15:00-16:00

The family of metric measure spaces can be endowed with the semigroup operation being the Cartesian product. The aim of this talk is to arrive at the generalisation of the fundamental theorem of arithmetic for metric measure spaces that provides a unique decomposition of a general space into prime factors. These results are complementary to several partial results available for metric spaces (like de Rham's theorem on decomposition of manifolds). Finally, the infinitely divisible and stable laws on the semigroup of metric measure spaces are characterised.

08/01/2018 - Nathanael Berestycki

15:00-16:00

The dimer model on a finite bipartite graph is a uniformly chosen perfect matching, i.e., a set of edges which cover every vertex exactly once. It is a classical model of mathematical physics, going back to work of Kasteleyn and Temeperley/Fisher in the 1960s, with connections to many topics including determinantal processes, random matrix theory, algebraic combinatorics, discrete complex analysis, etc.
A central object for the dimer model is a notion of height function introduced by Thurston, which turns the dimer model into a random discrete surface. I will discuss a series of recent results with Benoit Laslier (Paris) and Gourab Ray (Victoria) where we establish the convergence of the height function to a scaling limit in a variety of situations. This includes simply connected domains of the plane with arbitrary linear boundary conditions for the height, in which case the limit is the Gaussian free field, and Temperleyan graphs drawn on Riemannsurfaces. In all these cases the scaling limit is universal (i.e., independent of the details of the graph) and conformally invariant.
A key new idea in our approach is to exploit "imaginary geometry" couplings between the Gaussian free field and Schramm's celebrated SLE curves.

### December 2017

05/12/2017 - Anne Estrade

15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

In the present talk, we deal with a stationary isotropic random field $X:\R^d \to \R$ and we assume it is partially observed through some level functionals. We aim at providing a methodology for a test of Gaussianity based on this information. More precisely, the level functionals are given by the Euler characteristic of the excursion sets. On the one hand, we study the properties of these level functionals under the hypothesis that the random field $X$ is Gaussian. In particular, we focus on the mapping that associates to any $u$ the expected Euler characteristic of the excursion set above level $u$. On the other hand, we study the same level functionals under alternative distributions of $X$, such as chi-square or shot noise. The talk is based on a joint work with Elena Di Bernardino (CNAM Paris) and José R. Leon (Universidad Central de Venezuela).

### November 2017

28/11/2017 - Mateusz Majka

15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

We introduce the topic of transportation cost-information inequalities that characterize concentration of probability measures. We focus on measures that are distributions of solutions to stochastic differential equations, with noise consisting both of a diffusion and a jump component. We show how to use the Malliavin calculus to obtain such inequalities. In particular, we combine the Malliavin calculus approach with the coupling technique, which allows us to obtain results under relatively weak assumptions on the drift in the equation.

14/11/2017 - Reimer Kuehn

15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the probability of a particular node to remain part of the giant component, and in the expected size of small clusters containing that node. In
the vicinity of the percolation threshold, weakly non-linear analysis reveals that node-to-node heterogeneity is captured by the recently introduced notion of non-backtracking centrality. We supplement these results for fixed finite networks by a population dynamics approach to analyse random graph models in the infinite system size limit, also providing closed-form approximations for the large mean degree limit of Erdos-Renyi random graphs. Interpreted in terms of the application of percolation to real-world processes, our results shed light on the heterogeneous exposure of different nodes to cascading failures, epidemic spread, and information flow.

### October 2017

31/10/2017 - Damien Gayet

15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

The Fortuin-Kasteleyn-Ginibre inequality is a crucial tool in percolation theory. It says in particular that positive crossings of two overlapping rectangles are positively correlated. I will explain that a large family of discrete random models close to the Bernoulli percolation still satisfies percolation features, like the box-crossing property, even when they don't satisfy FKG. This applies to the antiferromagnetic Ising model with small parameter and certain discrete Gaussian fields with oscillating (but strongly decaying) 2-point correlation function. This is a joint work with Vincent Beffara.

26/10/2017 - Peter Sarnak (IAS)

15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

Nazarov and Sodin have shown that the zero set of a random real homogeneous polynomial in several variables and of large degree has many components and the same is true for the random harmonic such polynomial ("mono-chromatic waves") .We show that for each of these models the topologies of the components are equidistributed w.r.t a universal probability measure on the space of topologies.We examine possible connections to critical and super-critical percolation theory .
Joint works with I.Wigman and with Y.Canzani .

