# Past Events

### June 2018

### May 2018

24/05/2018-24/05/2018

Panos Vassiliou : Laws of large numbers for non-homogeneous Markov systems 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.

10/05/2018-10/05/2018

Tyler Helmuth: Recurrence of the vertex-reinforced jump process in two dimensions 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: Covariance identities for Poisson functionals 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): Skeletal stochastic differential equations for continuous-state branching process 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: Boltzmann-Gibbs measures associated with log-correlated fields and LQFT 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): Extinction time of CB-processes with competition in a Levy random environment 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: Random permutations and the Erdos-Mirsky problem 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): Couplings, metrics and contraction rates for Langevin diffusions 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): Squared Bessel processes and Poisson-Dirichlet interval partition evolutions 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): The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics 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: The Fyodorov-Bouchaud formula and Liouville conformal field theory 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: A relative anti-concentration inequality 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: Two Point Function for Critical Points of a Random Plane Wave 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: Non-Hermitian random Schroedinger operators 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: The semigroup of metric measure spaces and its infinitely divisible measure joint work with S.N. Evans (Berkeley) 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: The dimer model on Riemann surfaces 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: A test of Gaussianity based on the excursion sets of a random field 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: Transportation inequalities, Malliavin calculus and couplings 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: Heterogeneous micro-structure of percolation in sparse networks 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: Percolation without FKG 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): Toplogies of the zero sets of random real projective hyper-surfaces and monochromatic waves 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): Boundary effects in the evolution of interacting birth-and-death processes 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): Density functional theory and many-marginals optimal transport with Coulomb and Riesz costs 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: Capacity of random walk and Wiener sausage in 4 dimensions 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): Analytical and numerical approximations for maximum of weighted Brownian motion 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: Further topics: continuous time and heavy tails 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: Many-dimensional random walks 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: Lamperti's problem 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: Foster-Lyapunov criteria for Markov chains 10:00-12:00 KCL Department of Mathematics, Strand building, room S5.20

### May 2017

05/05/2017

Mateusz Majka: Couplings for SDEs driven by Levy processes and their applications 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: Stability of overshoots of recurrent random walks 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): Random walks on dynamical percolation 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): Scaling limit of the odometer in the divisible sandpile 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): Censored Markov Chains - a Powerful Tool in Probability Theory and its Applications 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): Excursion theory for Brownian motion indexed by the Brownian tree 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): Vertex Reinforced Jump Process, random Schroedinger operator and hitting time of Brownian motion 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): The dividend problem with a finite horizon 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): Heavy-tailed Stochastic Systems - Sample-path Large Deviations and Rare Event Simulation 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

Jose Manuel Arroyo (Universidad de Castilla-La Mancha): Addressing Uncertainty in Power System Operation and Planning via Two-Stage Adaptive Robust Optimization 16:00-16:45 KCL Department of Mathematics, Strand building, room S0.13

Power systems are increasingly exposed to uncertain aspects such as demand, system component availability, and renewable-based generation, among others. Given the crucial role played by power systems in nowadays society, it is essential to operate and plan this critical piece of infrastructure so that the balance between generation and consumption is guaranteed for all plausible uncertainty realizations.
Within this context, the presentation examines the use of two-stage adaptive robust optimization as a relevant tool to address the impact of uncertainty sources in the decision-making problems arising in power system operation and planning. Unlike alternative approaches to deal with uncertainty, neither accurate probabilistic information nor a discrete set of uncertainty realizations are required. Rather, uncertainty is modeled by decision variables within an uncertainty set. Hence, the size of the robust models does not depend on the dimension of the space of uncertainty realizations belonging to the uncertainty set, thereby providing a computationally efficient framework. In addition, an easy control of the degree of conservativeness can be implemented.
The resulting robust counterparts are instances of mixed-integer trilevel programming. Practical modeling aspects allow using effective decomposition-based techniques that guarantee finite convergence to optimality. Two applications are discussed, namely a security-constrained generation scheduling problem and a transmission network expansion planning problem under uncertain nodal injections.

