Past Stochastic Analysis Seminar

In this talk we summary some recent progress on limit theorems for the Wasserstein distance of empirical measures of Markov processes. For symmetric diffusion processes on Riemannian manifold possibly with reflecting or killing boundary, the sharp convergence rate is derived with renormalization limit formulated by using the spectrum of the generator. Moreover, a general framework is established to estimate the convergence rate in Wasserstein distance of empirical measures for ergodic Markov processes.

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

Two hundred years ago, Robert Brown observed the statistics of the motion of grains of pollen in water. It took almost one hundred years for Einstein and others to develop an effective theory describing this motion as that of a random walker. In this talk, I will challenge a key implication of this well established theory. When studying systems with very large numbers of particles diffusing together, I will argue that the Einstein random walk theory breaks down when it comes to predicting the statistical behavior of extreme particles—those that move the fastest and furthest in the system. In its place, I will describe a new theory of extreme diffusion which captures the effect of the hidden environment in which particles diffuse together and allows us to interrogate that environment by studying extreme particles. I will highlight one piece of mathematics that led us to develop this theory—a non-commutative binomial theorem—and hint at other connections to integrable probability, quantum integrable systems and stochastic PDEs.

In this talk, we give a construction of the Abelian Yang--Mills--Higgs measure on the two-dimensional torus via the variational approach initiated by Barashkov--Gubinelli. The construction is carried out through a disintegration of measures: we first construct the conditional Higgs measure given a rough gauge field, and then construct the gauge field marginal. This leads to iterated variational problems, one for the Higgs field and one for the gauge field. At the technical level, the starting point is the construction of the renormalised covariant Laplacian associated to a rough gauge field, together with the study of its resolvent. This allows us to define the covariant Gaussian free field, which serves as the reference Gaussian field for the conditional Higgs measure. Finally, we analyse the ratios of determinants that arise from the change-of-measure formula for Gaussian measures. This is joint work with Nikolay Barashkov, Ajay Chandra, Ilya Chevyrev, and Andreas Koller.
 

I will start the talk by discussing renormlisation group in perturbative algebraic quantum field theory (pAQFT) and its non-perturbative incarnation acting on the Buchholz-Fredenhagen dynamical C*-algebra. I will also explain how pAQFT can be used to derive functional renormlisation group (FRG) equations that generalize Wetterich equations to globally hyperbolic Lorentzian manifolds and arbitrary states (beyond the usual FRG in the vacuum).

I will review how the large N expansion can be used in the context of the renormalisation group to probe some strongly coupled regimes. In particular, I will discuss a work by Gawedzki and Kupiainen where the authors study the three-dimensional non-Gaussian infrared fixed point of Phi^4 in the case of a hierarchical model of rank-one covariance, and explain how their approach could generalise to more realistic models. 

This is a joint work with Ajay Chandra.  

I will discuss a 2-dimensional model of random walk in random environment known as line model. The environment is described by two independent families of i.i.d. random variables dictating rates of jumps in vertical, respectively horizontal directions, and whose values are constant along vertical, respect. horizontal lines. When jump rates are heavy-tailed in one of the directions, the random walk becomes superdiffusive in that direction, with an explicit scaling limit written as a two-dimensional Brownian motion time-changed (in one of the components) by a process introduced by Kesten and Spitzer in 1979. I will present ideas of the proof of this result, which relies on appropriate time-change arguments.  In the case of a fully degenerate environment, I will present a sufficient condition for non-explosion of the process (which is also believed to be sharp), as well as conjectures on the associated scaling limit.

This is based on joint work with J.-D. Deuschel (TU Berlin). 

In recent years, a more top-down approach to renormalisation for singular SPDEs has emerged within the theory of regularity structures, based on regularity structures of multi-indices. This approach adopts a geometric viewpoint, aiming to stably parametrise the solution manifold rather than the larger space of renormalised objects that typically arise in fixed-point formulations of the equation. While several works have established the construction of the renormalised data (the model) in this setting, less has been shown with regards to the corresponding solution theory since the intrinsic nature of the model leads to renormalised data that is too lean to apply Hairer’s fixed-point approach.