17/10/2017 - Vadim Shcherbakov (Royal Holloway)

15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

This talk concerns the long term behaviour of a continuous time Markov chain formed by two interacting birth-and-death processes. In absence of interaction the Markov chain is a pair of two independent birth-and-death processes, whose long term behaviour is well known. Presence of interaction can significantly affect the individual behaviour. We describe in detail the long term behaviour for a range of models of two interacting birth-and-death processes. The following effect is detected in some of these models. Namely, eventually one of the components tends to infinity, while the other component is confined to a finite set {0, 1,..., k}, where k is explicitly determined by the model parameters. Moreover, the bounded component of the corresponding embedded discrete time Markov chain takes any values from the above finite set infinitely many times. In particular, this effect is observed in a stochastic population model for competition between two species with Lotke-Volterra interaction.
The talk is based on joint work with M. Menshikov and S. Volkov.

10/10/2017 - Codina Cotar (UCL)

15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

Multi-marginal optimal transport with Coulomb cost arises as a dilute limit of density functional theory, which is a widely used electronic structure model. The number N of marginals corresponds to the number of particles. I will discuss the question whether Kantorovich minimizers must be Monge minimizers (yes for N = 2, open for N > 2, no for N =infinity), and derive the surprising phenomenon that the extreme correlations of the minimizers turn into independence in the large N limit. I will also discuss the next order term limit and the connection of the problem to Coulomb and Riesz gases.
The talk is based on joint works with Gero Friesecke (TUM), Claudia Klueppelberg (TUM), Brendan Pass (Alberta) and Mircea Petrache (MPI/ETH).

03/10/2017 - Perla Sousi

15:00-16:00 KCL Department of Mathematics, Strand Building, room S5.20

In four dimensions we prove a non-conventional CLT for the capacity of the range of simple random walk and a strong law of large numbers for the capacity of the Wiener sausage. This is joint work with Amine Asselah and Bruno Schapira.

### September 2017

28/09/2017 - Alex Novikov (UTS)

15:00-16:00 KCL Department of Mathematics Strand Campus room C16 East Wing (ask at reception)

Motivated by the use of weighted Kolmogorov-Smirnov tests in Gene Set Enrichment Analysis (GSEA) we consider the problem of finding fast and accurate approximations for nonlinear boundary crossing probabilities. A technique based on the approximations via exact lower and upper bounds with use of piecewise linear boundaries has been developed. An alternative approach based on solving the system of two integral equations will be also discussed.
In the numerical examples, in the context of GSEA, we compare the results obtained using these two techniques to Durbin's asymptotic approximations and Monte-Carlo simulation.

### July 2017

20/07/2017 - Mikhail Menshikov

10:00-12:00 KCL Department of Mathematics, Strand building, room S5.20

These lectures are about the Foster-Lyapunov or semimartingale method for studying the asymptotic behaviour of near-critical stochastic systems. The basic idea of the method is exhibiting a function of the underlying process with a one-dimensional image which satisfies locally a drift condition, which can be used to conclude about e.g. recurrence, transience, or positive-recurrence of the process. If the process is near-critical in the sense of being near some phase boundary in asymptotic behaviour, then the one-dimensional process arising from a suitable Lyapunov function is typically near-critical as well. The prototypical family of near-critical one-dimensional stochastic processes are processes with asymptotically-zero drift, studied in a seminal series of papers by Lamperti. Analysis of these one-dimensional process allows one to study, via the method of Lyapunov functions, asymptotic behaviour of many-dimensional Markov processes. The Lyapunov function method also enables one to study continuous-time Markov chains, and random walks with heavy-tailed increments. These lectures will lead a tour encompassing some of the key aspects of the above topics, and are based on the recently published book "Non-Homogeneous Random Walks" by Menshikov, Popov, and Wade, Cambridge University Press, 2016.

