Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms
The article stands in the long tradition of papers which deal with the close relationship between Dirichlet forms and Markov processes which was first discovered by M. Fukushima: 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, the authors construct an associated regular lower bounded semi-Dirichlet form \(\eta \) on \(L^{2}(E; m)\). This semi-Dirichlet form produces a Hunt process \(X^{0}\) on the space \(E\) whose jump behaviour is governed by \(k\). For an arbitrary open subset \(D \subset E\), they also construct a Hunt process \(X^{D,0}\) on \(D\) in an analogous manner. When \(D\) is relatively compact, the authors show that \(X^{D,0}\) is censored, that is, it admits no killing inside \(D\) and is killed only when the path approaches the boundary. In the particular case when \(E\) is the \(d\)-dimensional Euclidean space and \(m\) is the Lebesgue measure, a typical example of \(X^{0}\) is the stable-like process which was first constructed by R. F. Bass. This process is 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 \(D\) is examined in terms of the polarity of \(\partial D\) for the symmetric stable processes with indices that bound the variable exponent \(\alpha (x)\).
scientific article; zbMATH DE number 6032946