15/03/2017

John Moriarty (Queen Mary University of London): Energy imbalance market call options and the valuation of storage 14:45-15:30 KCL Department of Mathematics, Strand building, room S0.13

The use of energy storage to balance electric grids is increasing and, with it, the importance of operational optimisation from the twin viewpoints of cost and system stability. In this paper we assess the real option value of balancing reserve provided by an energy-limited storage unit. The contractual arrangement is a series of American-style call options in an energy imbalance market (EIM), physically covered and delivered by the store, and purchased by the power system operator. We take the EIM price as a general regular one-dimensional diffusion and impose natural economic conditions on the option parameters. In this framework we derive the operational strategy of the storage operator by solving two timing problems: when to purchase energy to load the store (to provide physical cover for the option) and when to sell the option to the system operator. We give necessary and sufficient conditions for the finiteness and positivity of the value function -- the total discounted cash flows generated by operation of the storage unit. We also provide a straightforward procedure for the numerical evaluation of the optimal operational strategy (EIM prices at which power should be purchased) and the value function. This is illustrated with an operational and economic analysis using data from the German Amprion EIM.

15/03/2017

Simon Tindemans (Imperial College London): Smart refrigerators: a distribution-referred approach to decentralised control 14:00-14:45 KCL Department of Mathematics, Strand building, room S0.13

The physical characteristics of refrigerators and other thermostatically controlled loads make them exceptionally suitable as a low-cost provider of flexibility to the grid: their power consumption can be shifted by 10s of minutes without noticeable effects on cooling performance. This flexibility can then be used for the provision of response and reserve services, to reduce extreme load levels and to alleviate ramping constraints. However, it is challenging to design a robust control scheme that respects the thermal limits imposed by individual appliances, but does not depend on complex, costly and invasive centralised control.
I will describe a decentralised control scheme that is based on a probabilistic representation of refrigerator states. Each appliance receives only one signal that describes the overall control intent (i.e. population power consumption). The appliance uses this signal to compute the appropriate switching actions, by considering its cooling requirements versus those of a virtual distribution of fridges with the same physical model and randomised phases. Subject to statistical independence of refrigerators prior to the control action, the law of total expectation guarantees that this control scheme results in the desired collective behaviour for a large number of appliances, even for heterogeneous populations. Monte Carlo simulations will be used to illustrate the results, and the control approach will be placed in the context of other approaches to decentralised control. This is joint work with Vincenzo Trovato and Goran Strbac.

15/03/2017

Robert Griffiths (Univeristy of Oxford): A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning 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.

15/03/2017

James R. Cruise (Hariot-Watt University): Control of storage for buffering uncertainty 11:45-12:30 KCL Department of Mathematics, King's building, room K2.40

Electricity supply and demand needs to be kept balanced at all times.
However, renewable generation in particular is both variable and
difficult to predict. We study the use of storage which is used to
buffer both fluctuations in market price and uncertainties arising
from forecast errors and other sudden shocks. We view the problem as
being formally one of stochastic dynamic programming (SDP), but show
how to recast the SDP recursion in terms of functions which, if known,
would reduce the associated optimisation problem to one which is
deterministic, except that it must be re-solved at times when shocks
occur. The functions required for this approach may be defined in
terms of a probabilistic coupling. In the case of a perfectly
efficient store facing linear buying and selling costs the functions
may be determined exactly; otherwise they may typically be estimated
to good approximation. The fact that the storage is also being used
for arbitrage, i.e. for making money by buying and selling, in general
has the effect that the above coupling occurs quickly, improving the
speed of exact solutions and improving the quality of approximate
ones. We give examples based on Great Britain electricity price data.
This is joint work with Stan Zachary.

