Stochastic Analysis Seminar

Abstract: We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint works with Amir Dembo, Tomoyuki Ichiba, Ioannis Karatzas, Soumik Pal and Ofer Zeitouni

It is a general belief that the heat kernels on fractals should exhibit highly oscillatory behaviors as opposed to the classical case of Riemannian manifolds.

For example, on a class of finitely ramified fractals, called (affine) nested fractals, a canonical ``Brownian motion" has been constructed and its transition density (heat kernel) $p_{t}(x,y)$ satisfies $c_{1} \leq t^{d_{s}/2} p_{t}(x,x) \leq c_{2}$ for $t \leq 1$ for any point $x$ of the fractal; here $d_{s}$ is the so-called spectral dimension. Then it is natural to ask whether the limit of this quantity as $t$ goes to 0 exists or not, and it has been conjectured NOT to exist by many people.

In this talk, I will present partial affirmative answers to this conjecture. First, for a general (affine) nested fractal, the non-existence of the limit is shown to be true for a ``generic" (in particular, almost every) point. Secondly, the same is shown to be valid for ANY point of the fractal in the particular cases of the $d$-dimensional standard Sierpinski gasket with $d\geq 2$ and of the $N$-polygasket with $N\geq 3$ odd, e.g. the pentagasket ($N=5$) and the heptagasket ($N=7$).

Abstract: First we provide a survey on the long-time behaviour of stochastic delay equations with bounded memory, addressing existence and uniqueness of invariant measures, Lyapunov spectra, and exponential growth rates.

Then, we study the very simple one-dimensional equation $dX(t)=X(t-1)dW(t)$ in more detail and establish the existence of a deterministic exponential growth rate of a suitable norm of the solution via a Furstenberg-Hasminskii-type formula.

Parts of the talk are based on joint work with Martin Hairer and Jonathan Mattingly.