Stochastic processes at Melbourne

Michael McAuley (TU Dublin): Geometric functionals of smooth Gaussian fields

31 July

Smooth Gaussian fields are widely used for modelling phenomena across scientific disciplines (e.g. in cosmology, medical imaging, quantum chaos and machine learning). In many of these areas, statistical analysis can be naturally related to the geometric properties of the field. Understanding these geometric properties has, in turn, motivated deep theoretical questions which require techniques from many different areas (e.g. differential/integral geometry, analysis, probability theory and mathematical physics) to answer.

In this talk I will give an overview of this research topic and touch on some recent progress in proving limit theorems (i.e. laws of large numbers and central limit theorems) for geometric functionals associated with Gaussian fields.

Based on joint work with Dmitry Beliaev (University of Oxford) and Stephen Muirhead (University of Melbourne)

Roxanne He (Melbourne): Cutoff for the SIS model with self-infection and mixing time for the Curie-Weiss-Potts model

7 August

In this talk, I present joint work with Malwina Luczak and Nathan Ross, where we study a variant of the classical Markovian logistic SIS epidemic model on a complete graph, which has the additional feature that healthy individuals can become infected without contacting an infected member of the population. This additional “self-infection” is used to model situations where there is an unknown source of infection or an external disease reservoir, such as an animal carrier population. In contrast to the classical logistic SIS epidemic model, the version with self-infection has a non-degenerate stationary distribution, and we show that it exhibits the cutoff phenomenon, which is a sharp transition in time from one to zero of the total variation distance to stationarity. At the end, I also briefly discuss joint work with Jackie Lok on the mixing time for the restricted Curie-Weiss-Potts model in the subcritical temperature regime, along with an application.

Maximilian Nitzschner (Hong Kong UST): Bulk deviation lower bounds for the simple random walk

14 August

In this talk we present large deviation lower bounds for the probability of certain bulk-deviation events depending on the occupation-time field of a simple random walk on the Euclidean lattice in dimensions larger or equal to three. As a particular application, these bounds imply an exact leading order decay rate for the probability of the event that a simple random walk covers a substantial fraction of a macroscopic body, when combined with a corresponding upper bound previously obtained by Sznitman. As a pivotal tool for deriving such optimal lower bounds, we recall the model of tilted walks which was first introduced by Li in order to develop similar large deviation lower bounds for the probability of disconnecting a macroscopic body from an enclosing box by the trace of a simple random walk. We then discuss a refined local coupling with the model of random interlacements which is used to locally approximate the occupation times of the tilted walk. Based on joint work with A. Chiarini (University of Padova).

Nathan Ross (Melbourne): Gaussian random field approximation via Stein’s method, with applications to wide random neural networks

21 August

We describe a general technique to derive bounds for Gaussian random field approximation with respect to a Wasserstein transport distance in function space, equipped with the supremum metric. The technique combines Stein’s method and infinite dimensional Gaussian smoothing, and we apply it to derive bounds on Gaussian approximations of wide random neural networks of any depth. The bounds are explicit in the widths and natural parameters of the neural network. The talk covers joint works with Krishnakumar Balasubramanian, Larry Goldstein, and Adil Salim; and A.D. Barbour and Guangqu Zheng.

Adrian Röllin (NU Singapore): When Markov processes collapse, and what to do about it

28 August

We consider sequences of Markov processes that exhibit increasingly strong drifts towards a subspace of the state space and therefore “collapse” in the limit. The mathematical challenge is to prove process-level convergence, since the generators of the processes “blow up” outside the subspace. We show that Lyapunov functions for Markov processes and the Meyer-Zheng topology are convenient tools in such situations. We provide two examples — the first is a Moran model in a homogeneously mixing population with random resampling rates and the second is a voter model on a dynamically evolving random graph. This is joint work with Siva Athreya and Frank den Hollander.

Jesse Goodman (U Auckland): Saddlepoint approximations for likelihoods

4 September

The saddlepoint approximation is a systematic method for converting a known generating function into an approximation for an unknown density function. Interpreted instead as an approximation to the unknown likelihood function, the saddlepoint approximation can be maximized to compute the saddlepoint MLE for a given observed value. This talk will explain how the saddlepoint approximation can be interpreted with a statistical lens, and describe a class of models with theoretical guarantees for the effect of using the saddlepoint MLE as a substitute for the unknown true MLE. The talk will also demonstrate new tools to simplify and automate the computation of saddlepoint MLEs and to quantitatively assess the amount of approximation error. Based on joint work with Godrick Oketch and Rachel Fewster.

