Forthcoming events in this series


Wed, 14 May 2025
11:00
L5

Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension

Quentin Guillaume Moulard
(Vienna School of Mathematics)
Abstract

The Stochastic Burgers Equation (SBE) was introduced in the eighties by van Beijren, Kutner and Spohn as a mesoscopic model for driven diffusive systems with one conserved scalar quantity. In the subcritical dimension d=1, it coincides with the derivative of the KPZ equation whose large-scale behaviour is polynomially superdiffusive and given by the KPZ Fixed Point, and in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to an anisotropic Stochastic Heat equation. At the critical dimension d=2, the SBE was conjectured to be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections. This talk is based on the work joint with Giuseppe Cannizzaro and Fabio Toninelli under the same name https://arxiv.org/abs/2501.00344, where we pin down the logarithmic superdiffusivity by identifying exactly the large-time asymptotic behaviour of the so-called diffusion matrix and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem. This is the first superdiffusive scaling limit result for a critical SPDE, beyond the weak coupling regime.

Wed, 07 May 2025
11:00
L5

On statistical stationary solutions to the Schrödinger Map Equation in 1D

Dr Emanuela Gussetti
(Bielefeld University)
Abstract

In this talk, we discuss the existence of statistically stationary solutions to the Schrödinger map equation on a one-dimensional domain, with null Neumann boundary conditions, or on the one-dimensional torus. To approximate the Schrödinger map equation, we employ the stochastic  Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish existence of a random initial datum, whose distribution is preserved under the dynamic of the deterministic equation. We explore the relationship between the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove existence of statistically stationary solutions to the binormal curvature flow.[https://arxiv.org/abs/2501.16499]

This is a joint work with Professor M. Hofmanová.

Wed, 30 Apr 2025
15:30
C1

Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills

Ilya Chevyrev
(University of Edinburgh)
Abstract

In this talk, I will describe a family of observables for 3D quantum Yang-Mills theory based on regularising connections with the YM heat flow. I will describe how these observables can be used to show that there is a unique renormalisation of the stochastic quantisation equation of YM in 3D that preserves gauge symmetries. This complements a recent result on the existence of such a renormalisation. Based on joint work with Hao Shen.

Wed, 30 Apr 2025
11:00
L5

Hydrodynamic limit of an active-passive lattice gas

Maria Bruna
(Mathematical Institute)
Abstract

In this talk, I will discuss a model mixture of active (self-propelled) and passive (diffusive) particles with non-reciprocal effective interactions (or forces that violate Newton’s third law). We derive the hydrodynamic PDE limit for the particle densities, which is not a Wasserstein gradient flow of any free energy, consistent with the microscopic model having non-equilibrium steady states. We study the emergence of collective behaviour, which includes phase separation and dynamical (travelling) steady states.

Wed, 12 Mar 2025
11:00
L4

Uniqueness of Dirichlet operators related to stochastic quantisation for the exp(φ)_{2}-model

Hiroshi Kawabi
(Keio University)
Abstract

In this talk, we consider Dirichlet forms related to stochastic quantisation for the exp(φ)_{2}-model on the torus. We show strong uniqueness of the corresponding Dirichlet operators by applying an idea of (singular) SPDEs. This talk is based on ongoing joint work with Hirotatsu Nagoji (Kyoto University).

Wed, 05 Mar 2025
11:00
L4

Scaling limits of stochastic transport equations on manifolds

Wei Huang
(Freie Universität Berlin)
Abstract

In this talk, I will present the generalization of scaling limit results for stochastic transport equations on torus by Flandoli, Galeati and Luo, to compact manifolds. We consider the stochastic transport equations driven by colored space-time noise(smooth in space, white in time) on a compact Riemannian manifold without boundary. Then we study the scaling limits of stochastic transport equations, tuning the noise in such a way that the space covariance of the noise on the diagonal goes to identity matrix but the covariance operator itself goes to zero, which includes the large scale analysis regime with diffusive scaling.

We obtain different scaling limits depending on the initial data. With space white noise as initial data, the solutions converge in distribution to the solution of a stochastic heat equation with additive noise. With square integrable initial data, the solutions of transport equation converge to the solution of the deterministic heat equation, and we give quantitative estimates on the convergence rate.

Wed, 26 Feb 2025
11:00
L4

Nonlinear rough Fokker--Planck equations

Fabio Bugini
(Technische Universitat Berlin)
Abstract

We present an existence and uniqueness result for nonlinear Fokker--Planck equations driven by rough paths. These equations describe the evolution of the probability distributions associated with McKean--Vlasov stochastic dynamics under (rough) common noise.  A key motivation comes from the study of interacting particle systems with common noise, where the empirical measure converges to a solution of such a nonlinear equation. 
Our approach combines rough path theory and the stochastic sewing techniques with Lions' differential calculus on Wasserstein spaces.

This is joint work with Peter K. Friz and Wilhelm Stannat.

Wed, 19 Feb 2025
11:00
L4

A new take on ergodicity of the stochastic 2D Navier-Stokes equations

Dr Jonas Tölle
(Aalto University)
Abstract

We establish general conditions for stochastic evolution equations with locally monotone drift and degenerate additive Lévy noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the e-property of the semigroup. Examples include the stochastic incompressible 2D Navier-Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic heat equation, as well as, stochastic semilinear equations such as the 1D stochastic Burgers equation.

