Conservativeness of diffusion processes with drift

Abstract:

We show the conservativeness of the Girsanov transformed diffusion process by drift $b\in L^p(\mathbb {R}^d\!\to \!\mathbb {R}^d)$ with $p\geq 4/(2-\sqrt {2\delta (\vert b\vert ^2)/\lambda })$ or $p>4d/(d+2)$, or $p=2$ if $\vert b\vert ^2$ is of the Hardy class with sufficiently small coefficient of energy $\delta (\vert b\vert ^2)<\lambda /2$. Here $\lambda >0$ is the lower bound of the symmetric measurable matrix-valued function $a(x):=(a_{i,j}(x))_{i,j}$ appearing in the given Dirichlet form. In particular, our result improves the conservativeness of the transformed process by $b\in L^d(\mathbb {R}^d\!\to \!\mathbb {R}^d)$ when $d\geq 3$.