Nick Beaton (Melbourne): Chemical distance for the half-orthant model

18 September

The half-orthant model is a partially oriented model of a random medium involving a parameter $p\in [0,1]$, for which there is a critical value $p_c(d)$ (depending on the dimension $d$) below which every point is reachable from the origin. We prove a limit theorem for the graph-distance (or "chemical distance") for this model when $p<p_c(2)$, and also when $1-p$ is larger than the critical parameter for site percolation in $\mathbb{Z}^d$. The proof involves an application of the subadditive ergodic theorem. Novel arguments herein include the method of proving that the expected number of steps to reach any given point is finite, as well as an argument that is used to show that the shape is "non-trivial" in certain directions. This is joint work with Mark Holmes and Xin Huang. (2)$,>

Rongfeng Sun (NU Singapore): The Critical 2D Stochastic Heat Flow: disordered system meets singular SPDE

25 September

We discuss recent progress in the study of the 2-dimensional stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation, which are critical singular stochastic partial differential equations (SPDEs) that lie beyond existing solution theories. Both the 2D SHE and KPZ undergo a phase transition, and the solution of the 2D SHE at the critical point leads to the so-called critical 2D stochastic heat flow. This provides a rare example of a model in the critical dimension and at the critical point with a non-Gaussian limit. Our approach is motivated by the scaling limits of disordered systems, in particular, the directed polymer model in random environment for which disorder is marginally relevant in 2D. Based on joint work with F. Caravenna and N. Zygouras.

Weijun Xu (Peking U) Belz Short Course: Ergodicity of diffusion processes

26 September, 2 October, 9 October

We discuss ergodicity problems for continuous time Markov processes, with a focus on diffusions. Along the way, we will naturally encounter some landmarks in the development of stochastic analysis, such as Ito's SDEs, Malliavin calculus, etc. We will also discuss what happens in infinite dimensional situations (SPDEs).

Tejas Iyer (Weierstrass Institute): Leadership in growth processes

2 October

Consider a model where N 'agents' possess 'values' subject to increase over time. More precisely, these values are represented by N_0 valued increasing processes, with random, independent waiting times between jumps. We show that the event that a single agent possesses the maximum value for all sufficiently large values of time (called 'leadership') occurs with probability zero or one, and provide necessary and sufficient conditions for this to occur. In the particular case when waiting times are mixtures of exponential distributions, we improve a well-established result on the 'balls in bins' model with feedback (a model also known as non-linear P\'olya urns), removing the requirement that the feedback function be bounded from below and also allowing random feedback functions. The result is closely related to results regarding series of independent random variables. During the talk we will elucidate this connection, and then, time permitted, outline the technicalities associated with the proofs.

arXiv ref: https://arxiv.org/abs/2408.11516

Mihai Gradinaru (Rennes 1): Lévy driven non-linear Langevin type equations

16 October

We will consider a one-dimensional kinetic stochastic model driven by a stable Lévy process, with a non-linear time-inhomogeneous drift. More precisely, a process $(v,x)$ is considered, where $x$ is the position of the particle and its velocity $v$ is the solution of a stochastic differential equation with a drift of the form $t^{-\gamma}F(v)$. The behaviour of the process $(v,x)$ will be described when the noise is small or with a fixed noise in large time.

Xi Geng (Melbourne): Long time asymptotics for the parabolic Anderson model in the hyperbolic space

23 October

In this talk, we establish the exact second-order moment asymptotics for the parabolic Anderson model in the hyperbolic space with a time-independent, regular, isometry-invariant Gaussian potential. Although the solution is defined under hyperbolic geometry, surprisingly it turns out that the fluctuation exponent is determined by an Euclidean variational problem which is insensitive to the underlying geometry. Heuristically, this is due to a curvature dilation effect: the geometry becomes asymptotically flat after suitable renormalisation in the derivation of the second-order asymptotics. On the other hand, the almost-sure asymptotics becomes drastically different from the Euclidean case due to the exponential volume growth in negative curvature.

This is based on a recent joint work with Weijun Xu (Peking University) as well as an ongoing project with Weijun and my PhD student Sheng Wang.

Illia Donhauzer (La Trobe): Superpositions of continuous autoregressive random fields (supCAR fields).

30 October

The talk will introduce the supCAR random fields that form a rich class of homogeneous and isotropic random fields constructed as superpositions of CAR (continuous autoregressive) random fields. The supCAR random fields possess infinitely divisible marginal distributions and flexible dependence structures (including long-range dependence). The talk will discuss the existence of the supCAR fields and their properties. Special emphasis will be given to new functional limit theorems for the supCAR fields and properties of the limit processes.

