# Applied Probability Seminar (2022-23)

###### Past Seminars: 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, Surprise**

**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**

###### Past Seminars: 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 2023 **

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 2023 **

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 2023 (Moved -> 24th Feb)**

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 2023 **

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 2023 **

Speaker: **Simon Gabriel**

**Title: **Diffusion limits of the condensed Inclusion Process

**Abstract**

The Inclusion Process is a stochastic particle system in which attractive interactions can lead to particle clusters of diverging size, when taking a thermodynamic limit. This phenomenon is known as condensation. Our aim is to study the limiting dynamics of the process, under appropriate rescaling of space and time.

We will see that the limit depends on the relative strength of the attractive interactions between particles and their individual random walk dynamics. When these two quantities are balanced, we derive the Poisson-Dirichlet diffusion as scaling limit, a well studied process in population genetics. On the other hand, if the random walk dynamics are relatively stronger, we derive a process that can be interpreted as Poisson-Dirichlet diffusion with infinite mutation rate. The talk is based on joint work with Paul Chleboun and Stefan Grosskinsky.

**24 February 2023 **

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).

**3 March 2023 **

Speaker: **Félix Foutel-Rodier**

**Title: **Crump-Mode-Jagers processes with interaction as epidemic models

**Abstract**

Crump-Mode-Jagers processes are population models where the ages at which an individual produces its offspring is allowed to take a general form, creating a complex dependence between the birth times in the population. I will introduce an extension of this model motivated by epidemiology, where some interaction among individuals is added to account for the finite number of hosts susceptible to the disease. The main result which I will present is a law of large numbers for the age structure of the population, which can be seen as a non-linear version of a corresponding result without interaction. Our proof relies on studying the local graph structure of infections around a typical individual, and has potential interesting consequences for contact-tracing that I will discuss.

This is joint work with Jean-Jil Duchamps and Emmanuel Schertzer.

**10 March 2023 **

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 2023 **

Speaker: **Pierre Patie **

**Title: **The van Dantzig problem and the Riemann hypothesis

**Abstract**

In this talk, we start by introducing the intriguing van Dantzig problem which consists in characterizing the subset of Fourier transforms of probability measures on the real line that remain invariant under the composition of two involutions.

We first focus on the so-called Lukacs class of solutions that is the ones that belong to the set of Laguerre-Pόlya functions which are entire functions with only real zeros. In particular, we show that the Riemann hypothesis is equivalent to the membership to the Lukacs class of the Riemann ξ function. We state several closure properties of this class including adaptation of known results of Pόlya, de Bruijn and Newman but also some new ones.

We proceed by presenting a new class of entire functions, which is in bijection with a set of continuous negative definite functions, that are solutions to the van Dantzig problem and discuss the possibility of the Riemann ξ function to belong to this class.

###### Past Seminars: Academic Year 2022-23, Term 3

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.

**28 April 2023 **

Speaker: **Federico Sau**

**Title: **Spectral gap of the symmetric inclusion process.

**Abstract**

In this talk, we consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on the same graph. In the regime in which the gamma-like reversible measures of the particle system are log-concave, our bounds match, yielding a version for the symmetric inclusion process of the celebrated Aldous' spectral gap conjecture --- originally formulated for the interchange process and proved by Caputo, Liggett and Richthammer (JAMS 2010). Finally, by means of duality techniques, we draw analogous conclusions for an interacting diffusion-like unbounded conservative spin system known as Brownian energy process, which may be interpreted as a spatial version of the Wright-Fisher diffusion with mutation. Based on a joint work with Seonwoo Kim (SNU, South Korea).

**5 May 2023 **

Speaker: **No Talk**

**Title: **

**Abstract**

**12 May 2023 **

Speaker: **Karen Habermann**

**Title: **Long-time existence of Brownian motion on configurations of two landmarks

**Abstract**

In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by n ≥ 2 distinct landmark points in R^d. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimisation problem which minimises a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold Q of n distinct landmark points in R^d can be endowed with a Riemannian metric g such that the above optimisation problem is equivalent to the geodesic boundary value problem for g on Q. Despite its importance for modelling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold (Q,g) is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterisation of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel.

**19 May 2023 **

Speaker: **Larbi Alili**

**Title:** Scaled skew Bessel processes and the time inversion property for SSMP on the real line

**Abstract**

First, I will introduce the class of scaled-skew Bessel processes. Then I will construct their semigroup densities, give different ways to construct them and show that they satisfy the time-inversion property. Second, I will characterise, under the conditions of L. Gallardo, S. Lawi and M. Yor , all Markov processes on the real line that satisfy the time inversion property. This is a joint work with A. Aylwin.

**26 May 2023 (Re-scheduled)**

Speaker: **Stefan Adams**

**Title: **TBA

**Abstract**

TBA

**2 June 2023 (Cancelled) **

Speaker: **Dario Spano**

**Title:** Pólya urns, duality and eigenstructure for Wright-Fisher-type processes

**Abstract**

Much of the tractability of neutral, parent-independent Wright-Fisher diffusions of population genetics comes from the coexistence of a number of properties: they are polynomial diffusions; they have a moment dual (Kingman’s coalescent); they are time-reversible. I will describe the interplay of the above-mentioned properties under a perspective inspired by Bayesian statistics, and will show how this analysis can help in characterising new classes of reversible diffusions, whose eigenstructure is driven by a generalised moment dual process.

**9 June 2023 **

Speaker: **Anastasia Papavasileiou**

**Title: **Statistical Inference for Differential Equations driven by any Gaussian Process – a general framework

**Abstract**

I will present a general framework for constructing a likelihood function for a continuously observed differential equation driven by general Gaussian processes. Rather than trying to construct the likelihood directly , the main idea is to construct a pathwise approximation of the realisation of the Gaussian process that is consistent with the data, conditioned on the model – so, the problem becomes equivalent to solving an `inverse problem’. I will discuss how to appropriately define the inverse problem and then I will present an algorithm for its solution.

**16 June 2023 **

Speaker: **Paul Jenkins**

**Title: **The mutual arrangement of Wright--Fisher diffusion path measures

**Abstract**

The Wright--Fisher diffusion is a fundamental model of evolution in which genetic drift acts on the same timescale as other evolutionary processes such as mutation and natural selection. Suppose you want to infer the parameters associated with these processes from an observed sample path. Then to write down the likelihood one first needs to know the mutual arrangement of two path measures under different parametrizations; that is, whether they are absolutely continuous, equivalent, singular, and so on. In this talk I will give a characterization of this mutual arrangement by using recent advances in the theory of separating times for diffusions. Along the way we will find some new zero-one type laws for the diffusion on its approach to, and emergence from, the boundary.

**23 June 2023 **

Speaker: **Harry Giles**

**Title: **Self-repelling Brownian polymer in the critical dimension

**Abstract**

The Self-Repelling Brownian Polymer is a type of weakly self-avoiding motion in R^d that was introduced separately by physicists (Amit, Parisi, Peliti, '83) and probabilists (Norris, Rogers, Williams, '87). In dimension d = 3 and higher, the process converges to Brownian motion under a diffusive rescaling, whereas, in the critical dimension, d = 2, the process is logarithmically super-diffusive. Nonetheless, we can show that in the limit we still recover Brownian motion, provided that the rescaling is done in the "weak coupling" sense, whereby as we zoom out, we also logarithmically tune down the strength of self-repulsion. In this talk, I will define the model and discuss the recent techniques that give this result. This is a work in progress with Giuseppe Cannizzaro.

**30 June 2023 **

Speaker:

**Title: **TBA

**Abstract**