Joint work with Gerardo Barrera (IST Lisboa), https://arxiv.org/abs/2412.01381

Wed, 29 Jan 2025
11:00
L4

Singularity of solutions to singular SPDEs.

Hirotatsu Nagoji
(Kyoto University)
Abstract

In this talk, we discuss the condition for the marginal distribution of the solution to singular SPDEs on the d-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearized equation. As applications of our result, we see the singularity of the Phi^4_3-measure with respect to the Gaussian free field measure and the border of parameters for the fractional Phi^4-measure to be singular with respect to the base Gaussian measure. This talk is based on a joint work with Martin Hairer and Seiichiro Kusuoka.

Wed, 22 Jan 2025
11:00
L6

Adapted Wasserstein distance between continuous Gaussian processes

Yifan Jiang
(Mathematical Institute)
Abstract
Adapted Wasserstein distance is a generalization of the classical Wasserstein distance for stochastic processes. It captures not only the spatial information but also the temporal information induced by the processes. In this talk, I will focus on the adapted Wasserstein distance between continuous Gaussian processes. An explicit formula in terms of their canonical representations will be given. These results cover rough processes such as fractional Brownian motions and fractional Ornstein--Uhlenbeck processes. If time permits, I will also show that the optimal coupling between two 1D additive fractional SDE is driven by the synchronous coupling of the noise.
We introduce a 'causal factorization' as an infinite dimensional Cholesky decomposition on Hilbert spaces. This naturally bridges the probabilistic notion 'causal transport' and the algebraic object 'nest algebra'.  Such a factorization is closely related to the (non)canonical representation of Gaussian processes which is of independent interest. This talk is based on a work-in-progress with Fang Rui Lim.
Wed, 04 Dec 2024
11:00
L4

Effective Mass of the Polaron and the Landau-Pekar-Spohn Conjecture

Chiranjib Mukherjee
(University of Münster)
Abstract

According to a conjecture by Landau-Pekar (1948) and by Spohn (1986), the effective mass of the Fröhlich Polaron should diverge in the strong coupling limit like a quartic power of the coupling constant. In a recent joint with R. Bazaes, M. Sellke and S.R.S. Varadhan, we prove this conjecture.

Wed, 20 Nov 2024
11:00
L4

Quadratic and $p^\mathrm{th}$ variation of stochastic processes through Schauder expansions

Yuchen Fan
(University of Oxford)
Abstract
We present a class of stochastic processes which admit a unique quadratic variation along any sequence of partitions $(\pi^n)_{n\geq 1}$ with $\sum_{n\geq 1}|\pi^n|<\infty$, which generalizes the previous results for finitely refining partitions. This class of processes contains some signed Takagi-Landsberg functions with random coefficients and standard Brownian motions, and these processes admit $\frac{1}{4}$-Hölder continuous version. We study the quadratic and $p^\mathrm{th}$ variation of signed Takagi-Landsberg functions with random coefficients. Finally, we seek some generalizations and applications of our results.


 

Wed, 13 Nov 2024
11:00
L4

Flow equation approach for the stochastic Burgers equation

Andrea Pitrone
(Mathematical Institute)
Abstract

I will present the basic idea of the flow equation approach developed by Paweł Duch to study singular stochastic partial differential equations. In particular, I will show how it can be used to prove the existence of a solution of the stochastic Burgers equation on the one-dimensional torus.

Wed, 06 Nov 2024
11:00
L4

Probabilistic Schwarzian Field Theory

Ilya Losev
(Cambridge University)
Abstract

Schwarzian Theory is a quantum field theory which has attracted a lot of attention in the physics literature in the context of two-dimensional quantum gravity, black holes and AdS/CFT correspondence. It is predicted to be universal and arise in many systems with emerging conformal symmetry, most notably in Sachdev--Ye--Kitaev random matrix model and Jackie--Teitelboim gravity.

In this talk we will discuss our recent progress on developing rigorous mathematical foundations of the Schwarzian Field Theory, including rigorous construction of the corresponding measure, calculation of both the partition function and a natural class of correlation functions, and a large deviation principle.

Wed, 23 Oct 2024
11:00
L4

Weak coupling limit for polynomial stochastic Burgers equations in $2d$

Da Li
(Mathematical Institute)
Abstract

We explore the weak coupling limit for stochastic Burgers type equation in critical dimension, and show that it is given by a Gaussian stochastic heat equation, with renormalised coefficient depending only on the second order Hermite polynomial of the nonlinearity. We use the approach of Cannizzaro, Gubinelli and Toninelli (2024), who treat the case of quadratic nonlinearities, and we extend it to polynomial nonlinearities. In that sense, we extend the weak universality of the KPZ equation shown by Hairer and Quastel (2018) to the two dimensional generalized stochastic Burgers equation. A key new ingredient is the graph notation for the generator. This enables us to obtain uniform estimates for the generator. This is joint work with Nicolas Perkowski.

Tue, 10 Oct 2023
11:00
Lecture Room 4, Mathematical Institute

DPhil Presentations

DPhil Students
Abstract

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time.