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

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 $\alpha(x)$.Comment: Published in at http://dx.doi.org/10.1214/10-AOP633 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On unique extension of time changed reflecting Brownian motions

Let $D$ be an unbounded domain in \RR^d with $d\geq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $\overline D$ is transient. Next assume that RBM $X$ on $\overline D$ is transient and let $Y$ be its time change by Revuz measure ${\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $\overline D$. We further show that if there is some $r>0$ so that $D\setminus \overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has a unique Martin boundary point at infinity.Comment: To appear in Ann. Inst. Henri Poincare Probab. Statis

Time changes of symmetric diffusions and Feller measures

We extend the classical Douglas integral, which expresses the Dirichlet integral of a harmonic function on the unit disk in terms of its value on boundary, to the case of conservative symmetric diffusion in terms of Feller measure, by using the approach of time change of Markov processes.Comment: Published at http://dx.doi.org/10.1214/009117904000000649 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Traces of symmetric Markov processes and their characterizations

Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In this paper we give a complete characterization of time changed processes of an arbitrary symmetric Markov process, in terms of the Beurling--Deny decomposition of their associated Dirichlet forms and of Feller measures of the process. In particular, we determine the jumping and killing measure (or, equivalently, the L\'{e}vy system) for the time-changed process. We further discuss when the trace Dirichlet form for the time changed process can be characterized as the space of finite Douglas integrals defined by Feller measures. Finally, we give a probabilistic characterization of Feller measures in terms of the excursions of the base process.Comment: Published at http://dx.doi.org/10.1214/009117905000000657 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org