# Past Seminars and Recordings

**Tuesday, May 21, 2024**

###### Evan Sorensen (Columbia University)

**Title**: Jointly invariant measures for the KPZ equation with periodic noise

**Abstract**:

We present an explicit coupling of Brownian bridges plus affine shifts that are jointly invariant for the Kardar-Parisi-Zhang equation with periodic noise. These are described as Pitman-like transforms of independent Brownian bridges. We obtain these invariant measures by working with a semi-discrete model known as the O’Connell-Yor polymer in a periodic environment. In that setting, the relevant Markov process is described by a system of coupled SDEs. We show how to transform this Markov process to an auxiliary Markov process with a more tractable invariant measure. We discuss connections of this method to works of Ferrari and Martin in the mid 2000s in the context of multi-species particle systems. Furthermore, we present an application of this work to give an explicit formula for the covariance function of a limiting Gaussian process obtained from the coupled stochastic heat equation. Based on forthcoming joint work with Ivan Corwin and Yu Gu.

**Tuesday, May 14, 2024**

###### Lucio Galeati (EPFL)

**Title**: Propagation of chaos for singular interacting particle systems driven by fBm

**Abstract**:

###### Michaela Szölgyenyi (University of Klagenfurt)

**Title**: Bicausal optimal transport for SDEs with irregular coefficients

**Abstract**:

**Tuesday, April 2, 2024**

###### Samuel Johnston(King’s College London)

**Title**: Free probability via entropic optimal transport

**Abstract**:

**Tuesday, March 26, 2024**

###### Matthew Rosenzweig (Carnegie Mellon University) [Online]

**Title**: Fluctuations around the mean-field limit for singular Riesz flows

**Abstract**:

**Tuesday, March 19, 2024**

###### Paweł Duch (Adam Mickiewicz University in Poznań)

**Title**: Construction of critical fermionic QFTs using Polchinski flow equation

**Abstract**:

**Wednesday, March 13, 2024: Maxwell Institute Probability and Stochastic Analysis Workshop**

###### Massimiliano Gubinelli (University of Oxford)

###### Title: Stochastic quantisation of the fractional model via flow equations

###### Abstract:

I will describe a recent result obtained in collaboration with P. Rinaldi and P. Duch. We construct the Euclidean fractional \Phi^4 quantum field via the method of stochastic quantisation. This involves obtaining global space-time solutions for a singular stochastic PDE of fractional parabolic type with cubic non-linearity. This goal is achieved via a novel method of flow equations introduced by Duch.

###### Jordan Stoyanov (Bulgarian Academy of Sciences)

###### Title: Characterizations of probability distributions: a bunch of old and new results

###### Abstract:

It is common for any specific distribution, say F, to try to find a property which is valid only for F. In any such a case we talk about a characterizing property and a characterization problem. Well-known are classical results for popular distributions such as Normal, Poisson, Exponential, Geometric, Gamma, etc. (Cramer, Raikov, Polya, Bernstein, Skitovich-Darmois). The work on such problems is going on and some new very recent results will be reported. We can also ask whether or not a distribution can be characterized by its moments, which is one of the fundamental questions in the classical moment problem (Chebyshev, Markov, Stieltjes). Besides well-known results (Hamburger, Hausdorff, Carleman, Hardy, Krein, Lin, rate of growth of moments), some very recent results on M-(in)determinacy of distributions will be presented. Useful illustrations and (sometime) surprising facts for specific continuous and/or discrete distributions will be given. Challenging open questions will be outlined.

###### Ofer Busani (University of Edinburgh)

###### Title: On the number of infinite geodesics in exceptional directions in the KPZ class

###### Abstract:

The KPZ class consists of many models of random growth interface. In many cases, the dynamics of these models can be studied via variational forms that give rise to metric-like spaces, which in turn, can be studied through geodesics. The study of infinite geodesics in the KPZ class has been studied intensively in the past 30 years. One central question is the following:

Given a direction, how many infinite geodesics that are asymptotically going in that direction are there? In this talk, I shall discuss what we know about infinite geodesics in the KPZ class and some recent developments regarding the question above.Based on several works with Marton Balazs, Timo Seppalainen and Evan Sorensen.

###### Paul Dobson (Heriot-Watt University)

###### Title: Piecewise deterministic sampling with splitting schemes

###### Abstract:

Piecewise deterministic Markov processes (PDMPs) received substantial interest in recent years as an alternative to classical Markov chain Monte Carlo algorithms. While theoretical properties of PDMPs have been studied extensively, their practical implementation remains limited to specific applications in which bounds on the gradient of the negative log-target can be derived. In order to address this problem, we propose to approximate PDMPs using splitting schemes and will discuss the properties of this approximation.

