The Polyak-Łojasiewicz inequality (PŁI) in $\mathbb{R}^d$ is a natural condition for proving convergence of gradient descent algorithms. In this talk, I will discuss an analogue of PŁI on the space of probability measures $\mathcal{P}(\mathbb{R}^d)$ and I will explain why it is a natural condition for showing exponential convergence of a class of birth-death processes. I will also discuss the connection between those birth-death processes and a class of interacting particle systems that can be used in the problem of approximate sampling and in mean-field optimization.

Over the last twenty years there has been significant progress in the well-posedness study of singular stochastic PDEs in both parabolic and dispersive settings. In this talk, I will discuss some convergence problems for singular stochastic nonlinear PDEs. In a seminal work, Da Prato and Debussche (2003) established well-posedness of the stochastic quantization equation, also known as the parabolic $\phi_2^{k+1}$-model in the two-dimensional case. More recently, Gubinelli, Koch, Oh, and Tolomeo proved the corresponding well-posedness for the canonical stochastic quantization equation, also known as the hyperbolic $\phi_2^{k+1}$-model in the two-dimensional case. In the first part of this talk, I will describe convergence of the hyperbolic $\phi_2^{k+1}$-model to the parabolic $\phi_2^{k+1}$-model. In the dispersive setting, Bourgain (1996) established well-posedness for the dispersive $\phi_2^4$-model (=deterministic cubic nonlinear Schrödinger equation) on the two-dimensional torus with Gibbsian initial data. In the second part of the talk, I will discuss the convergence of the stochastic complex Ginzburg-Landau equation (= complex-valued version of the parabolic $\phi_2^4$-model) to the dispersive $\phi_2^4$-model at statistical equilibrium.

Stein method is a three-step program to measure distance between a generic probability measure of that a given target probability distribution. The first step is to find a suitable (differential) Stein operator that characterises the target distribution. In this talk, we introduce the novel notion of an algebraic polynomial Stein operator and show that every polynomial Stein operator of a Gaussian polynomial target distribution in dimension one is algebraic. This opens a new door to find algorithmically Stein operators of the complex target distributions in higher Wiener chaoses. We also discuss the class PSO(N) of the polynomial Stein operators associated to the standard Gaussian distribution and how it connects Stein method to non-commutative algebra. The talk is based on a series of joint works with Dario Gasbarra (Helsinki) and Robert Gaunt (Manchester).

In this talk I will talk about the analytical properties of a class of stochastic shallow water (SRSW) models which includes SRSW models derived using the Location Uncertainty (Mémin, 2014) approach. We prove that there exists a unique maximal strong solution and a unique global weak solution, using a method which relies on a Cauchy approximating sequence. We show that although the strong solution exists in a high order Sobolev space, it suffices to prove the Cauchy property in L^2. Comparisons with other types of SRSW models will also be made, and we will discuss conditions under which the underlying strong solution can become global with positive probability.

This is joint work with Dan Crisan and Étienne Mémin and is partly based on Analytical Properties for a Stochastic Rotating Shallow Water Model under Location Uncertainty , to appear in Journal of Mathematical Fluid Mechanics.