# Applied Probability Seminars

This is an informal seminar to give academic staff, visitors, graduate students, etc., based or hosted in Warwick (any department), the opportunity to know more about each other’s research on topics of probability and its uses in related areas such as mathematical statistics, statistical physics, computer sciences, analysis, et cetera.

PhD students and postdocs are particularly encouraged to contribute, as well as any external speaker from other departments or universities that you would like to invite. Please contact Paul Chleboun (p.i.chleboun<at>warwick.ac.uk) if you wish to give a talk or if you have a name of speaker to suggest.

Below is a provisional schedule of the upcoming talks. Given the informal nature of the seminar, last minute changes may happen. Please join the Applied probability Seminar Teams channel to receive up to date information.

###### Academic Year 2022-23, Term 2

Venue: MB0.08 or Online (Teams link will be provided below each abstract)

Time: 11:00 unless otherwise stated - followed by lunch in the atrium.

**20 January 2022**

Speaker: **Martina Favero**

**Title: **Asymptotic behaviour of the Kingman coalescent

**Abstract**

The Kingman coalescent is a classical stochastic process modelling genealogies in mathematical population genetics. We study its large-sample-size asymptotic behaviour, assuming a finite-allele, parent-dependent mutation model. We start by showing that the sampling probabilities under the coalescent decay polynomially in the sample size. The degree of the polynomial depends on the number of types in the model, and its coefficient on the stationary density of a dual Wright-Fisher diffusion. Then, we focus on a convergence result for a sequence of Markov chains that are composed of type-counting, mutation-counting and cost components. The limiting process includes a deterministic part and Poisson processes. Finally, we illustrate how these results may be used to analyse asymptotic properties of some backward sampling algorithms based on the coalescent, in particular the asymptotic behaviour of importance sampling weights.

**27January 2022**

Speaker: **Sigurd Assing**

**Title: **One way to turn the primitive equations into stochastic dynamical systems for climate modelling

**Abstract**

I am going to motivate an established approach to climate modelling. The first step gives a rather complicated system of equations for so-called resolved and unresolved variables, and the second step is about simplifying a scaled version of this system of equations. The second step is based on some ad hoc assumptions and stochastic model reduction. I`ll tell what we (joint work with Flandoli and Pappalettera) were able to do about this, but I’ll also raise awareness of what we could not do. Some of this might be of interest to statisticians, too, as knowing the modelling mechanism would shed light on how data should be mapped to parameters and coefficients of climate models.

**3 February 2022**

Speaker: **David Croydon**

**Title: **Sub-diffusive scaling regimes for one-dimensional Mott variable-range hopping

**Abstract**

I will describe anomalous, sub-diffusive scaling limits for a one-dimensional version of the Mott random walk. The first setting considered nonetheless results in polynomial space-time scaling. In this case, the limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. I will outline how the proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces. The second setting considered concerns a regime that exhibits even more severe blocking (and sub-polynomial scaling). For this, I will describe how, for any fixed time, the appropriately-rescaled Mott random walk is situated between two environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Moreover, I will give an asymptotic description of the distribution of the Mott random walk between the barriers that contain it. This is joint work with Ryoki Fukushima (University of Tsukuba) and Stefan Junk (Tohoku University).

**10 February 2022**

Speaker: **Horatio Boedihardjo**

**Title: **Expected signature as a moment generating function for stochastic processes

**Abstract**

There are many ways to characterise stochastic processes, with the description in terms of finite dimensional distribution being one of the most classical. Motivated by the study of stochastic differential equations, the expected signature is an alternative way to characterise a stochastic process. In this talk, we will discuss some motivations and applications of expected signature (Lyons-Victoir's cubature method and Papavasiliou-Ladroue's Expected signature matching method). We will also discuss a number of recent progress in calculating expected signature of various processes and trying to understand the required moment condition for the expected signature to fully characterise the process. Joint work with Hao Ni, Joscha Diehl and Marc Mezzarobba.

