Abstract: In this talk, we will present recent results on the quantitative study stochastic interacting particle systems and of their mean field limit. We will start by exploring the relationship between the large $N$ limit of the constant in the Logarithmic Sobolev Inequality and the presence or absence of phase transitions for the mean field PDE. We will then present methods for computing both stable and unstable stationary states of the mean field PDE and for steering the dynamics towards a chosen steady state using optimal control methodologies. Finally, we will study the problem of inferring the interaction potential from discrete space-time observations of the McKean-Vlasov PDE.

Abstract: We consider a random graph in which vertices can have one of two possible colours. Each vertex switches its colour at a rate that is proportional to the number of vertices of the other colour to which it is connected by an edge. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same colour or not. The resulting double-dynamics is an example of co-evolution.
We prove that, in the limit as the graph size tends to infinity and the graph becomes dense, the graph process converges, in a suitable path topology, to a limiting Markov process that lives on a certain subset of the space of coloured graphons. In the limit, the density of each vertex colour evolves according to a Fisher-Wright diffusion driven by the density of the edges, while the underlying edge connectivity structure evolves according to a stochastic flow whose drift depends on the densities of the two vertex colours.

Joint work with Siva Athreya (Bangalore) and Adrian Röllin (Singapore).

Abstract: Regularisation by noise refers to the phenomenon of certain non-linear dynamical systems behaving better in the presence of a noisy (stochastic) perturbation compared to their deterministic counterpart. In this talk, we will discuss such phenomena for differential equations of the form
\begin{align*} dX_t &= f(X_t)dt, \\ X_0 &= x. \end{align*} It is well known that the above equation admits a unique solution if $f$ is Lipschitz continuous, and this is essential sharp: if $f$ is only $\alpha$-Hölder continuous for some $\alpha \in (0,1)$, one might have infinitely many solutions and if $f$ is not even continuous, it might not have solutions at all. We will consider equations of the form \begin{align} dX_t &= f(X_t)dt + \sigma(X_t)dB_t^H, \\ X_0 &=x, \end{align} where $B^H$ is a fractional Brownian motion of Hurst parameter $H \in (1/3,1/2)$. We will see that this equation admits a unique solution for quite irregular $f$, provided that $\sigma$ is bounded away from zero. More precisely, $f$ does not even need to be a function but merely a Schwartz distribution of regularity $\alpha > 1 - (1/2H)$ (notice that $\alpha$ can be negative) in the Besov scale $\mathcal{B}_{\infty,\infty}^{\alpha}$. We will discuss what is meant by a solution in this case and we will present the main ideas which rely on the theory of rough paths, Malliavin calculus, and the stochastic sewing lemma. Our result provides a multiplicative noise analogue to a result of Catellier-Gubinelli in 2016.

Joint work with M. Gerencsér, K. Lê, and C. Ling.