**Tuesday, March 5, 2024**

###### Michela Ottobre (Heriot-Watt University)

**Title**: **McKean-Vlasov S(P)Des with additive noise**

**Abstract**:

Many systems in the applied sciences are made of a large number of particles. One is often not interested in the detailed behaviour of each particle but rather in the collective behaviour of the group. An established methodology in statistical mechanics and kinetic theory allows one to study the limit as the number of particles N tends to infinity and to obtain a (low dimensional) PDE for the evolution of the density of particles. The limiting PDE is a non-linear equation, where the non-linearity has a specific structure and is called a McKean-Vlasov nonlinearity. Even if the particles evolve according to a stochastic differential equation, the limiting equation is deterministic, as long as the particles are subject to independent sources of noise. If the particles are subject to the same noise (common noise) then the limit is given by a Stochastic Partial Differential Equation (SPDE). In the latter case the limiting SPDE is substantially the McKean-Vlasov PDE + noise; noise is further more multiplicative and has gradient structure. One may then ask the question about whether it is possible to obtain McKean-Vlasov SPDEs with additive noise from particle systems. We will explain how to address this question, by studying limits of weighted particle systems, in a framework introduced by Kurtz and collaborators.

This is a joint work with L. Angeli, D. Crisan, M. Kolodziejzik.

###### William Salkeld(University of Nottingham)

**Title**: Locally interacting equations and their associated McKean-Vlasov marginal

**Abstract**:

In this talk, I will discuss the dynamics of a collection of Gaussian stochastic differential equations indexed by a locally finite graph. The drift of each individual equation is dependent only on the dynamics of the individual and their neighbours so that each SDE exhibits strong correlation with a small number of other SDEs via these local interactions.

**Tuesday, February 27, 2024**

###### Larisa Yaroslavtseva (University of Graz)

**Title**: On the complexity of strong approximation of SDEs with a non-Lipschitz drift coefficient

**Abstract**:

We study pathwise approximation of stochastic differential equations (SDEs) at a single time based on finitely many (sequential) evaluations of the driving Brownian motion. The classical assumption in the literature on numerical approximation of SDEs is global Lipschitz continuity of the coefficients of the equation. However, many SDEs arising in applications fail to have globally Lipschitz continuous coefficients. In this talk we focus on the case when the drift coefficient is not Lipschitz continuous. In particular, we discuss recent results on corresponding upper and lower error bounds for SDEs with a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity.

The talk is based on joint work with Simon Ellinger (University of Passau), Arnulf Jentzen (Univerity of M¨unster) and Thomas M¨uller-Gronbach (University of Passau).

**Tuesday, February 20, 2024**

###### Zhenfu Wang (BICMR-Peking University) [Online]

**Title**: Quantitative Propagation of Chaos for 2D Viscous Vortex Model on the Whole Space

**Abstract**:

We derive the quantitative estimates of propagation of chaos for the large interacting particle systems in terms of the relative entropy between the joint law of the particles and the tensorized law of the mean field PDE. We resolve this problem for the first time for the viscous vortex model that approximating 2D Navier-Stokes equation in the vorticity formulation on the whole space. We obtain as key tools the Li-Yau-type estimates and Hamilton-type heat kernel estimates for 2D Navier-Stokes on the whole space. This is based on a joint work with Xuanrui Feng from Peking University.

###### Renyuan Xu (University of Southern California) [Online]

**Title**: Some Mathematical Results on Generative Diffusion Models

**Abstract**:

Generative diffusion models, which transform noise into new data instances by reversing a Markov diffusion process in time, have become a cornerstone in modern generative models. A key component of these models is to learn the associated Stein’s score function. While the practical power of diffusion models has now been widely recognized, the theoretical developments remain far from mature. Notably, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. In this talk, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models and the accuracy of score estimation. Our analysis covers both the optimization and the generalization aspects of the learning procedure, which also builds a novel connection to supervised learning and neural tangent kernels.

This is based on joint work with Yinbin Han and Meisam Razaviyayn (USC).