**17 February 2022**

Speaker: **Karen Habermann**

**Title: **TBA

**Abstract**

TBA

**24 February 2022**

Speaker: **Simon Gabriel**

**Title: **TBA

**Abstract**

TBA

**3 March 2022**

Speaker: **Félix Foutel-Rodier**

**Title: **TBA

**Abstract**

TBA

**10 March 2022**

Speaker:**Aleks Mijatovic**

**Title: **Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity

**Abstract**

For a multidimensional driftless diffusion in an unbounded, smooth, sub-linear generalized parabolic domain, with oblique reflection from the boundary, we give natural conditions under which either explosion occurs, if the domain narrows sufficiently fast at infinity, or else there is superdiffusive transience, which we quantify with a strong law of large numbers. In this talk, I will describe this result and discuss some ingredients of its proof, based on novel semimartingale criteria for studying explosions and establishing strong laws, which are of independent interest. This is joint work with M. Menshikov and A. Wade.

**17 March 2022**

Speaker: **Pierre Patie **

**Title: **TBA

**Abstract**

TBA

###### Academic Year 2022-23, Term 1

**21 October 2022**

Speaker: **Avery Ching**

**Title: **A simple construction of martingale polynomials

**Abstract**

The martingale polynomials for several common Lévy processes are well-known to be constructed by generating functions. In this talk a simpler construction of these martingale polynomials is given by the method of D-modules. In this context, the computation of expectations becomes a divisibility problem, and the martingale property becomes the elementary "difference of two n-th power" formula.

Teams link to follow

**28 October 2022**

Speaker: **Tommaso Rosati**

**Title: **Synchronisation for stochastic conservation laws with Dirichlet boundary

**Abstract**

We give a gentle introduction to some dynamical properties of stochastic partial differential equations (SPDEs) and discuss how they relate to open problems in the study of infinite-dimensional dynamical systems and singular SPDEs. We will show how (surprisingly) Dirichlet boundary conditions can help overcome certain issues in the setting of stochastic conservation laws. Joint works with Ana Djurdjevac and Martin Hairer.

**4 November 2022**

Speaker: **Sigurd Assing, Sam Olesker-Taylor, Tessy Papavasiliou, Suprise**

**Title: Short talks**

**Abstract **

**11 November 2022**

Speaker: **Sam Olesker-Taylor**

**Title: **Metastability for Loss Network

**Abstract**

We consider a fully-connected loss network with dynamic alternative routing, each link of capacity K. Calls arrive to each link {i, j} at rate λ. If the link is full upon arrival, a third node k is chosen uniform and the call is routed via k: it uses a unit of capacity on both {i, k} and {k, j} if both have spare capacity; otherwise, the call is lost.

We analyse the asymptotics of the mixing time of this process, depending on the traffic intensity α := λ/K. In particular, we determine a phase transition at an explicit threshold αc: there is fast mixing if α < αc or α > 1, but metastability if αc < α < 1.

**18 November 2022**

Speaker: **Simone Floreani**

**Title: **Hydrodynamics for the partial exclusion process in random environment via stochastic duality

**Abstract**

In this talk, I present a partial exclusion process in random environment, a system of random walks where the random environment is obtained by assigning a maximal occupancy to each site of the Euclidean lattice. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. After recalling some results on the Bouchaud’s trap model, I will show that, when assuming that the maximal occupancies are heavy tailed and i.i.d., the hydrodynamic limit of the particle system is the fractional-kinetics equation.

The strategy of the proof is based on stochastic duality, a useful tool in probability theory which allows to study a Markov process (the one that interests you) via another one, called dual process, which is hopefully simpler. In the final part of the talk, I will focus on how some duality relations can be obtained in a more general framework.

**25 November 2022**

Speaker: **Martina Favero (Moved to next term)**

**Title: **TBA

**Abstract**

**2 December 2022**

Speaker:** Yuxi Jiang**

**Title: **Bernoulli factory and the exponential coin

**Abstract**

The Bernoulli factory problem aims at simulating a coin that lands on head with probability f(p), using an independent sequence of Bernoulli(p) simulations. The function f is given with the value of p unknown, and we can simulate from the Bernoulli(p) distribution as many times as needed. We are interested in the function f(p)=exp(-p), and in this talk, we are going to look at different approaches to simulate the exp(-p)-coin, and compare the relative performance of the algorithms.

**9 December 2022**

Speaker: **No Talk**

**Title: **

**Abstract**