Markov processes with Lipschitz semigroups

Abstract:

For $f$ a function on a metric space, let \[ \operatorname {Lip} f = \sup \limits _{x \ne y} |f(x) - f(y)|/d(x, y),\] and say that a semigroup ${P_t}$ is Lipschitz if $\operatorname {Lip} ({P_t}f) \leqslant {e^{Kt}}\operatorname {Lip} f$ for all $f$, $t$, where $K$ is a constant. If one has two Lipschitz semigroups, then, with some additional assumptions, the sum of their infinitesimal generators will also generate a Lipschitz semigroup. Furthermore a sequence of uniformly Lipschitz semigroups has a subsequence which converges in the strong operator topology. Examples of Markov processes with Lipschitz semigroups include all diffusions on the real line which are on natural scale whose speed measures satisfy mild conditions, as well as some jump processes. One thus gets Markov processes whose generators are certain integro-differential operators. One can also interpret the results as giving some smoothness conditions for the solutions of certain parabolic partial differential equations.