Path regularity for Feller semigroups via Gaussian kernel estimates and generalizations to arbitrary semigroups on \(C_0\)
Let \(X_t\) be a Feller process with values in a complete locally compact metric space \(E\). The author gives conditions for the paths of the process \(X_t\) to be locally Hölder continuous or to belong to some weighted Besov spaces. The conditions are formulated in terms of variations of paths for dyadic times \(t=j2^{-n}\), \(j=0,1,\ldots ,2^n\), \(n\in \mathbb N\), and can be reformulated in such a way that they make sense (as a special property) for arbitrary semigroups of operators on \(C_0(E)\). Then it is proved that the above conditions are satisfied if the semigroup satisfies Gaussian kernel estimates typical for Feller semigroups on fractals and for semigroups generated by elliptic operators on \(\mathbb R^d\) of order \(m\geq d\).
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