Transience and Non-explosion of Certain Stochastic Newtonian Systems

Abstract

We give sufficient conditions for non-explosion and transience of the solution $(x_t, p_t)$ (in dimensions $\geq 3$) to a stochastic Newtonian system of the form $$\begin{cases} dx_t= p_t \, dt , \\ dp_t= -\frac{\partial V(x_t) }{\partial x} \, dt - \frac{ \partial c(x_t) }{ \partial x} \, d\xi_t,\end{cases}$$ where $\{\xi_t\}_{t\geq 0}$ is a $d$-dimensional Lévy process, $d\xi_t$ is an Itô differential and $c\in C^2(\mathbb{R}^d,\mathbb{R}^d)$, $V\in C^2(\mathbb{R}^d,\mathbb{R})$ such that $V\geq 0$.