The 10th International Conference on
Stochastic Analysis and its Applications

Program (updated on Sept. 6)(PDF)

Slots of talks by Ni and Gubinelli are interchanged by the update on September 6.
The following table (written by Japan Standard Time (JST=UTC+9)) gives the schedule for live talks during the conference. See below for a list of the pre-recorded talks.
(time differences: BST=JST-8, EDT=JST-13, PDT=JST-16, CST=JST-1)

(Time   in JST)	Sept. 6 (Mon.)	Sept. 7 (Tue.)	Sept. 8 (Wed.)	Sept. 9 (Thu.)	Sept. 10 (Fri.)
8:50   -9:00	Opening Remarks
9:00   -9:40	Khoshnevisan	Fribergh	Gordina	Jabin	Takeda
10:00   -10:40	Shen	Sethuraman	Murugan (10:00-10:25)	Fukushima	Uemura (10:00-10:25)
11:00   -11:25	Zhu	Okamura	Inahama (10:40-11:05)	Ganguly	Jie-Ming Wang   (10:40-11:05)
(11:15   -11:40)			Shiraishi (11:15-11:40)		Kim (11:15-11:40)
	Lunch Break
15:00   -15:25	Jian Wang (14:40-15:20)	Hoshino	Free Discussion	Zhang	Feng-Yu Wang
15:40   -16:20	Barbu	Zambotti		Stannat	Grigor'yan
16:40   -17:05	Ottobre	Penington		Grothaus	Hartung
17:20   -18:00	Chevyrev (17:20-17:45)	Gubinelli		Ni	Bovier
18:00   -18:10					Closing

Pre-recorded talks by
  Xin Chen
  Sunil Chhita
  Shen Lin
  Leonid Mytnik
  Zhongmin Qian

Abstracts

Davar Khoshnevisan (University of Utah)
Phase analysis of a family of stochastic reaction-diffusion equations

We consider a reaction-diffusion equation driven by a multiplicative space-time white noise, for a large class of reaction terms that include well-studied examples, such as the Fisher-KPP and the Allen-Cahn equations. We prove that, in the “intermittent regime”: (1) If the equation is “sufficiently noisy”, then the resulting stochastic PDE has a unique invariant measure; and (2) If the equation is in a “low-noise regime”, then there are infinitely many invariant measures, and the collection of all invariant measures is a certain line segment in path space. This endeavor gives a proof of earlier predictions due to Zimmerman et al (2000), discovered first through experiments and computer simulations. The quoted terms will be defined carefully in the talk.
This is based on ongoing joint work with Carl Mueller (University of Rochester), Kunwoo Kim (Pohang University of Science and Technology), and Shang-Yuan Shiu (National Central University).

Hao Shen (University of Wisconsin-Madison)
Stochastic Yang-Mills process in 3D

The Yang-Mills model in 3D is much more singular than 2D. This poses several new challenges in constructing the stochastic quantization of the model. In this talk we will describe a few of these new challenges, and our strategies to solve the problems. Depending on the time, we will discuss our construction of gauge invariant observables via deterministic Yang-Mills flow, the interplay between renormalisation and symmetries, and the state space for the gauge equivariant Yang-Mills process.

Xiangchan Zhu (Chinese Academy of Science)
Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier-Stokes equations: existence and non-uniqueness

We are concerned with the three dimensional incompressible Navier-Stokes equations driven by an additive stochastic forcing. First, for every divergence free initial condition in L2 we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result in particular implies non-uniqueness in law. Second, we prove non-uniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality.