**Tuesday, February 13, 2024**

###### Oana Lang (Imperial College London)

**Title**: Local and global existence for transport and non-transport SPDEs

**Abstract**:

This talk will be focused on local and global properties for a class of stochastic partial differential equations (SPDEs) which includes equations coming from geophysical fluid dynamics. There are many SPDEs in the literature for which global existence of solutions does not hold, with several such examples coming from fluid dynamics. This has important consequences not just analytically, but also in applications. I will discuss conditions under which such equations can admit existence of a global solution and implications at their structural level. This is joint work with Dan Crisan.

**Tuesday, February 6, 2024**

###### Johannes Muhle-Karbe (Imperial College London)

**Title**: Optimal Contracts for Delegated Order Execution

**Abstract**:

**Tuesday, January 30, 2024**

###### Ying Zhang (The Hong Kong University of Science and Technology (Guangzhou))

**Title**: Langevin dynamics-based algorithm e-THeO POULA for stochastic optimization with discontinuous stochastic gradient

###### Abstract:

We introduce a new Langevin dynamics-based algorithm, called e-THO POULA, to solve optimization problems with discontinuous stochastic gradients which naturally appear in real-world applications such as quantile estimation, vector quantization, CVaR minimization, and regularized optimization problems involving ReLU neural networks. We demonstrate both theoretically and numerically the applicability of the e-THO POULA algorithm. More precisely, under the conditions that the stochastic gradient is locally Lipschitz in average and satisfies a certain convexity at infinity condition, we establish non-asymptotic error bounds for e-THeO POULA in Wasserstein distances and provide a non-asymptotic estimate for the expected excess risk, which can be controlled to be arbitrarily small. Two key applications in finance are provided, namely, multi-period portfolio optimization and transfer learning in multi-period portfolio optimization, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world datasets illustrate the superior empirical performance of e-THeO POULA compared to SGLD, ADAM, and AMSGrad in terms of model accuracy.

**Recording:** https://ed-ac-uk.zoom.us/rec/share/w06TSKfqlf2thqEKqBhWuXjvLFhQ-hmNwaRFk9hNbKFHCs74zDXzjtBM4gVTeJcD.SdlrZvqte4vYB0Vn

**Tuesday, January 23, 2024**

###### Xell Brunet Guasch (University of Edinburgh)

**Title**: Multi-Type Critical Birth-Death Process: A Model of Error-Induced Extinction

**Abstract:**

**Tuesday, January 16, 2024**

###### Catherine Wolfram (MIT) [Online]

**Title**: The dimer model in 3D

**Abstract: **

A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. I will explain how to formulate the large deviations principle in 3D, show simulations, try to give some intuition for why three dimensions is different from two. Time permitting, I will explain some of the ways that we use a smaller set of tools (e.g. Hall’s matching theorem or a double dimer swapping operation) in our arguments.

**Tuesday, December 5, 2023**

###### Maxim Raginsky (University of Illinois at Urbana-Champaign) [Online]

**Title**: A variational approach to sampling and path estimation in diffusion processes

**Abstract**:

Recently, diffusion models have achieved remarkable breakthroughs in generative modeling, with practical advances outpacing theoretical understanding. This has led to a surge of interest in the theory of diffusion processes, as well as in the fundamental links between stochastic control, nonequilibrium thermodynamics, and learning. In this talk, I will present a unified perspective on sampling, inference, and path estimation (or smoothing) in diffusion processes through the lens of free energy minimization. This variational perspective ties together the Gibbs variational principle with the Cameron-Martin-Girsanov change of measure and the Hamilton-Jacobi-Bellman optimal control formulation. Several existing results emerge as special cases, including the variational formulation of nonlinear estimation due to Newton and Mitter and the recent control-theoretic derivation of diffusion-based generative modeling due to Berner, Richter, and Ullrich.

**Corrected Slides:** **Raginsky.pdf**

**Tuesday, November 28, 2023**

###### Sarah-Jean Meyer(University of Oxford)

**Title**: The forward-backward approach to the sine-Gordon EQFT

**Abstract**:

We present a forward-backward approach to stochastic quantisation for the sine-Gordon Euclidean quantum field on the full space up to the second threshold, i.e. for . The basis of our method is a forward-backward stochastic differential equation (FBSDE) for a decomposition of the interacting Euclidean field along a scale parameter . This FBSDE describes the optimiser of a stochastic control representation of the Euclidean QFT introduced by Barashkov and Gubinelli and provides a scale-by-scale description of the interacting field without cut-offs. I will present the general idea of the approach, and illustrate how the FBSDE can be used effectively to study the sine-Gordon measure to obtain results such as large deviations, integrability, decay of correlations for local observables, Osterwalder–Schrader axioms and other properties.

