Brownian motion characterization of some Besov--Lipschitz spaces on domains (Q1781401)

Let \(\Omega\) be a bounded \(C^\infty\) domain in \(\mathbb R^n\). The subspaces \[ {}_{z} B^s_{pq} (\Omega) = \widetilde{B}^s_{pq} (\Omega) = \left\{ f \in B^s_{pq} (\mathbb R^n), \;\text{supp }f \subset \overline{\Omega} \right\} \] of the Besov spaces \(B^s_{pq} (\mathbb R^n)\) have been playing a role for decades in the theory of function spaces, in particular at present. It is well-known that the spaces \(B^s_{pq} (\mathbb R^n)\) can be characterized in terms of the Gauss--Weierstrass semigroup (thermic extension, Brownian motion in \(\mathbb{R}^n\)). It is the main aim of this paper to give a corresponding characterization of \({}_{z} B^s_{pq} (\Omega)\), restricted to \(1 \leq p,q \leq \infty\) and \(0<s<2\), in terms of Brownian motions on \(\Omega\).