19/07/2017 - Mikhail Menshikov

10:00-12:00 KCL Department of Mathematics, Strand building, room S5.20

These lectures are about the Foster-Lyapunov or semimartingale method for studying the asymptotic behaviour of near-critical stochastic systems. The basic idea of the method is exhibiting a function of the underlying process with a one-dimensional image which satisfies locally a drift condition, which can be used to conclude about e.g. recurrence, transience, or positive-recurrence of the process. If the process is near-critical in the sense of being near some phase boundary in asymptotic behaviour, then the one-dimensional process arising from a suitable Lyapunov function is typically near-critical as well. The prototypical family of near-critical one-dimensional stochastic processes are processes with asymptotically-zero drift, studied in a seminal series of papers by Lamperti. Analysis of these one-dimensional process allows one to study, via the method of Lyapunov functions, asymptotic behaviour of many-dimensional Markov processes. The Lyapunov function method also enables one to study continuous-time Markov chains, and random walks with heavy-tailed increments. These lectures will lead a tour encompassing some of the key aspects of the above topics, and are based on the recently published book "Non-Homogeneous Random Walks" by Menshikov, Popov, and Wade, Cambridge University Press, 2016.

18/07/2017 - Andrew Wade

10:00-12:00 KCL Department of Mathematics, Strand building, room S5.20

These lectures are about the Foster-Lyapunov or semimartingale method for studying the asymptotic behaviour of near-critical stochastic systems. The basic idea of the method is exhibiting a function of the underlying process with a one-dimensional image which satisfies locally a drift condition, which can be used to conclude about e.g. recurrence, transience, or positive-recurrence of the process. If the process is near-critical in the sense of being near some phase boundary in asymptotic behaviour, then the one-dimensional process arising from a suitable Lyapunov function is typically near-critical as well. The prototypical family of near-critical one-dimensional stochastic processes are processes with asymptotically-zero drift, studied in a seminal series of papers by Lamperti. Analysis of these one-dimensional process allows one to study, via the method of Lyapunov functions, asymptotic behaviour of many-dimensional Markov processes. The Lyapunov function method also enables one to study continuous-time Markov chains, and random walks with heavy-tailed increments. These lectures will lead a tour encompassing some of the key aspects of the above topics, and are based on the recently published book "Non-Homogeneous Random Walks" by Menshikov, Popov, and Wade, Cambridge University Press, 2016.

17/07/2017 - Andrew Wade

10:00-12:00 KCL Department of Mathematics, Strand building, room S5.20

### May 2017

05/05/2017 - Mateusz Majka

16:00-17:00 S5.20, Strand Building, Strand Campus, King's College London

We present a novel construction of a coupling of solutions to a certain class of SDEs with jumps, which includes SDEs driven by symmetric $\alpha$-stable processes and numerous other Levy processes with rotationally invariant Levy measures. Then we show how to use this coupling in order to obtain exponential convergence rates of solutions to such equations to their equilibrium, both in the standard $L^1$-Wasserstein and the total variation distances. As a second application of our coupling, we obtain some transportation inequalities, which characterize concentration of the distributions of these solutions, and which were previously known only under the global dissipativity assumption on the drift.

05/05/2017 - Vlad Vysotsky

15:00-16:00 S5.20, Strand Building, Strand Campus, King's College London

Take a one-dimensional random walk with zero mean increments, and consider the sizes of its overshoots over the zero level. It turns out that this sequence, which forms a Markov chain, always has a stationary distribution of a simple explicit form. The questions of uniqueness of this stationary distribution and convergence towards it are surprisingly hard. We were able to prove only the total variation convergence, which holds for lattice random walks and for the ones whose distribution, essentially, has density. We also obtained the rate of this convergence under additional mild assumptions. We will also discuss connections to related topics: local times of random walks, stability of reflected random walks, ergodic theory, and renewal theory. This is a joint work with Alex Mijatovic.

### April 2017

26/04/2017 - Perla Sousi (University of Cambridge)

16:00-17:00 University College London, 1-19 Torrington Place,Room 102, London WC1E 7HB

We study the behaviour of random walk on dynamical percolation. In this model, the edges of a graph are either open or closed and refresh their status at rate mu, while at the same time a random walker moves on G at rate 1, but only along edges which are open. On the d-dimensional torus with side length n, when the bond parameter is subcritical, the mixing times for both the full system and the random walker were determined by Peres, Stauffer and Steif. I will talk about the supercritical case, which was left open, but can be analysed using evolving sets (joint work with Y. Peres and J. Steif).