This talk is based on joint work with Massimiliano Gubinelli.

**Tuesday, November 21, 2023**

###### Gal Kronenberg (University of Oxford)

**Title**: Independent sets in random subgraphs of the hypercube

**Abstract**:

Independent sets in bipartite regular graphs have been studied extensively in combinatorics, probability, computer science and more. The problem of counting independent sets is particularly interesting in the d-dimensional hypercube , motivated by the lattice gas hardcore model from statistical physics. Independent sets also turn out to be very interesting in the context of random graphs.

The number of independent sets in the hypercube was estimated precisely by Korshunov and Sapozhenko in the 1980s and recently refined by Jenssen and Perkins.

In this talk we will discuss new results on the number of independent sets in a random subgraph of the hypercube. The results extend to the hardcore model and rely on an analysis of the antiferromagnetic Ising model on the hypercube.

This talk is based on joint work with Yinon Spinka.

**Tuesday, November 14, 2023**

###### Nikolas Nüsken (King’s College London)

**Title**: Transport meets Variational Inference: Controlled Monte Carlo Diffusions

**Abstract**:

Connecting optimal transport and variational inference, this talk is about a principled and systematic framework for score-based sampling and generative modelling centred around forward and reverse-time stochastic differential equations. In particular, I will discuss the novel Controlled Monte Carlo Diffusion (CMCD) methodology for Bayesian computation, a score-based annealing technique that crucially adapts both forward and backward dynamics in a diffusion model. Time permitting, we will discuss relationships to optimal control and Schroedinger bridges, as well as connections to the Jarzinsky and Crooks identities from statistical physics. This is joint work with F. Vargas, S. Padhy and D. Blessing.

**Tuesday, November 7, 2023**

###### Harprit Singh (University of Edinburgh)

**Title**: Singular SPDEs on Geometric Spaces

**Abstract**:

We shall discuss the solution theory to a large class of singular SPDEs in various nontranslation invariant settings. In particular, I shall present results on Riemannian manifolds and homogeneous Lie groups as well as related results for non-translation invariant parabolic and hypo-elliptic equations on Euclidean space. Based on joint work with M. Hairer, respectively A. Mayorcas.

**Tuesday, October 31, 2023**

###### Guillaume Barraquand (ENS)

**Title**: Stationary measures in the Kardar-Parisi-Zhang universality class

**Abstract**:

The Kardar-Parisi-Zhang (KPZ) equation is a nonlinear stochastic PDE introduced in Physics to model interface growth. In one dimension, the KPZ equation on the line admits a remarkably simple stationary measure: the Brownian motion. For the KPZ equation on an interval or a half-line, however, stationary measures are non-Gaussian and depend on the boundary conditions imposed. Their explicit description has been obtained only very recently, through the detailed study of integrable probabilistic models, such as ASEP or directed polymer models, that can be viewed as discretizations of the KPZ equation. We will first review these results, which are based on a series of works by several groups of authors. I will then outline a new method to compute the stationary distributions of stochastic integrable models on a lattice such as last passage percolation, or directed polymer models.

**Tuesday, October 24, 2023**

###### Roger Van Peski (KTH)

**Title**: New local limits in discrete random matrix theory

**Abstract**:

Random matrices over finite fields, integers, and p-adic integers have been studied since the late 1980s as natural models for random groups appearing in number theory and combinatorics. Recently it has also become clear that the theory has close structural parallels with singular values of complex random matrices, bringing new techniques from integrable probability and motivating new questions. In particular, in this talk I will explain recent results on the local limits of products of such matrices, which yield a new object that may be viewed as a discrete and non-determinantal analogue of (certain deformations of) the extended sine and Airy processes in classical random matrix theory.

**Tuesday, October 17, 2023**

**Ioannis Gasteratos (Imperial College London)**

**Title**: Large deviations of slow-fast systems driven by fractional Brownian motion

**Abstract**:

We consider a multiscale system of stochastic differential equations in which the slow component is perturbed by a small fractional Brownian motion with Hurst index and the fast component is driven by an independent Brownian motion. Working in the framework of Young integration, we use tools from fractional calculus and weak convergence arguments to establish a Large Deviation Principle (LDP) in the homogenized limit, as the noise intensity and time-scale separation parameters vanish at an appropriate rate. Our approach is based in the study of the limiting behavior of an associated controlled system. We show that, in certain cases, the non-local rate function admits an explicit non-variational form. The latter allows us to draw comparisons to the case which corresponds to the classical Freidlin-Wentzell theory. Moreover, we study the asymptotics of the rate function as and show that it is discontinuous at .

