Traces of harmonic functions, capacities, and traces of symmetric Markov processes
Let X be an m-symmetric Markov process on \(R^ d\) with infinitesimal generator A. For a non-negative locally integrable function \(\rho\) on [0,\(\infty)\), if there exists a diffusion process Y associated with the energy form \[ (g_ 1| g_ 2)_{\rho}\equiv \int^{\infty}_{- \infty}Dg_ 1(y)Dg_ 2(y)\rho (| y|)dy\quad (D=\partial /\partial y), \] then the trace (time changed process) \(\tilde Z\) of \(Z=(X,Y)\) to \(R^ d\equiv R^ d\times \{0\}\) is well-defined. If \(v(x,y)=\Pi f(x,y)\equiv E^{x,y}(f(Z_ T)),\) \(T=\inf \{t>0:\quad Z_ t\in R^ d\},\) then the generator B of \(\tilde Z\) is related to the Dirichlet form \(E_{\rho}\) of Z by \[ E_{\rho}(v)=-\int f(x)B\quad f(x)m(dx). \] Conversely, when \(\rho\) is given initially, to characterize the boundary Dirichlet form B is difficult in general. In this paper, considering the (generalized) Poisson integral \(P f(\cdot,y)=\exp (-y\sqrt{-{\mathcal A}})f\) instead of \(\Pi\) f, the author shows the existence of the Dirichlet form -(f,Cf) defined by \((- f,Cf)=E_{\rho}(Pf)\) under conditions on \(\rho\) including monotonicity conditions. The generator C is determined by a Bernstein function through subordination. This result is applied to show the q.e. existence of the boundary values of harmonic functions. Also \(E_{\rho}(Pf)=E_{\rho}(\Pi \quad \Gamma \quad f)\) holds for an automorphism \(\Gamma\) in \(L^ 2(m)\).
existence of the boundary values of harmonic functions
scientific article; zbMATH DE number 4119364
scientific article