Past Stochastic Analysis Seminar

In stochastic optimal control and conditional generative modelling, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and path-space integration by parts, that enables the development of methods robust to such singular rewards. This allows our approach to handle a broad range of applications, including classification, diffusion bridges, and conditioning without the need for artificial observational noise. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. 

We begin with motivation on how the study of SPDEs are relevant when interested in fluctuations of particle systems. 

We then present a law of large numbers, central limit theorem and large deviations principle for the generalised Dean--Kawasaki SPDE with truncated noise. 

Our main contribution is the ability to consider the equation on a general $C^2$-regular, bounded domain with Dirichlet boundary conditions. On the particle level the boundary condition corresponds to absorption and injection of particles at the boundary.

The work is based on discussions with Benjamin Fehrman and can be found at https://arxiv.org/pdf/2504.17094 

Conformal welding, the process of glueing together Riemann surfaces along their boundaries, has recently played a prominent role in probability theory. In this talk, I will discuss two examples, namely the welding associated with random Jordan curves (SLE(k) loops) and particularly their limit as k tends to zero, and the welding of random trees (such as the CRT).

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.

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.

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.

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.

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).

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.

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.

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

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.

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.

I will present TeXmacs (www.texmacs.org), a document preparation system with structured editing and high quality typography, designed to create technical documents. 

For more details see https://mgubi.github.io/docs/programming.html

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.

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.

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.

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.

Simple mathematical models have had remarkable successes in biology, framing how we understand a host of mechanisms and processes. However, with the advent of a host of new experimental technologies, the last ten years has seen an explosion in the amount and types of data now being generated. Increasingly larger and more complicated processes are now being explored, including large signalling or gene regulatory networks, and the development, dynamics and disease of entire cells and tissues. As such, the mechanistic, mathematical models developed to interrogate these processes are also necessarily growing in size and complexity. These detailed models have the potential to provide vital insights where data alone cannot, but to achieve this goal requires meeting significant mathematical challenges. In this talk, I will outline some of these challenges, and recent steps we have taken in addressing them.

It is well known that a rough path is uniquely determined by its signature (the collection of global iterated path integrals) up to tree-like pieces. However, the proof the uniqueness theorem is non-constructive and does not give us information about how quantitative properties of the path can be explicitly recovered from its signature. In this talk, we examine the quantitative relationship between the local p-variation of a rough path and the tail asymptotics of its signature for the simplest type of rough paths ("line segments"). What lies at the core of the work a novel technique based on the representation theory of complex semisimple Lie algebras. 

This talk is based on joint work with Horatio Boedihardjo and Nikolaos Souris

We discuss some results on integration of ``rough differential forms'', which are generalizations of classical (smooth) differential forms to similar objects involving Hölder continuous functions that may be nowhere differentiable. Motivations arise mainly from geometric problems related to irregular surfaces, and the techniques are naturally related to those of Rough Paths theory. We show in particular that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions (of Stratonovich or Ito type) and by summing over a refining sequence of partitions, leading to a two-dimensional extension of the classical Young integral, that coincides with the integral introduced recently by R. Züst. We further show that Stratonovich sums gives an advantage allowing to weaken the requirements on Hölder exponents, and discuss some work in progress in the stochastic case. Based on joint works with E. Stepanov, G. Alberti and I. Ballieul.

Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as high-temperature limits of thermal states in many-body quantum mechanics.

In our work, we apply a perturbative expansion in the interaction. This expansion is then analysed by means of Borel resummation techniques. In two and three dimensions, we need to apply a Wick-ordering renormalisation procedure. Moreover, in one dimension, our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the cubic nonlinear Schrödinger equation. This is based partly on joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein.

Neural networks are undoubtedly successful in practical applications. However complete mathematical theory of why and when machine learning algorithms based on neural networks work has been elusive. Although various representation theorems ensures the existence of the ``perfect’’ parameters of the network, it has not been proved that these perfect parameters can be (efficiently) approximated by conventional algorithms, such as the stochastic gradient descent. This problem is well known, since the arising optimisation problem is non-convex. In this talk we show how the optimization problem becomes convex in the mean field limit for one-hidden layer networks and certain deep neural networks. Moreover we present optimality criteria for the distribution of the network parameters and show that the nonlinear Langevin dynamics converges to this optimal distribution. This is joint work with Kaitong Hu, Zhenjie Ren and Lukasz Szpruch.