Conservation and Invariance Properties of Submarkovian Semigroups

Let ε be a Dirichlet form on L2(X) and Ω an open subset of X. Then one can define Dirichlet forms ε_D, or ε_N, corresponding to ε but with Dirichlet, or Neumann, boundary conditions imposed on the boundary ∂Ω of Ω. If S, S^D and S^N are the associated submarkovian semigroups we prove, under general assumptions of regularity and locality, that S_t φ= S^D_t φ for all φ∈L₂(Ω) and t>0 if and only if the capacity cap_Ω(∂Ω) of ∂Ω relative to Ω is zero. Moreover, if S is conservative, i.e. stochastically complete, then cap_Ω(∂Ω)=0 if and only if S^D is conservative on L₂(Ω). Under slightly more stringent assumptions we also prove that the vanishing of the relative capacity is equivalent to S^D_t φ=S^N_t φ for all φ∈L₂(Ω) and t>0.