Tohoku Mathematical Journal

Perturbation of Dirichlet forms and stability of fundamental solutions

Masaki Wada

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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.

Article information

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

Dates
First available in Project Euclid: 21 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1432229195

Digital Object Identifier
doi:10.2748/tmj/1432229195

Mathematical Reviews number (MathSciNet)
MR3350282

Zentralblatt MATH identifier
1328.60182

Subjects
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

Keywords
Dirichlet forms perturbation heat kernel Markov processes

Citation

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. https://projecteuclid.org/euclid.tmj/1432229195


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