Summer Symposium on Stochastic Systems 2026

In this talk, we study the averaging principle, central limit theorem, and diffusion approximation for a class of slow-fast stochastic differential equations with state-dependent switching. Our main techniques are based on the Poisson equation associated with the Markov chain, the Kolmogorov equation, and the martingale problem. The achieved convergence orders are all proven to be optimal through several examples. Lastly, we investigate the numerical approximation of such systems via the heterogeneous multiscale method. This talk is based on joint works with Yingchao Xie, Mingkun Ye and Zuozheng Zhang.

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.

We consider the cubic nonlinear Schrödinger equation on the three-dimensional ball with radial random initial data in a supercritical regime. We construct probabilistic strong solutions, substantially improving a previous result of Bourgain and Bulut. The criticality arises from a divergent contribution that, unlike for the NLS on the torus, cannot be removed by a global transformation. We overcome this obstruction by introducing a refined gauge transform which leaves the equation unchanged. I will then explain how this gauge transform can be incorporated into the random averaging operator ansatz to solve the problem. This is joint work with Nicolas Burq, Nicolas Camps, and Nikolay Tzvetkov.

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.

The criterion will be illustrated on the 2D Navier–Stokes system and the complex Ginzburg–Landau equations perturbed by bounded degenerate forces such as random Haar series. We will conclude with recent developments concerning the Lagrangian trajectories of a fluid particle: exponential mixing of the Lagrangian process and the strict positivity of the top Lyapunov exponent (Lagrangian chaos), obtained via controllability of the triple process formed by the velocity field, the particle, and its Jacobian.

Based on joint works with S. Kuksin and A. Shirikyan, and with D. Zhang and C. Zhou.

We study the scaling limit of the disordered pinning model in correlated Gaussian environments. The tail probability of the underlying renewal process has a polynomial decay with exponent $\alpha>0$. The covariance of the Gaussian disorder $\{\omega_n\}_{n\in\mathbb N}$ is given by $\cov(\omega_n,\omega_m)=|n-m|^{2H-2}$ with $H\in(0,1)$. Assuming $\alpha\in(0,\frac12)$, $H\in(\frac12,1)$ and $\alpha+2H>2$, we show that the partition function of the disordered pinning model, under an appropriate scaling, converges in distribution to the $L^1$-solution of the fractional stochastic heat equation driven by noise colored in time and delta in space. In particular, it is known that the solution is not $L^2$-integrable. This talk is based on joint work with Le Chen, Meng Wang and Ran Wei.

Joint work with Lorenzo Zambotti and Jonathan Mattingly. I will discuss unique ergodicity for the skew stochastic heat equation

$$\partial_t u = \Delta u + \delta_0(u) + \xi,$$

where $\delta_0$ is the Dirac delta and $\xi$ is space-time white noise. Since the drift of this SPDE is so singular, the standard methods for obtaining ergodicity do not work and 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 will review some recent results on the well-posedness theory and noise-regularization effect for stochastic Zakharov system in dimensions $d\geq 3$. Our focus is particularly directed at the large data regime where the global existence and long-time dynamics of deterministic Zakharov system is largely open. The noise-regularization effect on scattering is proved up to borderlines dictated by the regularity threshold of noise. These works are in joint with Sebastian Herr, Michael Röckner, Martin Spitz and Zhenqi Zhao.

For every $0<\kappa<\sqrt{5}-2$, we prove global existence for the two-dimensional generalized parabolic Anderson model on the whole plane $\mathbb{R}^2$ with nonlinearity $F\in C_b^2(\mathbb{R})$, driven by an enhanced noise $(\eta,\Psi)$. The noise $\eta$ has polynomially weighted spatial Besov–Hölder regularity $-1-\kappa$, and $\Psi$ is the corresponding renormalized second-order object. If $F''$ is globally Lipschitz, the solution is unique. The proof combines a weight-compatible annular high–low decomposition with a paracontrolled transport representation. The final remainder is estimated simultaneously in a weighted $L^\infty$ norm and in a higher-order weighted parabolic Hölder norm, using two strictly different polynomial weights. This weight gap absorbs the polynomial losses generated by the enhanced noise, the localization procedure, and the transport coefficient. Several refinements of earlier work allow the maximum-principle and Schauder estimates to yield a global a priori bound for a larger range of $\kappa$. Uniqueness is proved in a time-dependent exponentially weighted topology.

In this talk, we will show various functional inequalities for mean-field interacting particle systems and McKean–Vlasov SDEs under non-globally dissipative conditions. These inequalities include Poincaré, transportation, logarithmic Sobolev, concentration, and covariance inequalities. They provide powerful tools for characterizing exponential ergodicity, concentration phenomena, and error analysis for stochastic particle algorithms.

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 $W^{-1,\infty}$, 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 $\gamma > 1-1/(2H)$ 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.

Inspired by recent progress on the theory of regularization by noise for 2D fluid equations, we consider the stochastic 2D inviscid Boussinesq system driven by Kraichnan transport noise with parameter $\alpha\in(0,1/2)$. It is shown that the noise provides anomalous regularity for weak solutions with $L^p$ initial data, enabling us to prove pathwise uniqueness of solutions. This demonstrates that suitable noise improves the solution theory of the 2D inviscid Boussinesq system. The talk is based on a joint work with Dr. Shuaijie Jiao.

We study the convergence of a generic tamed Euler-Maruyama (EM) scheme for the kinetic type stochastic differential equations (SDEs) (also known as second order SDEs) with singular coefficients in both weak and strong probabilistic senses. We show that when the drift exhibits a relatively low regularity compared to the state of the art, the singular system is well-defined both in the weak and strong probabilistic senses. Meanwhile, the corresponding tamed EM scheme is shown to converge at the rate of 1/2 in both the weak and the strong senses.

In this talk, we study the two-dimensional quadratic nonlinear wave equation (NLW) with Gaussian initial data in the regime where one of the stochastic objects arising in the construction of solutions has infinite variance. This regime occurs before the critical regularity predicted by the probabilistic scaling heuristics of Deng–Nahmod–Yue (2023). Consequently, standard probabilistic methods for constructing solutions fail.

Our main result shows that, for suitably renormalized, frequency-truncated Gaussian initial data, the corresponding solutions to the NLW converge locally in time to solutions of a stochastic NLW with zero initial data.

In particular, an interesting phenomenon emerges: the randomness propagates from the initial condition to the equation itself, giving rise to an effective time-correlated space-time Gaussian noise whose covariance exhibits a Bessel-type structure. Our proof combines paracontrolled calculus, the fourth moment method, and multilinear dispersive smoothing.

This talk is based on joint work with Guopeng Li (BIT), Jiawei Li (University of Edinburgh), Tadahiro Oh (BIT), and Nikolay Tzvetkov (ENS Lyon).