20/04/2017 - Alessandra Cipriani (University of Bath)

16:00-17:00 University College London, 1-19 Torrington Place,Room 102, London WC1E 7HB

The divisible sandpile model, a continuous version of the abelian sandpile model, was introduced by Levine and Peres to study scaling limits of the rotor aggregation and internal DLA growth models. The dynamics of the sandpile runs as follows: to each site of a graph there is associated a height or mass. If the height exceeds a certain value then the site collapses by distributing the excessive mass uniformly to its neighbours. In a recent work Levine et al. addressed two questions regarding these models: the dichotomy between stabilizing and exploding configurations, and the behavior of the odometer (a function measuring the amount of mass emitted during stabilization). In this talk we will investigate further the odometer function by showing that, under appropriate rescaling, it converges to the continuum bi-Laplacian field or to an alpha-stable generalised field when the underlying graph is a discrete torus. Moreover we present some results about stabilization versus explosion for heavy-tailed initial distributions.
(With Rajat Subhra Hazra and Wioletta Ruszel)

### March 2017

31/03/2017 - Isaac Sonin (University of North Carolina at Charlotte)

17:00-18:00 S5.20, Strand Building, Strand Campus, King's College London

An important, though not well-known tool for the study of Markov chains (MCs) is the notion of a Censored (Embedded) MC. It is based on a simple and insightful idea of Kolmogorov and Doeblin: a MC observed only on a subset of its state space is again a MC with a reduced state space and a new transition matrix. The sequential application of this idea leads to an amazing variety of important algorithms in Probability Theory and its Applications.
In my talk I will briefly touch on a couple of related questions: Continue, Quit, Restart model and Insertion, a new operation for MCs; Some Remarks about Independence; Decomposition-Separation (DS) Theorem, describing the behaviour of a family of nonhomogeneous MCs defined by the sequence of stochastic matrices when there are no assumptions about this sequence; Tanks model of clearing in financial networks.

31/03/2017 - Jean-Francois Le Gall (University Paris-Sud Orsay)

16:00-17:00 S5.20, Strand Building, Strand Campus, King's College London

We develop an excursion theory for Brownian motion
indexed by the Brownian tree, which in many respects is analogous
to the classical Ito theory for linear Brownian motion. Each excursion is
associated with a connected component of the complement of the zero set of
the tree-indexed Brownian motion. Each such connected
component is itself a continuous tree, and we introduce a
quantity measuring the length of its boundary.
The collection of boundary lengths coincides with the collection
of jumps of a continuous-state branching process. Furthermore,
conditionally on the boundary lengths, the different excursions
are independent, and we determine their conditional distribution in terms
of an excursion measure which is the analogue of the Ito
measure of Brownian excursions. If time permits, we will
discuss applications to the Brownian map.

22/03/2017 - Christophe Sabot (Lyon)

13:00 Queen Mary University of London, Mathematics, Room W316 (Queens Building)

It is well-known that the first hitting time of 0 by a negatively drifted Brownian motion starting at $a>0$ has the inverse Gaussian law. Moreover, conditionally on this first hitting time, the BM up to that time has the law of a 3-dimensional Bessel bridge. In this talk, we will give a generalization of this result to a familly of Brownian motions with interacting drifts. The law of the hitting times will be given by the inverse of the random potential that appears in the context of the self-interacting process called the Vertex Reinforced Jump Process (VRJP). The spectral properties of the associated random SchrÃ¶dinger operator at ground state are intimately related to the recurrence/transience properties of the VRJP.
We will also explain some "commutativity" property of these BM and its relation with the martingale that appeared in previous work on the VRJP.
Work in progress with Xiaolin Zeng.

17/03/2017 - Tiziano De Angelis (University of Leeds)

17:00-18:00 S0.03, Strand Building, Strand Campus, King's College London

We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton-Jacobi-Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund's value at a time-dependent optimal boundary.
Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at $0$ and created at a rate proportional to its local time.
https://arxiv.org/abs/1609.01655

17/03/2017 - Bert Zwart (CWI Amsterdam)