15/03/2017

Eduardo Alejandro Martinez Cesena: Managing Uncertainty in Distribution Network Planning with Flexibility from Demand Response, Network Reconfiguration and Conservation Voltage Reduction Techniques 11:00-11:45 KCL Department of Mathematics, King's building, room K2.40

Demand growth is becoming more and more uncertain due to increasing electrification of heating and transports, improved energy efficiency, installation of photovoltaic systems, and so forth. These changes are challenging the adequacy of (i) traditional asset-based distribution network reinforcement solutions, such as feeder and substation reinforcements, which can be pricy and time intensive and (ii) passive fit-and-forget planning practices based on best-view forecasts that do not properly model uncertainty.
This tutorial provides an overview of emerging stochastic programming approaches for the planning of distribution networks under uncertainty developed at The University of Manchester in various UK and European projects (e.g., ADDRESS, C2C and Smart Street). These tools, specialize on managing risks and uncertainty by using flexibility from traditional line reinforcement and substation upgrade options, as well as from smart post-contingency demand response, active network reconfiguration and conservation voltage reduction options.
The different network planning approaches are demonstrated using examples based on real UK distribution networks where the different smart solutions were tested.

15/03/2017

Bert Zwart (CWI Amsterdam): Reliability of power grids 09:45-10:30 KCL Department of Mathematics, King's building, room K6.63

In a well-designed network, events such as line failures or blackouts should be rare. I will give an overview of some results and challenges in this domain from my own perspective. I am particularly intrigued in the way renewables can influence the occurrence of failures and
focus on the problem of developing computationally feasible chance constraints for such events that could be used for planning taking into account the uncertainty of, for example, wind energy. The results are also intended as a first step towards a qualitative understanding of the propagation of multiple failures, which can lead to large blackouts.

15/03/2017

Goran Strbac (Imperial College London): TBA 09:00-09:45 KCL Department of Mathematics, King's building, room K6.63

08/03/2017

Neil O'Connell (University of Bristol) : From longest increasing subsequences to Whittaker functions and random polymers 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

23/02/2017

Franco Flandoli (Universita di Pisa): Regularization by noise in infinite dimensions 16:00-17:00 KCL Department of Mathematics, Strand Building S-3.18

After a short review of results and ideas in finite dimensions, two
main directions of research on regularization by noise in infinite dimensions will be discussed and compared, one based on Kolmogorov equations in Hilbert spaces, the other on specific argument applicable to examples. Fluid dynamics lives in the second class; the 3D Euler equations and their linear analog, a linear vector valued advection equation, will be discussed.

23/02/2017

Tusheng Zhang (University of Manchester): Global solutions of stochastic heat equations 14:30-15:30 KCL Department of Mathematics, Strand Building S-3.18

In this talk I will present some recent results on global existence of solutions of stochastic heat equations with super linear drifts and multiplicative space-time noise.

23/02/2017

Dave Applebaum (Sheffield University): Levy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space 13:30-14:30 KCL Department of Mathematics, Strand Building S-3.18

L\'{e}vy processes are stochastic processes with independent and stationary increments.
In this talk we will discuss some recent results on the study of the properties of L\'{e}vy processes taking values in the dual of a nuclear space.
In simple terms, a nuclear space is an infinite dimensional space that shares many properties of finite dimensional spaces. However, it is its dual space that has been of
most importance in the study of probability measures and stochastic processes, both from a theoretical perspective as well as for applications.
The first step in our program will be to prove the L\'{e}vy-It\^{o} decomposition, which describes the structure of a L\'{e}vy process as a sum of four components.
This decomposition is of importance for later study of stochastic integrals with respect to L\'{e}vy processes and applications to the study
of stochastic partial differential equations with L\'{e}vy noise. Second, we will establish the correspondence between infinitely divisible measures and L\'{e}vy
processes in the dual of a nuclear space. Finally, as a by-product of our results, we will prove the L\'{e}vy-Khintchine formula for the characteristic function of
infinitely divisible measures on the dual of a nuclear space.
(This is based entirely on PhD work by my former student Christian Fonseca Mora, now at the University of Costa Rica).

