The Annals of Probability

Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms

Masatoshi Fukushima and Toshihiro Uemura

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Let E be a locally compact separable metric space and m be a positive Radon measure on it. Given a nonnegative function k defined on E × E off the diagonal whose anti-symmetric part is assumed to be less singular than the symmetric part, we construct an associated regular lower bounded semi-Dirichlet form η on L2(E; m) producing a Hunt process X0 on E whose jump behaviours are governed by k. For an arbitrary open subset DE, we also construct a Hunt process XD,0 on D in an analogous manner. When D is relatively compact, we show that XD,0 is censored in the sense that it admits no killing inside D and killed only when the path approaches to the boundary. When E is a d-dimensional Euclidean space and m is the Lebesgue measure, a typical example of X0 is the stable-like process that will be also identified with the solution of a martingale problem up to an η-polar set of starting points. Approachability to the boundary ∂ D in finite time of its censored process XD,0 on a bounded open subset D will be examined in terms of the polarity of ∂ D for the symmetric stable processes with indices that bound the variable exponent α(x).

Article information

Ann. Probab., Volume 40, Number 2 (2012), 858-889.

First available in Project Euclid: 26 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J75: Jump processes 31C25: Dirichlet spaces
Secondary: 60G52: Stable processes

Jump-type Hunt process semi-Dirichlet form censored process stable-like process


Fukushima, Masatoshi; Uemura, Toshihiro. Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms. Ann. Probab. 40 (2012), no. 2, 858--889. doi:10.1214/10-AOP633.

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