Jian Wang (Fujian Normal University)
Convergence rates for inhomogeneous Markov chains from stochastic algorithms

We consider limit behaviors for the following inhomogeneous Markov chains originated from stochastic algorithms \[ X_{n+1} = X_{n} + r_{n} b(X_{n}) + h(r_{n+1}) \xi_{n}, \] where $b:\mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ is a vector-valued function, $(r_{n})_{n \geq 1}$ is a strictly positive and decreasing sequence such that $\lim_{n \to \infty} r_{n} = 0$ and $\sum_{n=1}^{\infty} r_{n} = \infty $, and $(\xi_{n})_{n \geq 1}$ is a sequence of i.i.d. $d$-dimensional random variables. Under a proper choice of the scaling function $h(r)$, we show the processes $(X_{n})_{n \geq 1}$ converge weakly to SDEs driven by Lévy noises, and the associated quantitative bounds on convergence rates are also established. This is based on a joint work with Dr. Lujing Huang (Fujian Normal University).

Viorel Barbu (Romanian Academy)
Exact and asymptotic controllability of nonlinear Fokker-Planck equations

Nonlinear Fokker-Planck equations with controller in drift term are asymptotically controllable for a certain class of target probability densities. In the case of controller in diffusion term one obtains the exact controllability in finite time with suitable feedback controllers.

Michela Ottobre (Heriot Watt University)
Uniform in time approximations of stochastic dynamics

Complicated models, for which a detailed analysis is far out of reach, are routinely approximated via a variety of procedures; this is the case when we use multiscale methods, when we take many particle limits and obtained a simplified, coarse-grained dynamics, or, simply, when we use numerical methods. While approximating, we make an error which is small over small time-intervals but it typically compounds over longer time-horizons. Hence, in general, the approximation error grows in time so that the results of our ``predictions" are less reliable when we look at longer time-hormizons. However this is not necessarily the case and one may be able to find dynamics and corresponding approximation procedures for which the error remains bounded, uniformly in time. Whether this is possible or not will depend on both the dynamics to approximate and on the chosen approximation procedure. I will show how the approach we take to produce uniform in time results is very broad and, while this talk will focus on approximations produced via averaging methods, I will demonstrate how such an approach can be used for all of the approximation procedures mentioned above. Furthermore I will emphasize how this approach relies on producing appropriate derivative estimates for Markov semigroups. This is based on a series of joint works with a number of people: L. Angeli, J. Barre', D. Crisan, P. Dobson, I. Souttar and E. Zatorska.

Ilya Chevyrev (University of Edinburgh)
Path functions and homogenisation

In this talk, I will introduce a geometric way to solve differential equations with jumps based on the notion of path functions. This method generalises Marcus solutions to stochastic differential equations and has a natural extension to rough path space. As an application, I will show how homogenisation theorems for fast-slow dynamical systems can be stated and proven using path functions.

Alexander Fribergh (Université de Montréal)
The ant in high-dimensional labyrinths

One of the most famous open problem in random walks in random environments is to understand the behaviour of a simple random walk on a critical percolation cluster, a model known as the ant in the labyrinth. We will present new results on the scaling limit for the simple random walk on the critical graphs in high dimensions. This result can be applied from critical branching random walks and critical lattice trees. We believe these results will extend to critical percolation as well. This is a joint work with G. Ben Arous and M. Cabezas.

Sunder Sethuraman (University of Arizona)
On microscopic derivation of a continuum mean-curvature flow

We derive a continuum mean-curvature flow as a scaling limit of a class of zero-range + Glauber interacting particle systems. The zero-range part moves particles while preserving particle numbers, and the Glauber part allows birth and death of particles, while favoring two levels of particle density. When the two parts are seen together in certain (different) time-scales, and the Glauber part is ‘bi-stable’, a mean-curvature interface flow, incorporating a homogenized ‘surface tension’ reflecting microscopic rates, between the two levels of particle density, can be captured as a limit of the mass empirical density. This is work with Perla El Kettani, Tadahisa Funaki, Danielle Hilhorst, and Hyunjoon Park.

