Markov Processes with Equal Capacities

P. J. Fitzsimmons
Additional contact information
P. J. Fitzsimmons: University of California

Journal of Theoretical Probability, 1999, vol. 12, issue 1, 271-292

Abstract: Abstract Let X and $$\hat X$$ be transient standard Markov processes in weak duality with respect to a σ-finite measure m. Let (Y, Ŷ, μ) be a second “dual pair” with the same state space E as (X, $$\hat X$$ , m). Let Cap X and Cap Y be the 0-order capacities associated with (X, $$\hat X$$ , m) and (Y, Ŷ, μ), and let V and $$\hat V$$ denote the potential kernels for Y and Ŷ. Assume that singletons are polar with respect to both X and Y, and that semipolar sets are of capacity zero for both dual pairs. We show that if Cap X (B)=Cap Y (B) for every Borel subset of E then there is a strictly increasing continuous additive functional D=(D t) t≥0 of (X, $$\hat X$$ , m) such that $$U_D (x,dy) + \hat U_D (x,dy) = V(x,dy) + \hat V(x,dy)$$ with the exception of a capacity-zero set of x's. Here U D (resp. Û D) is the potential kernel of the time-changed process $$X_{D^{ - 1} (t)}$$ (resp. $$\hat X_{D^{ - 1} (t)} )$$ , t≥0. In particular, if both X and Y are symmetric processes, then the equality of the capacities Cap X and Cap Y implies that X and Y are time changes of one another. This derivation rests on a generalization of a formula of Choquet concerning the “differentiation” of capacities. In the symmetric case, our main result extends a theorem of Glover et al.(23)

Keywords: Capacity; quasi-continuous; symmetric process; duality; hitting probability (search for similar items in EconPapers)
Date: 1999
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1023/A:1021713114477 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:12:y:1999:i:1:d:10.1023_a:1021713114477

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

DOI: 10.1023/A:1021713114477

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 ().