Joe Yukich (Lehigh) Belz Short Course: Asymptotic analysis of statistics of random geometric structures

15 January

Recent years have seen the development of tools leading to the limit theory of statistics of geometric structures built on possibly correlated spatial data, as the number of data points tends to infinity.  Malliavin-Stein methods, Poincare inequalities, and stabilization are the tools lying at the heart of these methods.

We introduce and survey these methods and show that they lead to laws of large numbers, variance asymptotics and central limit theorems for functionals of random graphs on point processes, random convex hulls, and random deposition models.

Yuzuru Inahama (Kyushu): Wong-Zakai approximation of density functions

6 March

In this talk we prove the Wong-Zakai approximation of probability density functions of solutions at a fixed time of rough differential equations driven by fractional Brownian rough path with Hurst parameter H (1/4<H≤1/2). Besides rough path theory, we use Hu-Watanabe’s approximation theorem in the framework of Watanabe’s distributional Malliavin calculus. When H=1/2, the random rough differential equations coincide with the corresponding Stratonovich-type stochastic differential equations. Even for that case, our main result seems new.

Renjie Feng (Sydney): Determinantal point processes on spheres: Multivariate linear statistics

13 March

I will talk about the multivariate linear statistics (also known as U-statistics) of determinantal point processes on unit spheres. I will first present a graphical representation for the cumulants of the multivariate linear statistics, extending the famous Soshnikov’s formula for the univariate case. Then I will explain how we derive the 1st and 2nd Wiener chaos using this graphical representation. We computed sphere cases as introductory examples, but the method can be applied to any other determinantal point processes. This is based on the joint work with F. Goetze and D. Yao.

Emma Horton (Warwick): Genealogies of branching Markov processes

20 March

Branching processes are pertinent to understanding many different real world processes such as cell division, population growth and neutron transport. In particular, understanding their genealogical structures can prove useful for parameter estimation, Monte Carlo simulations and scaling limits. In this talk we discuss a decomposition of the branching process known as the many-to-few formula, which allows one to understand the behaviour of a branching process in terms of a weighted subtree. I will then give two applications of this decomposition to demonstrate its use in understanding the genealogical structure of the branching process.

Vincent Liang (Melbourne): On boundary crossing probabilities of diffusion processes

27 March

We discuss two results related to the probability $F(g_-,g_+)$ that a general time-inhomogeneous diffusion process $X$ stays between two curvilinear boundaries $g_-$ and $g_+$ (possibly with $g_{\pm} = \pm \infty$) during a finite time interval. Joint work with K. Borovkov.

Nadia Sidorova (UCL): Edge-reinforced branching random walk on the triangle

10 April

Edge-reinforced random walk (ERRW) is a random process on the vertices of a graph that is more likely to cross the edges it has visited in the past. Depending on the strength of the reinforcement, one-dimensional ERRW can either exhibit localisation (eventually moving back and forth across a single edge) or remain transient. We consider a model where a single ERRW is replaced by an exponentially growing number of random particles, and we study its localisation properties on the simple triangle graph. Using the dynamical systems approach we analyse the frequencies with which the edges are traversed and prove their almost sure convergence. We discuss the scenarios when those frequencies become negligible for one or two edges (dominance). We also discuss the situation when an edge stops being traversed entirely (monopoly). This is a joint work with Giordano Giambartolomei.

Konstantin Borovkov (Melbourne): Large deviation probabilities for random walks: Light vs heavy tails

17 April

Random walks are important models for real-life processes, including claim surplus processes for insurance companies, where light-tailed jump distributions are usually assumed for life insurance, and heavy-tailed – for non-life one. We will present the fundamentals of the large deviations theory for random walks, touching on both light- and heavy-tailed cases, and outline results on the asymptotics of the probabilities of remote curvilinear boundary crossing by the walks. We will also discuss large deviation results in the case of right-censored jumps in the random walk. Part of the work was joint with A. Chong.

Greg Markowsky (Monash): Ways in which the geometry of plane domains is reflected in the distribution of Brownian motion exit times

24 April

The distribution of the exit time of Brownian motion from a plane domain carries a great deal of information about the shape and size of the domain. There are beautiful connections between this fact and classical complex analysis. I will present some recent results on this, and discuss some open problems and conjectures.