Kazuki Okamura (Shizuoka University)
Some results for range of random walk on graph with spectral dimension two

We consider the range of the simple random walk on graphs with spectral dimension two. We give a form of strong law of large numbers under a certain uniform condition, which is satisfied by not only the square integer lattice but also a class of fractal graphs. Our results imply the strong law of large numbers on the square integer lattice established by Dvoretzky and Erdös. As an application, we establish a form of law of the iterated logarithms for lamplighter random walks in the case that the spectral dimension of the underlying graph is two.

Masato Hoshino (Osaka University)
Paracontrolled calculus and regularity structures

We prove a general equivalence statement between the notions of models and modelled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. The construction of a modelled distribution from a paracontrolled system is explicit, and takes a particularly simple form in the case of the regularity structures introduced by Bruned, Hairer and Zambotti for the study of singular stochastic partial differential equations. This talk is based on a joint work with Ismaël Bailleul (Université Rennes 1).

Lorenzo Zambotti (Sorbonne Université)
Hairer's Reconstruction Theorem without Regularity Structures

This talk, based on joint work with Francesco Caravenna, is devoted to Martin Hairer's Reconstruction Theorem, which is one of the cornerstones of his theory of Regularity Structures. Our aim is to give a new self-contained and elementary proof of this Theorem, together with some applications, including a characterization, based on a single arbitrary test function, of negative Hölder spaces. We present the Reconstruction Theorem as a general result in the theory of distributions that can be understood without any knowledge of Regularity Structures themselves, which we do not even need to define.

Sarah Penington (University of Bath)
Brownian bees in the infinite swarm limit

Consider a system of $N$ particles moving according to Brownian motions and branching at rate one. Each time a particle branches, the particle in the system furthest from the origin is killed. I will discuss the large $N$ and large time behaviour of the system, which is related to solutions of a novel non-linear free boundary partial differential equation. Based on joint work with Julien Berestycki, Éric Brunet and Jim Nolen.

Massimiliano Gubinelli (Universität Bonn)
A variational method for Euclidean quantum fields

I will explain how Euclidean quantum fields like the $\Phi^{4}_{3}$ field can be studied via a variational formulation involving a scale-by-scale description of the fluctuations. This formulation is connected with ideas from renormalization group and also to stochastic analysis, bridging two important areas of probability theory.

Masha Gordina (University of Connecticut)
Ergodicity for Langevin dynamics with singular potentials

We will discuss Langevin dynamics of $N$ particles on $\mathbb{R}^{𝑑}$ interacting via a singular repulsive potential, such as the Lennard-Jones potential, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted Sobolev norm. The proof relies on an explicit construction of a Lyapunov function using a modified Gamma calculus. In contrast to previous results for such systems, our results imply geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. This is based on the joint work with F. Baudoin and D. Herzog, as well a recent preprint with E. Camrud, D. Herzog and G. Stoltz.

Mathav Murugan (University of British Columbia)
On the comparision between jump processes and subordinated diffusions

A well known method to obtain heat kernel estimates and Harnack inequalities for jump processes is to compare the given jump process with a subordinated diffusion process. On any space that admits a ‘nice diffusion’ that satifies sub-Gaussian heat kernel bounds, we show that a large family of jump processes have jump kernel comparable to that of a subordinated diffusion process. This is joint work with Guanhua Liu (Tsinghua University).

Yuzuru Inahama (Kyushu University)
Stochastic Flows and Rough Differential Equations on Foliated Spaces

Stochastic differential equations (SDEs) on compact foliated spaces were introduced a few years ago. As a corollary, a leafwise Brownian motion on a compact foliated space was obtained as a solution to an SDE. In this work we construct stochastic flows associated with the SDEs by using rough path theory, which is something like a “deterministic version” of Ito's SDE theory. (This is a joint work with Kiyotaka SUZAKI.)

