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

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

2012-04-13

arxiv.org/abs/1204.2944v1

Annals of Probability 2012, Vol. 40, No. 2, 858-889

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP633 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP633

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 $E\times E$ 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 $\eta$ on $L^2(E;m)$ producing a Hunt process $X^0$ on $E$ whose jump behaviours are governed by $k$. For an arbitrary open subset $D\subset E$, we also construct a Hunt process $X^{D,0}$ on $D$ in an analogous manner. When $D$ is relatively compact, we show that $X^{D,0}$ 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 $X^0$ is the stable-like process that will be also identified with the solution of a martingale problem up to an $\eta$-polar set of starting points. Approachability to the boundary $\partial D$ in finite time of its censored process $X^{D,0}$ on a bounded open subset $D$ will be examined in terms of the polarity of $\partial D$ for the symmetric stable processes with indices that bound the variable exponent $\alpha(x)$.

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