Perturbation of dirichlet forms and stability of fundamental solutions

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