16:00-17:00 S0.03, Strand Building, Strand Campus, King's College London

Many rare events in man-made networks exhibit heavy-tailed features. Examples are file sizes, delays and financial losses, but also magnitudes of systemic events, such as the size of a blackout in a power grid. The theory of rare events in the heavy-tailed case is not as well developed as it is for light-tailed systems: apart from a few isolated examples, it is restricted to events that are caused by a single big jump. In this work, we develop sample-path large deviations for random walks and Levy processes in the heavy-tailed case that go beyond such restrictions. We show that for such systems, the rare event is not characterized by the solution of a variational problem as it would be in the light-tailed case, but by an impulse control problem. These insights are used to develop a generic importance sampling technique that has bounded relative error, is applicable to any continuous functional of a (collection of) random walks, and is tested on applications arising in finance, insurance, and queueing networks. Joint work with Jose Blanchet, Chang-Han Rhee, and Bohan Chen.

15/03/2017 - Robert Griffiths (Univeristy of Oxford)

13:00-14:00 Queen Mary University of London, Mathematics, Room W316 (Queens Building)

The Wright-Fisher diffusion process with recombination models the haplotype frequencies in a population where a length of DNA contains $L$ loci, or in a continuous model where the length of DNA is regarded as an interval $[0,1]$. Recombination may occur at any point in the interval and split the length of DNA. A typed dual process to the diffusion, backwards in time, is related to the ancestral recombination graph, which is a random branching coalescing graph. Transition densities in the diffusion have a series expansion in terms of the transition functions in the dual process. The history of a single haplotype back in time describes the partitioning of the haplotype into fragments by recombination. The stationary distribution of the fragments is of particular interest and we show an efficient way of computing this distribution. This is joint research with
Paul A. Jenkins, University of Warwick, and Sabin Lessard, Universite de Montreal.

08/03/2017 - Neil O'Connell (University of Bristol)

13:00-14:00 Queen Mary University of London, Mathematics, Room W316 (Queens Building)

The Robinson-Schensted-Knuth (RSK) correspondence is a combinatorial bijection which plays an important role in the theory of Young tableaux and provides a natural framework for the study of longest increasing subsequences in random permutations and related percolation problems. I will give some background on this and then explain how a birational version of the RSK correspondence provides a similar framework for the study of GL(n)-Whittaker functions and random polymers.

### February 2017

17/02/2017 - Zbigniew Palmowski (University of Wroclaw)

16:00-17:00 S0.12, Strand Building, Strand Campus, King's College London

In this talk we present the solutions of so-called exit problems for
a (reflected) spectrally negative one-dimensional L\'evy process
exponentially killed with killing intensity depending on
the present state of the process. We will also analyze respective resolvents.
All identities are given in terms of new generalizations of scale functions.
Particular cases concern $\omega(x)=q$ when we derive classical exit
problems and $\omega(x)=q \mathbf{1}_{(a,b)}(x)$ producing Laplace
transforms of occupation times of intervals until first passage times.
We will show how derived results can be applied to find bankruptcy
probability in so-called Omega model, where bankruptcy occurs at rate
$\omega(x)$ when the surplus L\'evy process process is at level $x<0$.
Finally, we demonstrate how to get some exit identities for a
spectrally positive self-similar Markov processes.
The main idea of all proofs relies on classical fluctuation identities
for L\'evy process, the Markov property and some basic properties of a
Poisson process.
The talk is based on [1].
[1] B. Li and Z. Palmowski (2016) Fluctuations of Omega-killed spectrally negative L\'evy

03/02/2017 - Zhen Wu (Shandong University)

17:00-18:00 S0.12, Strand Building, Strand Campus, King's College London

This talk is concerned with backward stochastic differential equations (BSDEs) coupled
by a finite-state Markov chains which has a two-time scale structure i.e. the states of
the Markov chain can be divided into a number of groups so that the chain jumps rapidly
within a group and slowly between the groups. In this talk, we give a convergence result
as the fast jump rate goes to infinity, which can be used to reduce the complexity of the
original problem. This method is also referred to as singular perturbation.
The first result is the weak convergence of the BSDEs with two-time-scale BSDEs.
It is proved that the solution of the original BSDE system converges weakly under the
Meyer-Zheng topology. The limit process is a solution of aggregated BSDEs. The results
are applied to a set of partial differential equations and used to validate their convergence
to the corresponding limit system.
And then we focus on the optimal switching problem for regime-switching model with
two-time-scale Markov chains. Under the two-time-scale structure, we prove the convergence
of the value functions (variational inequalities) and obtain the optimal switching
strategy by virtue of the oblique reflected BSDEs with Markov chains.
Numerical examples are given for the problem to demonstrate the approximation results.
joint work with Ran Tao and Qing Zhang