**Tuesday, October 10, 2023**

**Raphaël Maillet (CEREMADE) [Online]**

**Title**: Stochastic McKean-Vlasov Equation with common noise

**Abstract**:

In this talk, we explore the long-term behaviour of solutions to a nonlinear McKean-Vlasov equation with common noise. This equation arises naturally when studying the collective behaviour of interacting particles driven by both individual and common noise. Our main focus is to understand how the presence of common noise affects the stability of the system. We will first present the results for the case without common noise and then discuss how the introduction of common noise can enhance stability in certain respects.

**Tuesday, October 3, 2023**

**Kohei Suzuki (Durham University)**

**Title**: Curvature Bound of the Dyson Brownian Motion

**Abstract**:

The Dyson Brownian Motion (DBM) is an eigenvalue process of a particular Hermitian matrix-valued Brownian motion introduced by Freeman Dyson in 1962, which has been one of the central subjects in the random matrix theory. In this talk, we study the DBM from a geometric perspective. We show that the infinite particle DBM possesses a lower bound of the Ricci curvature à la Bakry-Émery. As a consequence, we obtain various quantitative estimates of the transition probability of the DBM (e.g., the local spectral gap, the local log-Sobolev, and the dimension-free Harnack inequalities) as well as the characterisation of the DBM as the gradient flow of the Boltzmann entropy in a particular Wasserstein-type space.

**September 26, 2023**

**Ofer Busani (University of Edinburgh)**

###### Title: Scaling limit of multi-type stationary measures in the KPZ class

**Abstract**:

The KPZ class is a very large set of models that are meant to describe random growth interfaces. It is believed that upon scaling, the long time behavior of members in this class is universal and is described by a limiting random object, a Markov process called the KPZ fixed-point. The (one-type) stationary measures for the KPZ fixed-point as well as many models in the KPZ class are known – it is a family of distributions parametrized by some set that depends on the model. For the k-type stationary distribution with intensities is a coupling of one-type stationary measures of indices that is stationary with respect to the model dynamics. In this talk we will present recent progress in our understanding of the multi-type stationary measures of the KPZ fixed-point as well as the scaling limit of multi-type stationary measures of two families of models in the KPZ class: metric-like models (e.g. last passage percolation) and particle systems (e.g. exclusion process). Based on joint work with Timo Seppalainen and Evan Sorensen.

**June 14, 2023**

**Yufei Zhang (LSE)**

###### Title: Exploration-exploitation trade-off for continuous-time reinforcement learning

**Abstract**:

Recently, reinforcement learning (RL) has attracted substantial research interests. Much of the attention and success, however, has been for the discrete-time setting. Continuous-time RL, despite its natural analytical connection to stochastic controls, has been largely unexplored and with limited progress. In particular, characterising sample efficiency for continuous-time RL algorithms remains a challenging and open problem.

In this talk, we develop a framework to analyse model-based reinforcement learning in the episodic setting. We then apply it to optimise exploration-exploitation trade-off for linear-convex RL problems, and report sublinear (or even logarithmic) regret bounds for a class of learning algorithms inspired by filtering theory. The approach is probabilistic, involving analysing learning efficiency using concentration inequalities for correlated continuous-time observations, and applying stochastic control theory to quantify the performance gap between applying greedy policies derived from estimated and true models.

**May 3, 2023: North British Probability Workshop**

**Neil O’Connell (University College Dublin) [Online]**

###### Title: A Markov **chain** on reverse plane partitions

**Abstract:**

I will discuss a natural Markov chain on reverse plane partitions

which is closely related to the Toda lattice.

**Athanasios Vasileiadis (University of Nice)**

**Title: Learning linear quadratic Mean Field Games. **

**Abstract: **

In this talk we are going to use common noise as a form of exploration to learn the Nash equilibrium of a Linear Quadratic Mean Field Game. We develop a scheme based on Fictitious Play and study its convergence to the analytical solution of the limit system. We illustrate the analysis by numerical implementations using regression on a basis of Hermit polynomials and also artificial neural networks.