Phillip Yam (City U Hong Kong): Mean Field Games, their FBSDEs and Master Equations

1 May

Modeling collective behaviors of individuals in account of their mutual interactions arisen in various physical or sociological dynamical systems have been one of the major problems in the history of mankind. To resolve this matter, a completely different macroscopic approach inspired from statistical physics had been gradually developed in the last decade, which eventually leads to the primitive notion of mean field game theory. In this talk, we shall introduce a theory of global-in-time well-posedness fora general class of mean field game problems, which include as an example setting with quasi-convex payoff functions as long as the mean field sensitivity is not too large.

Aram Perez (Monash): Multivariate Non-Normal Stein’s Method with Application to the Critical, Mean-Field O(N) Model

8 May

Stein’s Method is a powerful tool for studying distances between distributions of random variables. It has been used widely in the context of statistical mechanics to obtain approximations for various thermodynamic quantities. We will discuss an application of Stein’s Method to the O(N) model, a model of magnetism, and show how recent advances in Stein’s theory allow us to study the limiting behaviour of the magnetisation at the phase transition.

Kazutoshi Yamakazi (Queensland): A series expansion formula of the scale matrix

14 May

We present a new series expansion formula for the scale matrix of Markov additive processes with a constant drift and general finite-activity one-sided jumps. This formula generalizes the series expansion formula for the scale function derived in Landriault & Willmot (Scand. Actuar. J., 2020) for the Cramer-Lundberg process. We discuss its applications in ruin theory and sequential testing. This work is a collaboration with J. Ivanovs (Aarhus University).

Mario Kieburg (Melbourne): Stable and Group Invariant Hermitian Random Matrices

22 May

The classification and characterisation of stable random variables is one of the central questions in classical probability theory. Starting with the classical central limit theorem for Gaussian random variables, it was soon extended by Lévy to α-stable distributions for univariate statistics. Those results were generalised to vectors by Rvačeva which completed the multi-variate case. As flat symmetric matrix spaces are vector spaces, one could think that the problem is solved for those, as well. However, matrices give rise to other intriguing quantities such eigenvalues, singular values and eigenvectors whose statistics have a non-trivial relation to the matrix entries. In a series of works, Jiyuan Zhang and I have studied the classification of joint probability densities of the eigenvalues for unitarily invariant Hermitian random that are stable with respect to matrix addition. We have analysed their domain of attraction and the optimal rate of convergence when generating those from a stable random vector. In the talk, I will report about these results and what the future directions of research will be.

Simon Harris (Auckland): Universal genealogies of samples from Galton-Watson trees with heavy-tailed offspring

29 May

Consider some population evolving stochastically in time. Conditional on the population surviving until some large time T, take a sample of particles from those alive. What does the ancestral tree drawn out by this sample look like? Some special cases were known, e.g. Durrett (1978), O’Connell (1991), but we will discuss an approach behind some more recent advances for Bienyame-Galton-Watson (BGW) processes conditioned to survive. In near-critical settings with finite offspring variance, the same universal limiting sample genealogy always appears up to some deterministic time change. This genealogical tree has the same binary tree topology as a Kingman coalescent, but where the coalescent (or split) times can be represented as a mixture of IID times - this very roughly interpreted as a mixture of time changed ‘slowed down’ Kingman coalescents. In contrast, in critical infinite variance offspring settings, we find that more complex universal limiting sample genealogies emerge that exhibit multiple-mergers, these being driven by massive birth events within the underlying population. The key tool in our proofs is a change of measure involving k distinguished particles, also known as spines, which corresponds to k-size biasing and discounting by the population size. Some ongoing work and open problems will also be mentioned. This talk is based on work in collaboration with M.Roberts (Bath), S.Johnston (KCL) in AAP (2020), with J.C.Pardo (CIMAT), S.Johnston in AOP (2024), and with S.Palau (UNAM), J.C.Pardo (2022+).

Allan Sly (Princeton): Transience for the interchange process in dimension 5

12 June

The interchange process $\sigma_T$ is a random permutation valued stochastic process on a graph evolving in time by transpositions on its edges at rate 1. On $Z^d$, when $T$ is small all the cycles of the permutation $\sigma_T$ are finite almost surely. In dimension $d \geq 3$ Toth conjectured that infinite cycles appear when $T$ is large. The cycles can be interpreted as a random walk which interacts with its past and we give a multi-scale proof establishing transience of the walk (and hence infinite cycles) when $d\geq 5$. In a finite volume we establish Poisson-Dirichlet statistics for the largest cycles.