23/02/2017

Jentzen Arnulf (ETH Zurich): On approximation algorithms for stochastic ordinary differential equations (SDEs) and stochastic partial differential equations (SPDEs) 11:00-12:00 KCL Department of Mathematics, Strand Building S-3.18

In this lecture I intend to review a few selected recent results on numerical approximations for stochastic ordinary differential equations (SDEs) and stochastic partial differential equations (SPDEs). Key goals of the lecture are to provide links to real world applications of such equations and to present challenging open problems for numerical approximations of such equations. The lecture includes content on lower and upper error bounds, on strong and weak convergence rates, on Cox-Ingersoll-Ross (CIR) processes, on pricing models for financial derivatives, on the parabolic Anderson model, stochastic Burgers equations, and other parabolic SPDEs, as well as on stochastic Wave equations and other hyperbolic SPDEs. We illustrate our results by some numerical simulations and we also calibrate the Heston derivative pricing model to real exchange market prices of financial derivatives on the stocks in the American Standard & Poor's 500 (S&P 500) stock market index.

23/02/2017

Francesco Russo (ENSTA-ParisTech): BSDEs, cadlag martingale problems and Follmer-Schweizer decomposition under basis risk 10:00-11:00 KCL Department of Mathematics, Strand Building S-3.18

The aim of this talk consists in introducing a new formalism for the
deterministic analysis associated with backward stochastic differential
equations
driven by general c\`adl\`ag martingales.
When the martingale is a standard Brownian motion,
the natural deterministic analysis is provided by the solution of a semilinear PDE of parabolic type.
A significant application concerns the hedging problem under basis risk of a contingent claim $g(X_T,S_T)$,
where $S$ (resp. $X$) is an underlying price of a traded (resp. non-traded but observable) asset,
via the celebrated F\"ollmer-Schweizer decomposition. We revisit the case when the couple of price processes
$(X,S)$ is a diffusion and we provide explicit expressions when $(X,S)$ is an exponential of additive processes.

17/02/2017-17/02/2017

Jason Miller (University of Cambridge): Convergence of the self-avoiding walk on random quadrangulations to SLE_{8/3} on \sqrt{8/3}-Liouville quantum gravity. 17:30-18:30 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

Let (Q,\lambda) be a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk. We prove that (Q,lambda) converges in the scaling limit to a certain \sqrt{8/3}-Liouville quantum gravity surface decorated by an independent chordal SLE_{8/3}. The scaling limit can equivalently be described as the metric gluing of two independent instances of the Brownian half-plane. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the Gromov-Hausdorff topology to curve-decorated metric measure spaces. This is joint work with E. Gwynne.

17/02/2017

Zbigniew Palmowski (University of Wroclaw): Fluctuations of Omega-killed spectrally negative L'evy processes 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

17/02/2017-17/02/2017

Djalil Chafai (University Paris-Dauphine): Concentration for Coulomb gases and Coulomb transport inequalities 16:00-17:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

This talk will present a recent joint work with Mylene Maida and Adrien Hardy on the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest.

17/02/2017-17/02/2017

Nadia Sidorova (UCL): Delocalising the parabolic Anderson model 15:00-16:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. 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 (including Pareto potentials) it is localised at just one point. In the talk, we discuss a natural modification of the parabolic Anderson model on Z, which exhibits a phase transition between localisation and delocalisation. This is a joint work with Stephen Muirhead and Richard Pymar.

17/02/2017-17/02/2017

Roman Kotecky (University of Warwick): Metastability for a model on continuum 11:30-12:30 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

17/02/2017-17/02/2017

Stefan Adams (University of Warwick): Variational problems for Laplacian interface models in $ (1+1) $ dimensions 10:00-11:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

We obtain variational problems for the free energy of a Laplacian interface model which is a Hamiltonian system with a bi-Laplacian operator. We study scaling limits and the corresponding large deviation principles perturbed by an attractive force towards the origin to complete the microscopic-macroscopic transition. In particular we analyse the critical situation that the rate functions admit more than one minimiser leading to concentration of measure problems. The interface models are a class of linear chain models with Laplacian interaction and appear naturally in the physical literature in the context of semi-flexible polymers. We discuss these connections as well as the ones with the related gradient models. These random fields are a class of model systems arising in the studies of random interfaces, critical phenomena, random geometry, field theory, and elasticity theory. If time permits we outline open questions in higher dimensions, that is $ (d+m) $ -dimensional models and their large deviation principles.