**Recording:** https://ed-ac-uk.zoom.us/rec/share/U6UmjO9M6cvpULjTi1balg3UvtFQy-C0pueruoivPCYvWUmeboa1k_0rVi9a5HM.f7EkReDHRV5W0lPH

**Stefano Bruno (University of Edinburgh)**

###### Title: Optimal Regularity in Time and Space for Stochastic Porous Medium Equations

**Abstract:**

**April 26, 2023:**

** Benjamin Fehrman (University of Oxford)**

**Title:** Non-equilibrium fluctuations and parabolic-hyperbolic PDE with irregular drift

**Abstract: **

Non-equilibrium behavior in physical systems is widespread. A statistical description of these events is provided by macroscopic fluctuation theory, a framework for non-equilibrium statistical mechanics that postulates a formula for the probability of a space-time fluctuation based on the constitutive equations of the system. This formula is formally obtained via a zero noise large deviations principle for the associated fluctuating hydrodynamics, which postulates a conservative, singular stochastic PDE to describe the system far-from-equilibrium. In this talk, we will focus particularly on the fluctuations of the zero range process about its hydrodynamic limit. We will show how the associated MFT and fluctuating hydrodynamics lead to a class of conservative SPDEs with irregular coefficients, and how the study of large deviations principles for the particles processes and SPDEs leads to the analysis of parabolic-hyperbolic PDEs in energy critical spaces. The analysis makes rigorous the connection between MFT and fluctuating hydrodynamics in this setting, and provides a positive answer to a long-standing open problem for the large deviations of the zero range process.

**April 19, 2023:**

**Patrik Ferrari (University of Bonn)**

**Title: Space-time correlations in the KPZ universality class**

**Abstract:**

###### Recording: https://ed-ac-uk.zoom.us/rec/share/TE70TZkOomtJPuzZzQbM1cv3B5dl1cZowwDWjvPPlezJI0ryvySyFANfHi8tF8mN.4dBa1PTDRP02Pl1t?startTime=1681912646000

**March 29, 2023:**

** Mao Fabrice Djete (École Polytechnique) [Online]**

###### Title: Non–regular McKean–Vlasov equations and calibration problem in local stochastic volatility models

###### Abstract:

###### Recording: https://ed-ac-uk.zoom.us/rec/share/UCav231Kx_fHc0Lc_v_dw_CxNRMbeHNFXuRATW2F6pxsVYT-v_2HSIoXY-1PTjaa.OmjpNkASe9pORBBT?startTime=1680098573000

**Christian Bayer (WIAS Berlin) [Online]**

###### Title: Optimal stopping with signatures

###### Abstract:

We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process X.

We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature associated to X, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a(deterministic) optimization problem depending only on the (truncated) expected signature. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically.

The only assumption on the process X is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. of financial or electricity markets. (Based on joint work with Paul Hager, Sebastian Riedel, and John Schoenmakers)

###### Recording: https://ed-ac-uk.zoom.us/rec/share/UCav231Kx_fHc0Lc_v_dw_CxNRMbeHNFXuRATW2F6pxsVYT-v_2HSIoXY-1PTjaa.OmjpNkASe9pORBBT?startTime=1680102531000

**March 15, 2023: **

**Guangqu Zheng**** (University of Liverpool)**

###### Title: Pathwise local well-posedness of stochastic nonlinear Schrödinger equations with multiplicative noises.

###### Abstract:

In this talk, I will present a recent work on the stochastic nonlinear Schrödinger equation (SNLS) with multiplicative noises. The equation was first proposed in physics literature, and has attracted much attention in the mathematical community. It has been studied by various researchers such as de Bouard, Debussche, Barbu, Röckner, either in Ito framework or via specific transform. As a result, it is not pathwise. The major difficulty towards a pathwise theory lies at how to give a meaning to the product of the noise and an unknown in presence of deficiency of temporal regularities. Despite some development in the parabolic setting by Gubinelli-Tindel (2010) and by Hairer-Pardoux (2015), the dispersive problem remains unsolved.

In this talk, I will present a joint work with T. Oh (Edinburgh), which established the first pathwise local well-posedness result for SNLS with multiplicative noises in the spatially smooth setting. This work exploits Gubinelli-Tindel’s framework of algebraic integration, in combination with random matrix/tensor estimates from a recent work by Deng-Nahmod-Yue (2022), and in particular, the notion of operator-valued controlled rough paths adapted to the Schrödinger flow is introduced.

