Perturbation of Dirichlet forms and stability of fundamental solutions

Abstract

Let $\{X_{t}\}_{t \geq 0}$ be the $\alpha$-stable-like or relativistic $\alpha$-stable-like process on $\boldsymbol{R}^{d}$ generated by a certain symmetric jump-type regular Dirichlet form $(\mathcal{E, F})$. It is known in [5-7] that the transition probability density $p(t, x, y)$ of $\{X_{t}\}_{t \geq 0}$ admits the two-sided estimates. Let $\mu$ be a positive smooth Radon measure in a certain class and consider the perturbed form $\mathcal{E}^{\mu}(u, u) = \mathcal{E}(u, u) - (u, u)_\mu$. Denote by $p^{\mu}(t, x, y)$ the fundamental solution associated with $\mathcal{E}^{\mu}$. In this paper, we establish a necessary and sufficient condition on $\mu$ for $p^{\mu}(t, x, y)$ having the same two-sided estimates as $p(t, x, y)$ up to positive constants.