Daisuke Shiraishi (Kyoto University)
Convergence of three-dimensional loop-erased random walk in the natural parameterization

In 2005, Gady Kozma proved that three-dimensional loop-erased random walk (LERW) has a well-defined scaling limit as a set. In stark contrast to two-dimensional SLE theory, Kozma's proof does not identify what this scaling limit is, and many of the fine properties of 3d LERW remain poorly understood. In this talk, I will explain a new result which strengthens Kozma's theorem, proving that the scaling limit of 3d LERW is well-defined as a curve, i.e., that the length-parameterized discrete LERW converges under rescaling to some time-parameterized version of Kozma's scaling limit. Furthermore, I will show that the limiting parameterization can be computed as a function of the set. This plays a key role in a joint work with Angel, Croydon and Hernandez-Torres where it is proved that the scaling limit of the three-dimensional uniform spanning tree is well-defined as a metric space. This talk is based on a joint work with Xinyi Li (Peking University).

Pierre-Emmanuel Jabin (Pennsylvania State University)
Mean-field limit for non-exchangeable multi-agent systems

We are studying the asymptotic behavior of multi-agent or many-particle systems as the number of agents increases. In a major difference with classical mean-field limits however, agents are not identical and in particular, which pair of agents interact and how much is prescribed by a possibly time-dependent graph of interactions. This type of models are used in a wide range of applications from coupled oscillators to networks of biological neurons. Those applications require a large number of agents or particles, making the mean-field limit attractive. But the usual notion of propagation of chaos cannot hold anymore which forces major changes in the classical theory. This is a joint work with D. Poyato and J. Soler.

Ryoki Fukushima (University of Tsukuba)
Number of paths in oriented percolation as zero temperature limit of directed polymer

We prove that the free energy of directed polymer in Bernoulli environment converges to the growth rate for the number of open paths in super-critical oriented percolation as the temperature tends to zero. Our proof is based on rate of convergence results which hold uniformly in the temperature. We also prove that the convergence rate is locally uniform in the percolation parameter inside the super-critical phase, which implies that the growth rate depends continuously on the percolation parameter. Based on a joint work with Stefan Junk (AIMR Tohoku University).

Shirshendu Ganguly (UC Berkeley)
Fractal geometry, noise-sensitivity and chaos in last passage percolation

Planar last passage percolation models [LPP] are canonical examples of stochastic growth and random geometry in the Kardar-Parisi-Zhang universality class, where one considers oriented paths between points in a random environment accruing the integral of the noise along itself as its weight. Given the endpoints, the extremal path with the maximum weight is termed as the geodesic. These models can also be viewed as examples of disordered systems admitting complex energy landscapes. In this formulation, the geodesic is the ground state, the configuration with lowest energy, lying at the base of the deepest valley. In this talk we will review various recent results about such models related to fractal geometry, noise-sensitivity and chaos. The results on fractal geometry focus on the coupling structure of the geodesics as the endpoints are varied and consider the exceptional set of points admitting multiple geodesics, by relating them to certain natural random measures with fractal supports. The notion of noise-sensitivity and chaos is closely related to the phenomenon of multiple valleys which these LPP models and many other important examples in statistical mechanics such as spin glasses exhibit. More precisely, the abundance of near-ground states (at the base of valleys) suggests that the ground state alters significantly when the disorder of the model is slightly perturbed. Obtained via a refined understanding of such near ground states, we will report a recent result computing the critical exponent, which we expect to be universal, that governs the onset of chaos in a dynamical version of a canonical LPP model. The talk is based on a number of works jointly with subsets of Riddhi Basu, Erik Bates, Alan Hammond and Milind Hegde.

Xicheng Zhang (Wuhan University)
Singular kinetic equations and applications

In this paper we study singular kinetic equations by paracontrolled distribution method introduced in [GIP15]. We first develop paracontrolled calculus in the kinetic setting, and then use it to establish the global well-posedness for linear singular kinetic equations under the assumptions that the products of singular terms are well-defined. We also demonstrate how the required products can be defined in the case that singular term is a Gaussian random field by probabilistic calculations. Interestingly, although the terms in the zeroth order Wiener chaos of regularization approximation are not zero, they converge in suitable weighted Besov spaces and thus no renormalization is required. As applications the global well-posedness for a nonlinear kinetic equation with singular kernels is obtained by the entropy method. Moreover, we also solve the martingale problem for nonlinear kinetic distribution dependent stochastic differential equations with singular drifts. (This is a joint work with Zimo Hao, Rongchan Zhu and Xiangchan Zhu).

