On the Generalized Feynman–Kac Transformation for Nearly Symmetric Markov Processes

Li Ma and Wei Sun ()
Additional contact information
Li Ma: Concordia University
Wei Sun: Concordia University

Journal of Theoretical Probability, 2012, vol. 25, issue 3, 733-755

Abstract: Abstract Suppose that X is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on L 2(E;m). For $u\in D(\mathcal{E})$ , we have Fukushima’s decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$ . In this paper, we investigate the strong continuity of the generalized Feynman–Kac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$ . Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$ . Denote by J 1 the dissymmetric part of the jumping measure J of $(\mathcal{E},D(\mathcal{E}))$ . Under the assumption that J 1 is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant α 0≥0 such that $\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}$ for every t>0. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on L 2(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J 1 is finite.

Keywords: Non-symmetric Dirichlet form; Generalized Feynman–Kac semigroup; Strong continuity; Lower semi-bounded; Beurling–Deny formula; LeJan’s transformation rule; 31C25; 60J57 (search for similar items in EconPapers)
Date: 2012
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-010-0318-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:25:y:2012:i:3:d:10.1007_s10959-010-0318-3

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-010-0318-3

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().