03/02/2017 - Monique Jeanblanc (Evry)

16:00-17:00 S0.12, Strand Building, Strand Campus, King's College London

In this presentation, we prove that a random time $\tau$
on a filtered probability space $(\Omega, \ff, \P)$ can written
as the infimum of two random times: the first one avoids $\ff$
stopping times and the second one is thin, i.e. its graph is
included in the union of graph of $\ff$-stopping times. This
allows us to give a condition so that any $\ff$ martingale is a
semi martingale in the filtration $\ff$ progressively enlarges
with $\tau$. We give examples of applications to default times.
Joint work with Anna Aksamit and Tahir Choulli

### January 2017

27/01/2017 - Ivar Ekeland (Universite Paris-Dauphine)

16:00-17:00 K0.16, King's Building, Strand Campus, King's College London

We present an infinite-horizon model for a commodity market. The supply at each period is random and i.i.d. with known distribution. It is traded between storers, processors and speculators. Each agent wants to maximise the short-term profit. We seek an optimal Markovian strategy for each agent. Of course, the profit realized between t and t+1 depends not only on the realized supply at time t+1, but also on the demand at time t+1, that is on the strategies of all the other agents. This leads to an equilibrium problem which has some unusual features. I will show how to solve it, and provide and algorithm and some numerical features.

### December 2016

02/12/2016 - Luciano Campi (LSE)

17:00-18:00 K-1.56, King's Building, Strand Campus, King's College London

We consider a symmetric N-player game with weakly interacting diffusions and an absorbing set. We study the existence of Nash equilibria of the limiting mean-field game and establish, under a non-degeneracy condition of the diffusion coefficient, that the latter provides nearly optimal strategies for the N-player game. Moreover, we provide an example of a mean-field game with absorption whose Nash equilibrium is not a good approximation of the pre-limit game.
This talk is based on a joint work with Markus Fischer (Padua University).

02/12/2016 - Tusheng Zhang (University of Manchester)

16:00-17:00 K-1.56, King's Building, Strand Campus, King's College London

We introduce a discretization/approximation scheme for reflected
stochastic partial differential equations driven by space-time white noise through
systems of reflecting stochastic differential equations.
To establish the convergence of the scheme, we study the existence and uniqueness of solutions of Skorohod-type deterministic systems
on time-dependent domains. We also need to establish the convergence of an approximation scheme for deterministic parabolic obstacle problems. Both are of independent interest on their own.

### November 2016

18/11/2016 - Mathew Joseph (University of Sheffield)

17:00-18:00 K0.16, King's Building, Strand Campus, King's College London

We give a discrete space- discrete time approximation of the stochastic heat equation
by replacing the Laplacian by the generator of a discrete time random walk and approximating
white noise by a collection of i.i.d. mean 0 random variables. We give a few applications of this
approximation, including fluctuations around the characteristic line for the harness process and the random average process.

18/11/2016 - Sam Cohen (University of Oxford)

16:00-17:00 K0.16, King's Building, Strand Campus, King's College London

In practice, stochastic decision problems are often based on statistical estimates of probabilities. We all know that statistical error may be significant, but it is often not so clear how to incorporate it into our decision making. In this talk, we will look at one approach to this problem, based on the theory of nonlinear expectations. We will consider the large-sample theory of these estimators, and also connections to `robust statistics' in the sense of Huber.

04/11/2016 - Pierre Del Moral, INRIA (Bordeaux-Sud Ouest Research Center)