**March 8, 2023:**

**Emanuela Gussetti (Universität Bielefeld)**

###### Title: An application of rough paths theory to the study of the stochastic Landau-Lifschitz-Gilbert equation

###### Abstract:

The Landau-Lifschitz-Gilbert equation is a model describing the magnetisation of a ferromagnetic material. The stochastic model is studied to observe the role of thermal fluctuations. We interpret the linear multiplicative noise appearing by means of rough paths theory and we study existence and uniqueness of the solution to the equation on a one dimensional domain . As a consequence of the rough paths formulation, the map that to the noise associates the unique solution to the equation is locally Lipschitz continuous in the strong norm . This implies a Wong-Zakai convergence result, a large deviation principle, a support theorem.

We prove also continuity with respect to the initial datum initial datum in , which allows to conclude the Feller property for the associated semigroup. We also discuss briefly a Path-wise central limit theorem and a moderate deviations principle for the stochastic LLG: in this case the rough paths formulation leads to a path-wise convergence, not easily reachable in the classical It‘s calculus setting.

The talk is based on a joint work with A. Hocquet [https://arxiv.org/abs/2103.00926] and on [https://arxiv.org/abs/2208.02136].

###### Recording: https://ed-ac-uk.zoom.us/rec/share/NqgUULB9KL3wRdB8YFe6VZosZn-62-YCMOggGmLgqEdmrtntXRddzTdjPmpITqE0.6AiqbM7rditaTm6k

**February 15, 2023: **

**Laure Dumaz (ENS)**

###### Title: Some aspects of the Anderson Hamiltonian in 1D.

###### Abstract:

In this talk, I will present several results on the Anderson Hamiltonian with white noise potential in dimension 1. This operator formally writes « – Laplacian + white noise ». It arises as the scaling limit of various discrete models and its explicit potential allows for a detailed description of its spectrum. We will discuss localization of its eigenfunctions as well as the behavior of the local statistics of its eigenvalues. Around large energies, we will see that the eigenfunctions are delocalized and follow a universal shape given by the exponential of a Brownian motion plus a drift, a behavior already observed by Rifkind and Virag in tridiagonal matrix models. Based on joint works with Cyril Labbé.

**February 8, 2023:**

** Fabian Germ (University of Edinburgh)**

**Title**:** On conditional densities of partially observed jump diffusions**

Abstract: I will present the filtering problem for a partially observed jump diffusion system Z=(X,Y) driven by Wiener processes and Poisson martingale measures. We will derive the filtering equations, describing the time evolution of the normalised conditional distribution and the unnormalised conditional distribution of the unobservable signal X t given the observations Y. Recent results on L_p-calculus for systems with jumps have made it possible to prove existence and regularity properties of (unnormalised) conditional densities in certain Sobolev spaces under fairly general assumptions. We will see such conditions and derive existence and regularity results based on SPDEs for the (unnormalised) conditional density processes.

This is joint work with Istvan Gyongy.

**Recording**: https://ed-ac-uk.zoom.us/rec/share/5uXeGb84iHFMOpq_GOgZyUgL94dzfYx9Xq5DNsAnrOq4dVVqgexFX0CYTUcoNAIE.lsJ9WNK-nN5QZ7w7

**November 24, 2022**

**Mo Dick Wong (Durham University)**

**Title: What can we hear about the geometry of an LQG surface?**

Abstract: The Liouville quantum gravity (LQG) surface, formally defined as a 2-dimensional Riemannian manifold with conformal factor being the exponentiation of a Gaussian free field, is closely related to random planar geometry as well as scaling limits of models from statistical mechanics. In this talk we consider the bounded, simply connected setting, and explain the Weyl’s law for the asymptotics of the eigenvalues of the (random) Laplace-Beltrami operator. Our approach is purely probabilistic and relies on the study of the so-called Liouville Brownian motion, the canonical diffusion process on an LQG surface. This is a joint work with Nathanael Berestycki.

**James Foster (University of Bath) -online talk **

**Title: High order splitting methods for stochastic differential equations**

Abstract: In this talk, we will discuss how ideas from rough path theory can be leveraged to develop high order numerical methods for SDEs. To motivate our approach, we consider what happens when the Brownian motion driving an SDE is replaced by a piecewise linear path. We show that this procedure transforms the SDE into a sequence of ODEs – which can then be discretized using an appropriate ODE solver. Moreover, to achieve a high accuracy, we construct these piecewise linear paths to match certain “iterated” integrals of the Brownian motion. At the same time, the ODE sequences obtained from this path-based approach can be interpreted as a splitting method, which neatly connects our work to the existing literature. For example, we show that the well-known Strang splitting falls under this framework and can be modified to give an improved convergence rate. We will conclude the talk with a couple of examples, demonstrating the flexibility and convergence properties of our methodology.

