Occupation time densities for stable-like processes and other pure jump Markov processes (Q1109425)

Suppose \(X_ t\) is a pure jump Markov process on the real line with generator \[ A f(x)=\int [f(x+h)-f(x)-f'(x)h\cdot 1_{[-1,1]}(h)]\nu (x,dh). \] The author considers the problem of determining what conditions on the jump measure \(\nu\) imply the existence of a local time for X. Clearly, \(\nu\) must have sufficiently many small jumps. In the case when \(\nu (x,dh)=| h|^{-(1+\alpha (x))}\), i.e. when X behaves at x like a stable process with index \(\alpha\) (x) then inf \(\alpha\) (x)\(>1\) plus Dini continuity of \(\alpha\) imply the existence of a local time. A previous example published by the author shows that a continuity assumption on \(\alpha\) cannot be eliminated.