17:00-18:00 K0.19, King's Building, Strand Campus, King's College London

The Ensemble Kalman filter is a sophisticated and powerful data assimilation method for filtering high dimensional problems arising in fluid mechanics and geophysical sciences. This Monte Carlo method can be interpreted as a mean-field McKean-Vlasov type particle interpretation of the Kalman-Bucy diffusions. In contrast to more conventional particle filters and nonlinear Markov processes these models are designed in terms of a diffusion process with a diffusion matrix that depends on particle covariance matrices. Besides some recent advances on the stability of nonlinear Langevin type diffusions with drift interactions, the long-time behaviour of models with interacting diffusion matrices and conditional distribution interaction functions has never been discussed in the literature. One of the main contributions of the article is to initiate the study of this new class of models The article presents a series of new functional inequalities to quantify the stability of these nonlinear diffusion processes. In the same vein, despite some recent contributions on the convergence of the Ensemble Kalman filter when the number of sample tends to infinity very little is known on stability and the long-time behaviour of these mean-field interacting type particle filters. The second contribution of this article is to provide uniform propagation of chaos properties as well as Lp-mean error estimates w.r.t. to the time horizon. Our regularity condition is also shown to be sufficient and necessary for the uniform convergence of the Ensemble Kalman filter. The stochastic analysis developed in this article is based on an original combination of functional inequalities and Foster-Lyapunov techniques with coupling, martingale techniques, random matrices and spectral analysis theory. This is joint work with Julian Tugaut.

04/11/2016 - Denis Denisov (University of Manchester)

16:00-17:00 K0.19, King's Building, Strand Campus, King's College London

We consider a one-dimensional Markov chain with asymptotically zero drift and finite second moments of jumps . In the transient case we will prove an integral renewal theorem. Then we connect the renewal theorem with asymptotic behaviour of the tail of the stationary measure in the positive recurrent case. This is a joint work with D. Korshunov and V. Wachtel.

### October 2016

21/10/2016 - Loic Chaumont (University of Angers)

18:00-19:00 K0.16, King's Building, Strand Campus, King's College London

We conjecture that any probability distribution on the real line can be characterized by the sole data of its upward space-time
Wiener-Hopf factor. We prove that this result holds for large classes of distributions. We also prove that the following stronger
result holds in many cases: the sole knowledge of the measure and the convolution product of this measure by itself both
restricted to the positive half line are actually sufficient to determine the measure. This is a joint work with Ron Doney (Manchester University).

21/10/2016 - Ben Hambly (University of Oxford)

17:00-18:00 K0.16, King's Building, Strand Campus, King's College London

A randomly trapped random walk on a graph is a simple random walk in which the holding time at a given vertex is an independent sample from a probability measure determined by the trapping landscape, a collection of probability measures
indexed by the vertices. This is a time change of the simple random walk. For the constant speed continuous time random walk, the landscape is an exponential distribution with rate 1 for all vertices. For the Bouchaud trap model it is an exponential random variable at each vertex but where the rate is chosen from a heavy tailed distribution. In one dimension the possible scaling limits are time changes of Brownian motion and include the fractional kinetics process and the Fontes-Isopi-Newman (FIN) singular diffusion. We extend this analysis to put these models in the setting of resistance forms, a framework that includes finitely ramified fractals. In particular we will construct a FIN diffusion as the limit of the Bouchaud trap model and the random conductance model on fractal graphs. We will establish heat kernel estimates for the FIN diffusion extending what is known even in the one-dimensional case.

19/10/2016 - Ashkan Nikeghbali (University of Zurich)

16:00-17:00 Queen Mary University of London, Mathematics, Room M103

It is standard in random matrix theory to study weak convergence of the eigenvalue point process, but how about almost sure convergence? In this talk we introduce a way to couple all dimensions of random unitary matrices together to prove a quantitative strong convergence for eigenvalues for random unitary matrices. Then we show how this can give some remarkable simple answers to important questions related to moments and ratios of characteristic polynomials of random unitary matrices (and insight in some conjectures related to the Riemann zeta function).

07/10/2016 - Wilfrid Kendall (University of Warwick)

17:00-18:00 K0.16, King's Building, Strand Campus, King's College London

In this talk I will discuss the use of Dirichlet forms to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite- dimensional distributions. (Joint with Giacomo Zanella and Mylene Bedard)

07/10/2016 - Terry Lyons (University of Oxford)

16:00-17:00 K0.16, King's Building, Strand Campus, King's College London

Rough Path theory is the extension of Newtonia Calculus to the context of highly oscillatory systems. It provides a rigorous framework for discussing and analysing the function theory on such systems.
Classical calculus is intimately related to the theory of smooth functions and particularly to Taylor Series. The equivalent idea is crucial in rough path theory and leads to the linkage of modern combinatorial algebra and Hopf structures to detailed computations of machine learning of practical application in finance and beyond.