Tohoku Mathematical Journal
2014
December
SECOND SERIES VOL. 66, NO. 4

Tohoku Math. J.
66 (2014), 523-537

Title PERTURBATION OF DIRICHLET FORMS AND STABILITY OF FUNDAMENTAL SOLUTIONS

Author Masaki Wada

(Received September 25, 2013, revised October 28, 2013)
Abstract. Let $\{X_{t}\}_{t \geq 0}$ be the $\alpha$\textit{-stable-like} or \textit{relativistic} $\alpha$\textit{-stable-like} process on $\boldsymbol{R}^{d}$ generated by a certain symmetric jump-type regular Dirichlet form $(\mathcal{E, F})$. It is known in \cite{CK03, CKK11} 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.

Mathematics Subject Classification. Primary 60J45; Secondary 60J75, 35J10, 60J35, 31C25.
Key words and phrases. Dirichlet forms, perturbation, heat kernel, Markov processes.

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