17/02/2017-17/02/2017

Ben Leimkuhler (University of Edinburgh): Stochastic differential equations and numerical methods for multimodal Gibbs sampling 09:00-10:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

Problems in molecular simulation and data analytics demand new types of sampling algorithms to efficiently traverse the landscapes of models with energetic and entropic barriers. I will compare several approaches based on modified stochastic differential equations which can provide enhanced sampling efficiency. I will also highlight the importance of numerical method design in obtaining optimal performance.

16/02/2017-16/02/2017

Johannes Zimmer (University of Bath): Particles and the geometry/thermodynamics of macroscopic evolution 17:30-18:30 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

One often aims to describe the collective behaviour of an infinite number of particles by the differential equation governing the evolution of their density. The theory of hydrodynamic limits addresses this problem. In this talk, the focus will be on linking the particles with the geometry of the macroscopic evolution. Zero-range processes will be used as guiding example. The geometry of the associated hydrodynamic limit, a nonlinear diffusion equation, will be derived. Large deviations serve as a tool of scale-bridging to describe the many-particle dynamics by partial
differential equations (PDEs) revealing the geometry as well. Finally, we will discuss the near-minimum structure, studying the fluctuations around the minimum state described by the deterministic PDE.

16/02/2017-16/02/2017

Hendrik Weber (University of Warwick): Equilibration for the dynamical $\Phi^4$ model 16:00-17:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

In this talk I will discuss the long term behaviour of the stochastic PDE
\partial_t \phi = \Delta \phi - \phi^3 + \xi,
where $\xi $ denotes space-time white noise and the space
variables $x $ takes values in the $d$ dimensional torus for
either $d=2,3$. This equation was proposed in the eighties
by Parisi and Wu to give a dynamical construction of the
Euclidean $\Phi^4$ quantum field theory which (at least formally)
arises as the invariant measure of this SPDE.
Due to the irregularity of the driving white noise, the constructing
solutions to the SPDE was an open problem for many years -
the construction of short time solutions in the more difficult three dimensional case
was accomplished by Hairer only a few years ago.
In this talk I will go back to Parisi and Wu's original question and
study the long term behaviour of solutions. In the two dimensional case
$d=2$ I will show that solutions converge to equilibrium exponentially
fast. I will also outline the proof of a similar statement in the three dimensional
case.
This is based on joint work with Pavlos Tsatsoulis and Jean-Christophe
Mourrat.

16/02/2017-16/02/2017

Jean-Dominique Deuschel (TU Berlin): A local limit theorem for 2-d conductance model with application to gradient interface model. 15:00-16:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

We consider a symmetric random walk in random environment
and show the convergence of the corresponding rescaled
2-dimensional potential. Using the random walk representation
this yields the asymptotic of the variance in the 2-d
anharmonic gradient interface model.

16/02/2017-16/02/2017

Mathieu Lewin (University Paris-Dauphine): Mean-field limits for bosons and fermions 11:30-12:30 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

In this talk I will review recent works in collaboration with Soeren Fournais, Phan Thanh Nam, Nicolas Rougerie and Jan Philip Solovej on the mean-field limit for quantum systems. This is a regime in which the number of particles $N$ tends to infinity and the interaction strength behaves as 1/N. In particular I will insist on the difference between bosons and fermions, and make some connections with results for classical gases.

16/02/2017-16/02/2017

Eric Cances (Ecole des Ponts ParisTech and INRIA): Incommensurate and disordered quantum systems 10:00-11:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

After recalling the standard mathematical formalism used to model disordered materials such as doped semiconductors, alloys, or amorphous materials, and classical results about random Schroedinger operators (Anderson localization), I will present a tight-binding model for computing the electrical conductivity of multilayer 2D materials. All these models fall into the scope of the mathematical framework, based on non-commutative geometry, introduced by Bellissard to study the electronic properties of aperiodic systems. I will finally present numerical calculations of the electronic conductivity of 1D incommensurate bilayer systems as a function of the lattice constant ratio and the Fermi level. The plot of the so-obtained function is reminiscent of Hofstadter's butterfly.

