Summer Symposium on Stochastic Systems 2026

In this talk, we establish a Bismut–Elworthy–Li (BEL) type formula which provides a probabilistic representation (relying only on the corresponding SDE) for quasi-linear PDEs with gradient-dependent nonlinearities. The BEL formula is proved using approximations of PDEs and a stochastic fixed point equation. Moreover, based on the above BEL formula, we construct a Monte Carlo approximation method for quasi-linear PDEs. The rate of convergence and error analysis are obtained as well.

In this talk, we discuss heat kernel estimates for local and nonlocal Schrödinger operators with supercritical killing potentials. A prototypical example is −(−Δ)^s − κ|x|^{−2(s+γ)}, κ, γ > 0. We present both probabilistic and analytic approaches to the study of these operators. We highlight the fundamental differences in the results for the local and nonlocal settings and examine how they differ from those in the critical case (γ = 0). The talk is based on joint work with Renming Song (UIUC) and Panki Kim (SNU).

A basic question in the theory of randomly forced PDEs is the uniqueness and mixing of the stationary measure: does the law of the solution converge, as time goes to infinity, to a limit that is independent of the initial state? When the noise acts directly on all determining modes of the dynamics, this question is by now well understood. The degenerate case, when the noise is bounded and acts directly only on a few low Fourier modes, is far more delicate and is the subject of this mini-course.

We will present an approach in which the ergodic properties of the stochastic system are derived from controllability properties of the associated deterministic one. The main result is an abstract criterion for uniqueness of the stationary measure and exponential mixing in the dual-Lipschitz metric, valid for a class of discrete-time Markov processes obtained by restricting the flow to integer times. The assumptions are of two kinds: approximate controllability of the nonlinear system by controls belonging to the support of the noise, and density of the image of the linearised operator. The proof is based on a coupling argument that exploits these controllability properties.

Joint work with Lorenzo Zambotti and Jonathan Mattingly. I will discuss unique ergodicity for the skew stochastic heat equation ∂_t u = Δu + δ_0(u) + ξ, where δ_0 is the Dirac delta and ξ is space-time white noise. Since the drift of this SPDE is so singular, the standard methods for obtaining ergodicity do not work and the strong Feller property is not clear. This problem was submitted to the second batch of the First Proof project, which tests AI systems on unpublished research problems. The human proof is based on asymptotic strong Feller and coupling arguments. To our great surprise, AI systems also solved this problem and found two proofs which are different from ours. One uses a novel “fake” Girsanov entropy estimate, and the other uses the order-preservation property in a very tricky way. I will explain the main ideas behind all three approaches.

In this talk, we study distributional properties of diffusion processes with singular divergence-free drifts. We discuss when the law of such a diffusion is mutually singular with respect to Wiener measure, and when the classical Varadhan formula remains valid at critical regularity. In particular, we show that the Varadhan formula holds under suitable Lorentz-space assumptions. We also present examples revealing a phase transition: at more singular distributional levels, such as W^{−1,∞}, the Euclidean Varadhan constant may fail.

We investigate the long-time behavior of singular SDEs driven by fractional Brownian motion whose drift consists of a dissipative Lipschitz term and a singular term of regularity γ > 1 − 1/(2H) in Besov-Hölder scales. We establish well-posedness and, through a Markovian enhancement, existence of an invariant measure. If the singular contribution is sufficiently small, we prove exponential contraction of solutions, and thereby, uniqueness of the invariant measure. Our methods rely on uniform pathwise estimates which utilise together the dissipativity of the drift and the regularisation effect of the noise. Some recent progress on extending the results to polynomial-type dissipative drifts will be discussed. Based on joint works with Konstantinos Dareiotis and El Mehdi Haress.