(joint work with Gonçalo dos Reis and Calum Strange)

###### Recording: https://ed-ac-uk.zoom.us/rec/share/uButTuqdAo8UX9g7sAdxsqVvyrtZZD229GFD5dcDmkpXchz8LdiTnn4cwTdQmUeF.HYlI6rng5spV3jEN

**November 17, 2022**

**Jiawei Li (University of Edinburgh) **

**Title: On distributions of random vorticity and velocity fields in turbulent flows**

###### Recording: https://ed-ac-uk.zoom.us/rec/share/o3IpQRfmxuAFsIftwzB_s1MSKQLH1BgGfZWT5YOsTjxTck9eNwhYfl77Wzj3HaBw.GygPxXqkjSXhQ5NC

**November 10, 2022**

**Mustazee Rahman (Durham University)**

**Title: Scaling limit of the second class particle**

Abstract: The totally asymmetric simple exclusion process (TASEP) is a microscopic analogue of Burgers equation. The characteristics of Burgers equation play an important role in its solution. The second class particle in TASEP mirrors the role of a microscopic characteristic. Although much is known about the macroscopic behaviour of the second class particle, it is of interest to understand its fluctuations. I will discuss the behaviour of the second class particle when the initial conditions of TASEP converge under KPZ scaling. The second class particle has a scaling limit, which can be described in terms of geometric concepts arising in the KPZ universality class. Joint work with Balint Virag.

###### Recording: https://ed-ac-uk.zoom.us/rec/share/Qt_qW6IHGW4K0reWxjoAIgNgcs48gQ_siLccucMbHGnOgn9IEXRP2aVq2wSrH_gy.02PSxYcCxnIAJXg1

**November 3, 2022**

**Alexey Korepanov (University of Paris)**

**Title: Almost sure approximation by Brownian motion for Bunimovich flowers** Abstract: I will talk about approximation of processes which appear on Bunimovich flowers (deterministic dynamical systems where a point particle moves inside a table made of several concave arcs, well known for slow memory loss), and about their approximation by a Brownian motion by reconstructing both on the same probability space so that they are close. This is a rather special result about a rather particular system, but in a way it completes a long line of research. I’ll attempt to introduce the topic and explain the kind of problems we are dealing with. This is a joint work in progress with C.Cuny, J.Dedecker and F.Merlevede.

###### Recording: https://ed-ac-uk.zoom.us/rec/share/KN21bP67RtH98DAUfANo_MtXCtOGvMnri6VbzF8mi3cAUtDydkJwhakSd39QrjJt.x-5UG9mACtygtg4Y

**October 27, 2022**

**Samuel Cohen (University of Oxford)**

**Title: Neural-SDEs and Market Models of Options **

Abstract: Measuring market risk for option portfolios is challenging; it requires modelling joint dynamics of liquid options under the real world measure, simulating realistic trajectories of risk factors, and evaluating option prices efficiently under a large of amount of risk scenarios. In this talk we will construct an arbitrage-free neural-SDE market model for European call options. This involves building neural networks satisfying certain structural assumptions. Through backtesting analysis, we show that our models are more computationally efficient and accurate for evaluating Value-at-Risk (VaR) of option portfolios, with better coverage performance and less procyclicality than standard filtered historical simulation approaches.

###### Recording: https://ed-ac-uk.zoom.us/rec/share/u7V5IkCcWe8mWoDeWQaDlTP2EW4zDWbz9YBq5PDzmtC9wEJNc2laui6r9vasFh58.xo4YiUvTBeWmKlTI

**Karen Habermann (University of Warwick)**

**Title: A polynomial expansion for Brownian motion and the associated fluctuation process **

Abstract: We start by deriving a polynomial expansion for Brownian motion expressed in terms of shifted Legendre polynomials by considering Brownian motion conditioned to have vanishing iterated time integrals of all orders. We further discuss the fluctuations for this expansion and show that they converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. We then link the asymptotic convergence rates of approximations for Brownian Lévy area which are based on the Fourier series expansion and the polynomial expansion of the Brownian bridge to these limit fluctuations. We close with a general study of the asymptotic error arising when approximating the Green’s function of a Sturm-Liouville problem through a truncation of its eigenfunction expansion, both for the Green’s function of a regular Sturm-Liouville problem and for the Green’s function associated with the classical orthogonal polynomials.

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