Wilhelm Stannat (TU Berlin)
McKean-Vlasov SDEs in Nonlinear Filtering

Consider the problem of estimating a Markovian signal observed with independent noise in continuous time. I will discuss two classes of ensemble-based filtering algorithms that have been proposed in the literature for the numerical approximation of the associated conditional distribution and discuss their asymptotic behaviour in the infinite ensemble limit in terms of limiting mean-field processes on the level of the ensemble members and corresponding propagation of chaos results. I will also identify stochastic partial differential equations driving the distribution of the mean-field processes and perform a comparison with the Kushner-Stratonovich equation. The talk is based on joint work with T. Lange, S. Pathiraja and S. Reich.

References:
[1] S. Pathiraja, S. Reich, W. Stannat: McKean-Vlasov SDEs in nonlinear filtering, arXiv:2007:12658, to appear in SIAM J. Control Optim.
[2] T. Lange, W. Stannat: Mean field limit of Ensemble Square Root Filters - discrete and continuous time, arXiv:2011.10516, published online in Foundations of Data Science.
[3] S. Pathiraja, W. Stannat: Analysis of the feedback particle filter with diffusion map based approximation of the gain, manuscript TU Berlin 2021, submitted.

Martin Grothaus (TU Kaiserslautern)
Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

Motivated by problems from Industrial Mathematics we further developed the concepts of hypocoercivity. The original concepts needed Poincaré inequalities and were applied to equations in linear finite-dimensional spaces. Meanwhile we can treat equations in manifolds or even infinite dimensional spaces. The condition giving micro- and macroscopic coercivity we could relax from Poincaré to weak Poincaré inequalities. In this talk an overview and many examples are given.

Hao Ni (University College London)
Conditional Sig-Wasserstein Generative models to generate realistic synthetic time series

Wasserstein Generative adversarial networks (WGANs) have been very successful in generating samples, from seemingly high dimensional probability measures. However, these methods struggle to capture the temporal dependence of joint probability distributions induced by time-series data. Furthermore, training WGANs is computational expensive due to the min-max formulation of the loss function. To overcome these challenges, we integrate Wasserstein GANs with mathematically principled and efficient path feature extraction called the signature of a path. The signature of a path is a graded sequence of statistics that provides a universal description for a stream of data, and its expected value characterises the law of the time-series model. In particular, we a develop new metric, (conditional) Sig-W1, that captures the (conditional) joint law of time series models, and use it as a discriminator. The signature feature space enables the explicit representation of the proposed discriminators which alleviates the need for expensive training. We validate our method on both synthetic and empirical dataset and observe that our method consistently and significantly outperforms state-of-the-art benchmarks with respect to measures of similarity and predictive ability.

Masayoshi Takeda (Kansai University)
Recurrent Dirichlet Forms, Critical Schrödinger Forms and Optimal Hardy Type Inequalities

The objectives of my talk are the following two:
1) We characterize the criticality of Schrödinger forms using the notion of extended Schrödinger spaces, which is a generalized notion of extended Dirichlet spaces. By applying this characterization, we give a method to construct critical Schrödinger forms by subtracting a certain positive potential from subcritical ones.
2) The criticality of Schrödinger forms are related to optimal Hardy-type inequalities (cf. B. Devyver, M. Fraas and Y. Pinchover). The criticality of Schrödinger forms and the recurrence of Dirichlet forms are related through $h$-transform. By using these relations, we rediscover several optimal $L^2$ Hardy type inequalities by $h$-transform of recurrent Dirichlet forms.