In this talk, I will discuss past and ongoing work with L. Broux and F. Otto addressing this issue for the Phi^4 equation in its full subcritical regime. We establish local and global well-posedness within the framework of regularity structures of multi-indices; first in a space-time periodic setting and subsequently in domains with Dirichlet boundary conditions.

We give an introduction to a rigorous renormalization group analysis of the sine-Gordon model with a focus on deriving the lowest-order beta function.

The dynamical Phi^4_3 model is a stochastic partial differential equation that arises in quantum field theory and statistical physics. Owing to the singular nature of the driving noise and the presence of a nonlinear term, the equation is inherently ill-posed. Nevertheless, it can be given a rigorous meaning, for example, through the framework of regularity structures. On compact domains, standard arguments show that any solution converges to the equilibrium state described by the unique invariant measure. Extending this result to infinite volume is highly nontrivial: even for the lattice version of the model, uniqueness holds only in the high-temperature regime, whereas at low temperatures multiple phases coexist.

We prove that, when the mass is sufficiently large or the coupling constant sufficiently small (that is, in the high-temperature regime), all solutions of the dynamical Phi^4_3 model in infinite volume converge exponentially fast to the unique stationary solution, uniformly over all initial conditions. In particular, this result implies that the invariant measure of the dynamics is unique, exhibits exponential decay of correlations, and is invariant under translations, rotations, and reflections.

Joint work with Martin Hairer, Jaeyun Yi, and Wenhao Zhao.

I will discuss the paper "Construction of Gross-Neveu model using Polchinski flow equation" by Pawel Duch (https://arxiv.org/abs/2403.18562).

In this talk, we will explore the emerging role of generative AI in mathematical research. Building on insights from the “Malliavin–Stein experiment”, carried out in collaboration with Charles-Philippe Diez and Luis Da Maia, we will discuss our experience and reflect on how AI might influence the way mathematics is conceived, proven, and created.

The 2d (massless) Sinh-Gordon model is amongst the simplest 2d quantum field theories that are expected to be integrable (= infinitely many symmetries), but without conformal symmetry. In this talk I will explain a rigorous construction of this model and its vertex correlations (= Laplace transforms) on the infinite cylinder using probability theory. A fundamental role is played by the Sinh-Gordon Hamiltonian and I will explain how the theory of Gaussian multiplicative chaos can be used to analyze this linear map. This talk will be based on joint work with Colin Guillarmou and Vincent Vargas.

We derive a lower bound, independent of the initial condition, for the solution of the KPZ equation on the torus, using its representation as the value function of a stochastic control problem.

With the same techniques we also prove a bound for its oscillation, again independent of initial conditions, which is related to Harnack's inequality for the (rough) heat equation.

This tutorial will be a hands-on introduction to proving theorems in Lean, using its mathematical library Mathlib. It will not assume any previous knowledge about formal theorem provers. We will discover the Lean language, learn how to read a statement and a proof, and learn the essential "tactics" one can use to prove theorems in Lean.

Participants should come with a computer, and it would be best if they could install Lean before the tutorial by following the instructions at https://lean-lang.org/install/ . The installation should be easy and takes only a few minutes.

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. 

In this talk I will discuss some recent results that allow to approximate a real singularity given by polynomial equations of degree d (e.g. the zero set of a polynomial, or the number of its critical points of a given Morse index) with a singularity which is diffeomorphic to the original one, but it is given by polynomials of degree O(d^(1/2)log d).
The approximation procedure is constructive (in the sense that one can read the approximating polynomial from a linear projection of the given one) and quantitative (in the sense that the approximating procedure will hold for a subset of the space of polynomials with measure increasing very quickly to full measure as the degree goes toinfinity).

The talk is based on joint works with P. Breiding, D. N. Diatta and H. Keneshlou