Tohoku Mathematical Journal

Perturbation of Dirichlet forms and stability of fundamental solutions

Masaki Wada

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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.

Article information

Tohoku Math. J. (2), Volume 66, Number 4 (2014), 523-537.

First available in Project Euclid: 21 May 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 60J75: Jump processes 35J10: Schrödinger operator [See also 35Pxx] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 31C25: Dirichlet spaces

Dirichlet forms perturbation heat kernel Markov processes


Wada, Masaki. Perturbation of Dirichlet forms and stability of fundamental solutions. Tohoku Math. J. (2) 66 (2014), no. 4, 523--537. doi:10.2748/tmj/1432229195.

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