Toshihiro Uemura (Kansai University)
Homogenization of Diffusion Processes with Drifts via Unfolding

We consider the homogenization problem with singular drifts using the semi-Dirichlet form theory and Unfolding operators methods to show the convergence of the solutions of Dirichlet problems. We could also characterize the limit of the solutions which solves a Dirichlet problem.

Jie-Ming Wang (Beijing Institute of Technology)
Boundary Harnack principle for diffusion with jumps

Under some mild conditions, the boundary Harnack principle with explicit boundary decay rate is established for a large class of diffusion with jumps on $C^{1, 1}$ open sets, where the jumping intensity function is allowed to be degenerate in some part of $\mathbb{R}^d$. Our approach uses the conservative Feller process associated with the non-local operator, by deriving sharp two-sided Green function estimates on bounded $C^{1,1}$ domains and establishing the Martin integral representation of harmonic functions on these domains, which are of independent interest. This talk is based on a joint work with Professor Z.-Q. Chen.

Panki Kim (Seoul National University)
Green function estimates and Boundary Harnack principles for non-local operators whose kernels degenerate at the boundary

In this talk, we discuss the potential theory of Markov processes with jump kernels decaying at the boundary of the half space. The boundary part of kernel is comparable to the product of three terms with parameters appearing as exponents in these terms. The constant $c$ in the killing term can be written as a function of a parameter $p$ which is strictly increasing in $p$. We establish sharp two-sided estimates on the Green functions of these processes for all admissible values of $p$ and parameters in the boundary part of kernel. Depending on the regions where parameters and $p$ belong, the estimates on the Green functions are different. In fact, the estimates have three different forms depending on the regions the parameters belong to. As applications, we completely determine the region of the parameters where the boundary Harnack principle holds or not. This talk is based on joint works with Renming Song and Zoran Vondraček.

Feng-Yu Wang (Tianjin University)
Wasserstein convergence for empirical measures of diffusion processes

Sharp convergence rate in Wasserstein distance is presented for diffusion processes on compact manifolds, SDEs and semilinear SPDEs.

Alexander Grigor'yan (University of Bielefeld)
Analysis and heat kernels on ultra-metric spaces

Given an ultra-metric space $M$ with a reference measure and a cumulative distribution function on $[0,\infty)$, we construct a symmetric Markov semigroup whose generator $L$ can be regarded as a non-local Laplace operator on such a space. We obtain explicit formulas and estimates for the heat kernel of $L$, which are similar to that for symmetric stable Levy processes in the Euclidean spaces. In a particular case when $M$ is the space of $p$-adic numbers (or its power), this construction recovers the Taibleson Laplacian. We apply this theory also to study of the Vladimirov Laplacian. Even in this well-established setting several results are new.

Lisa Hartung (Universität Mainz)
The structure of the extreme levels in the 2d discrete Gaussian free field

In this talk we take a closer look at the extreme level sets of the 2d discrete Gaussian free field. The extremes can be described by a decorated Poisson point process. The Poisson points correspond to extreme local maxima and the decoration to field around these extreme local maxima. We obtain quantitative estimates on the number of sites at which the field is above a certain fixed distance from the order of the global maximum. We also study, for a randomly chosen particle from an extreme level set, the spatial distance and height difference to the extreme local maximum. Based on joint work with M. Fels and O. Louidor.

Anton Bovier (Bonn University)
Branching Brownian motion with self repulsion

We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an $\epsilon$-neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise description of the particle numbers and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion equation with a time dependent reaction term. This is joint work with Lisa Hartung.