16/02/2017-16/02/2017

Benjamin Stamm (Aachen University): Continuum solvation models for the modelling of electrostatic interaction between solvent and solute molecules 09:00-10:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

The large majority of chemically interesting phenomena take place in liquid phase, where the environment (e.g., solvent) can play a crucial role in determining the structure, the properties and the dynamics of the system to be studied. In a practical context, accounting for all solvent molecules explicitly mat be infeasible due to the complexity of the underlying equations. A particular choice to reduce the complexity is to model the solvent to be a polarisable continuum medium. The resulting electrostatic energy contribution to the solvation energy can be computed by solving a Poisson-type interface problem.
To design a fast and efficient electrostatic solver is a delicate task as the electrostatic potential only decays slowly, i.e. with a rate 1/r, towards infinity. We refer to integral equations on the interface between the solvent and the solute in order to discretize the problem using a new domain decomposition paradigm for integral equations.

15/02/2017-15/02/2017

Augusto Gerolin (University Jyvaskyla): A counterexample in SCE Density Functional Theory 17:30-18:30 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

The Strictly-Correlated-Electrons (SCE) density functional theory (SCE DFT) approach, originally proposed by Michael Seidl, is a formulation of density functional theory, alternative to the widely used Kohn-Sham DFT, especially aimed at the study of strongly-correlated systems. Following the talk of Gero Friesecke, we will recall briefly a link between SCE DFT and Multi-marginal Optimal Transport (OT) Theory and discuss the main issues of SCE DFT through the OT framework. Finally, we will present a counterexample of the existence of a "Seild map" for a class of radially symmetric densities in \R^3.

15/02/2017-15/02/2017

Simone di Marino (Indam, Scuola Normale Superiore, Pisa): DFT, multimarginal optimal transport and Lieb-Oxford inequalities 16:00-17:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

We first review the Density Functional Theory and its link with
multimarginal optimal transportation. Then we will focus on the
Lieb-Oxford inequality which is an estimate from below of the optimal
transportation cost; we will show that this inequality is exact in the
limit N to infinity for the one dimensional case.

15/02/2017-15/02/2017

Gero Friesecke (TU Munich): Density functional theory and optimal transport with Coulomb cost 15:00-16:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

15/02/2017-15/02/2017

Christoph Ortner (University of Warwick): Separability and Locality of Energy for the Tight-Binding Model and some Applications 11:30-12:30 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

I will review some recent results on the locality of interaction in
the tight-binding model (treated as a toy-model for quantum
chemistry). Specifically, I will show how one can decompose the
density of states into spatially localised contributions. I will show
two applications of this technique: (1) a proof of equivalence of
canonical and grand-canonical ensembles for the electrons; (2)
construction of multi-scale methods with controlled approximation
errors. (joint work with Huajie Chen and Jianfeng Lu); (3) a
generalisation of Brillouin-zone sampling to incommensurate layers of
2D lattices.
Time permitting I will also discuss ongoing work (with Hong
Duong) on an analogous decomposition of free energy in the harmonic
approximation.

15/02/2017-15/02/2017

Haakan Hedenmalm (KTH Stockholm): Bloch functions, asymptotic variance, and geometric zero packing 10:00-11:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

Abrikosov's analysis from 1957 of type II superconductors involves
an energy functional which when suitable localized becomes the problem
of determining the minimal L^4 norm of a section given that the L^2 is
fixed. The problem is essentially one of determining the location of the zeros
of a wave function in the lowest Landau level with minimal L^4 average given
the L^2 average. This model problem has been considered by F. Nier et al.
It is believed that the equilateral triangular configuration of zeros is optimal.
Here we find a relation between an analogous hyperbolic geometry problem
and a sought-after constant in quasiconformal theory. We prove that a related
hyperbolic density is positive which then gives that the quasiconformal constant
is <1.
We also discuss the general minimization problem on compact surfaces,
depending on the genus.

15/02/2017-15/02/2017

Roland Bauerschmidt (University of Cambridge): Eigenvectors and spectral measure of random regular graphs of fixed degree 09:00-10:00 University College London, 1-19 Torrington Place, 115 Galton Lecture Theatre, London WC1E 7HB

I will discuss results on the delocalisation of eigenvectors and the
spectral measure of random regular graphs with large but fixed degree. Our
approach combines the almost deterministic structure of random regular graphs
at small distances with random matrix like behaviour at large distances.

