LP@K
London Probability at King's

Past Events

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
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.

May 2017

05/05/2017
Mateusz Majka: Couplings for SDEs driven by Levy processes and their applications
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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 Schrödinger 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
London Probability Seminar
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
London Probability Seminar
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
Probabilistic Methods for Energy Networks
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
Probabilistic Methods for Energy Networks
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
Probabilistic Methods for Energy Networks
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
London Probability Seminar
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
Probabilistic Methods for Energy Networks
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
Probabilistic Methods for Energy Networks
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
Probabilistic Methods for Energy Networks
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
Probabilistic Methods for Energy Networks
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
London Probability Seminar
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
Workshop on Infinite Dimensional Probability
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
Workshop on Infinite Dimensional Probability
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
Workshop on Infinite Dimensional Probability
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)
Workshop on Infinite Dimensional Probability
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
Workshop on Infinite Dimensional Probability
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.
New trends in Mathematical Physics at the interface of Analysis and Probability
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
London Probability Seminar
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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.
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
New trends in Mathematical Physics at the interface of Analysis and Probability
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
Workshop on New Directions in Ergodic Stochastic Control and its Applications
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
Workshop on New Directions in Ergodic Stochastic Control and its Applications
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
Workshop on New Directions in Ergodic Stochastic Control and its Applications
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
Workshop on New Directions in Ergodic Stochastic Control and its Applications
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
Workshop on New Directions in Ergodic Stochastic Control and its Applications
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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
London Probability Seminar
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.