Xin Chen (Shanghai Jiao Tong University)
Quenched Invariance Principle for a class of random conductance models with long-range jumps

We study random walks on $\mathbb{Z}^{d}$ (with $d \geq 2$) among stationary ergodic random conductances $\{C_{xy} \colon x,y \in \mathbb{Z}^{d}\}$ that permit jumps of arbitrary length. Our focus is on the Quenched Invariance Principle (QIP) with diffusive scaling which we establish under a moment condition on $\sum_{x \in \mathbb{Z}^{d} } C_{0,x} |x|^{2}$ and $1/C_{0,x}$ for $x$ neighboring the origin by a combination of corrector methods, functional inequalities and heat-kernel technology. The talk is based on a joint paper with Marek Biskup, Takashi Kumagai and Jian Wang.

Sunil Chhita (Durham University)
GOE Fluctuations for the maximum of the top path in ASMs

Alternating sign matrices (ASMs) are square matrices with entries in $\{0,\pm 1\}$ whose rows and columns sum to $1$ and whose nonzero entries alternate in sign along every row and column. They are closely related to the six-vertex or square-ice model in statistical mechanics, plane partitions and domino tilings of the Aztec diamond. In this talk, I will present a height function representation of the ASMs and show that the maximum of the topmost level line, for uniformly random ASMs, converges to the GOE Tracy-Widom distribution after appropriate scaling. This is based on joint work with Arvind Ayyer and Kurt Johansson.

Shen Lin (Sorbonne Université)
Scaling limits of tree-valued branching random walks

We consider a branching random walk taking its values in the $\mathtt{b}$-ary rooted tree $\mathbb{W}_{\mathtt{b}}$ (i.e. the set of finite words written in the alphabet $\{1, \ldots, \mathtt{b}\}$, with $\mathtt{b} \geq 2$). The branching random walk is indexed by a uniform ordered rooted tree with $n$ vertices. The jumps of the branching random walk are those of a nearest-neighbour null-recurrent random walk on $\mathbb{W}_{\mathtt{b}}$ (reflection at the root of $\mathbb{W}_{\mathtt{b}}$ and otherwise : probability $1/2$ to move closer to the root of $\mathbb{W}_{\mathtt{b}}$ and probability $1/(2\mathtt{b})$ to move away from the root towards one of the $\mathtt{b}$ sites above). We denote by $\mathcal{R}_{\mathtt{b}}(n)$ the set of all sites in $\mathbb{W}_{\mathtt{b}}$ that are visited by the branching random walk. We first establish a law of large numbers for $\#\mathcal{R}_{\mathtt{b}}(n)$, and we also prove that if we equip $\mathcal{R}_{\mathtt{b}}(n)$ (seen as a random subtree of $\mathbb{W}_{\mathtt{b}}$) with its graph-distance $d_{\mathtt{gr}}$, then as $n$ tends to infinity, the random metric space $(\mathcal{R}_{\mathtt{b}}(n), n^{-\frac{1}{4}} d_{\mathtt{gr}})$, equipped with its normalised empirical measure converges to the reflected Brownian cactus, a variant of the Brownian cactus introduced by N. Curien, J.-F. Le Gall and G. Miermont (Ann. IHP 2013). Based on joint work with Thomas Duquesne (Sorbonne Université), Robin Khanfir (Sorbonne Université) and Niccolò Torri (Université Paris-Nanterre). See: arxiv:2104.07314

Leonid Mytnik (Technion)
On the speed of a front for stochastic reaction-diffusion equations

We study the asymptotic speed of a random front for solutions to stochastic reaction-diffusion equations with Wright-Fisher noise proportional to $\sigma$. Under some conditions on the drift, we show the existence of the speed of the front and derive its asymptotics depending on $\sigma$.

Zhongmin Qian (University of Oxford)
Semi-linear parabolic equations on the Sierpinski gasket

I will report some recent results, joint work with Dr Xuan Liu, on a class of semi-linear parabolic type equations defined in terms of the Dirichlet form on the Sierpinski gasket. I will review the fundamental result on backward stochastic differential equations initiated by Pardoux and Peng from a general view-point, and apply the general theory to the study of non-linear “parabolic” equations associated with Dirichlet forms.