03/02/2017

Zhen Wu (Shandong University): Backward stochastic differential equations coupled with two-time-scale Markov chains and applications in optimal switching problem 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): Classification of random times and application to credit risk modellling 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): A rational expectations equilibrium for commodity markets 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.

20/01/2017

Huyen Pham (University Paris Diderot): Ergodicity of robust switching control and nonlinear system of quasi variational inequalities 16:00-17:00 Franklin-Wilkins Building 1.60, Waterloo Campus, King's College London

We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems.
We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term.
Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities.
Based on joint work with E. Bayraktar and A. Cosso.

20/01/2017

Hao Xing (LSE): Asset pricing under optimal contract 14:30-15:30 Franklin-Wilkins Building 1.60, Waterloo Campus, King's College London

We consider the problem of finding equilibrium asset prices in a financial market in which a portfolio manager (Agent) invests on behalf of an investor (Principal), who compensates the manager with an optimal contract. We extend one of the two models in Buffa, Vayanos and Woolley (2014), BVW (2014), by allowing general contracts. In particular, the optimal contract rewards Agent for taking specific risk of individual assets and not only the systematic risk of the index by making use of the quadratic variation of the deviation of the portfolio return from the return optimal when investing only in the index. Similarly to BVW (2014), we find that the stocks in large supply have high risk premia, while the stocks in low supply have low risk premia, and this effect is stronger as agency friction increases. However, by using the risk-incentive optimal contract, the sensitivity of the price distortion to agency frictions is of an order of magnitude smaller compared to the price distortion in BVW (2014), where only contracts linear in portfolio value and the benchmark are allowed.
This is a joint work with Jaksa Cvitanic.

20/01/2017

Ying Hu (Universite de Rennes 1): Linear Quadratic Mean Field Game with Control Constraint 13:30-14:30 Franklin-Wilkins Building 1.60, Waterloo Campus, King's College London

We study a class of linear-quadratic (LQ) mean-field games in which the individual control process is constrained in a closed convex subset of full space.The decentralized strategies and consistency condition are represented by a class of mean-field forward-backward stochastic differential equation (MF-FBSDE) with projection operators on the closed convex subset. The wellposedness of consistency condition system is obtained using the monotonicity condition method. The related $\epsilon$-Nash equilibrium property is also verified. This is a joint work with
Jianhui Huang and Xun Li.

20/01/2017

Sam Cohen (Oxford University): Fun, Games and Graphs with EBSDEs 11:00-12:00 Franklin-Wilkins Building 2.48, Waterloo Campus, King's College London

When studying ergodic games, or ergodic control systems with jumps, one often struggles with the fact that the comparison theorem does not hold without extra conditions. In this talk, we will look at some settings in which these problems arise, and some methods to get around them.

20/01/2017

Thaleia Zariphopoulou (University of Texas at Austin): Long-horizon optimal investments under forward performance criteria 10:00-11:00 Franklin-Wilkins Building 2.48, Waterloo Campus, King's College London

In this talk I will discuss the long-term behavior of optimal portfolio
functionals under forward performance criteria. Among others, I will show
that the temporal and spatial limits do not coincide, as it is the case in
the classical expected utility setting. I will provide representative
examples and also discuss the limiting behavior of the optimal processes
(wealth and investment) as well as connections with ergodic control and
ergodic BSDE.

### December 2016

02/12/2016

Luciano Campi (LSE): N-player games and mean field games with absorption 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) : Lattice Approximations of Reflected Stochastic Partial Differential Equations Driven by Space-Time White Noise 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): A discrete approximation to the stochastic heat equation 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): Data driven nonlinear expectations for statistical uncertainty 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): On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters 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): Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift 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): On distributions determined by their upward, space-time Wiener-Hopf factor 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): Scaling limits for randomly trapped random walks 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): Some remarkable applications of coupling and strong convergence for the circular unitary ensemble 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): A Dirichlet Form approach to MCMC Optimal Scaling 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): Rough Paths, Hopf Algebras and Chinese Handwriting 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.