PERTURBATION OF DIRICHLET FORMS AND STABILITY OF FUNDAMENTAL SOLUTIONS

Let {X-t}(t >= 0) be the alpha-stable-like or relativistic alpha-stable-like process on R-d generated by a certain symmetric jump-type regular Dirichlet form (epsilon, 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 mu be a positive smooth Radon measure in a certain class and consider the perturbed form epsilon(mu)(u, u) = epsilon(u, u) - (u, u)(mu). Denote by p(mu)(t, x, y) the fundamental solution associated with epsilon(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.