Self-intersection local times for Volterra Gaussian processes in stochastic flows with interaction

Abstract

In this paper, we study self-intersection local times for a stochastic process $x(u(\cdot),t)$, where $u$ is a Gaussian process of the form $u(t)=\int^t_0k(t,s)\mathrm{d}{w(s)}$, $k$ is a deterministic kernel of the Volterra type, $w$ is a Wiener process, and $x$ is a solution to the \emph{equation with interaction}. Equations with interaction are a class of interacting particle system described by stochastic differential equations whose coefficients depend on a random measure (initial distribution of particles) transformed by the flow of solutions. Considering the occupation measure of $u$ as the initial condition for the equation with interaction allows us to define a stochastic flow with interaction driven by self-intersection local times of the process $u$. The study of such stochastic differential equations whose coefficients carry information about the geometric properties of curves is new. They previously appeared only for deterministic differential equations and smooth curves, where the geometric characteristics typically considered are length, curvature, and so on. In this paper, we prove the existence of multiple self-intersection local times for the process $x(u(\cdot),t)$ and establish a ``change of variable formula" that allows us to describe self-intersection local times for the process $x(u(\cdot),t)$ in terms of the weighted self-intersection local times for the process $u.$ We describe the corresponding asymptotics of the self-intersection local times for $x(u(\cdot),t)$ for large $t$. Moreover, the existence of weighted self-intersection local times is established for a large class of unbounded weights, which is of independent interest.