Conservativeness of diffusion processes with drift

We show the conservativeness of the Girsanov transformed diffusion process by drift $b\in L^p(\mathbb{R} ^d\to\mathbb{R} ^d)$ with $p\geq4/(2-\sqrt{2\delta(\vert b\vert^2)/\lambda})$ 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 $d\geq3$.