Representation of \(a\)-harmonic functions in Lipschitz domains
Let \((X_t, P^x)\) be the standard symmetric \(\alpha\)-stable Lévy process in \({\mathbb R}^n\), \(0<\alpha<2\). For \(A\subset {\mathbb R}^n\), let \({\tau}_A=\inf \{t\geq 0: X_t\notin A \}\) denote the first exit time from \(A\). Let \(u\geq 0\) be a Borel measurable function on \({\mathbb R}^n\), and let \(V\subset {\mathbb R}^n\) be an open subset. Then (i) \(u\) is \(\alpha\)-harmonic in \(V\) if for every bounded open set \(B\) with \(\overline{B}\subset V\), \(u(x)=E^x u(X_{{\tau}_B})\), \(x\in B\); (ii) \(u\) is regular \(\alpha\)-harmonic in \(V\), if \(u(x)= E^x u(X_{{\tau}_V})\), \(x\in V\); (iii) \(u\) is singular \(\alpha\)-harmonic in \(V\), if \(u\) is \(\alpha\)-harmonic in \(V\), and \(u(x)=0\), \(x\in V^{c}\). By the strong Markov property, regular \(\alpha\)-harmonic functions are \(\alpha\)-harmonic. Singular \(\alpha\)-harmonic functions in \(V\) are precisely harmonic functions for the subprocess \(X^V\) killed upon leaving \(V\) (extended to be equal to zero outside \(V\)). Let \(D\) be a bounded Lipschitz domain in \({\mathbb R}^n\). The author proves that every (nonnegative) \(\alpha\)-harmonic function in \(D\) decomposes into a unique sum of a regular and a singular \(\alpha\)-harmonic function in \(D\). The main goal of the paper is to show that each part admits its Martin representation. Let \(u\geq 0\) be a regular \(\alpha\)-harmonic function in \(D\). Then \(u(x)=\int_{D^c}P(x,y)u(y) dy\), \(x\in D\), where \(P(x,y)\), \(x\in D\), \(y\in D^c\), is the Poisson kernel, i.e., the density of the \(\alpha\)-harmonic measure. Let \(u\geq 0\) be a singular \(\alpha\)-harmonic function in \(D\). Then there exists a unique finite nonnegative Borel measure \(\mu\) on \(\partial D\) such that \(u(x)=\int_{\partial D} K(x,Q)\mu(dQ)\), \(x\in {\mathbb R}^n\). The mapping \((x,Q)\mapsto K(x,Q)\), \(x\in D\), \(Q\in \partial D\), is the Martin kernel with respect to the killed subprocess \(X^D\). This representation shows that for a bounded Lipschitz domain \(D\), the Martin boundary of the killed subprocess \(X^D\) can be identified with the Euclidean boundary of \(D\). Similar results were independently obtained in [\textit{Z.-Q. Chen} and \textit{R. Song}, J. Funct. Anal. 159, 267-294 (1998)].
scientific article; zbMATH DE number 1347343