2009/03/31 by A. F. M. ter Elst, Derek W. Robinson, ter Elst, A. F. M. +1
Mathematics · #31C15 #31C25 #35Hxx #35J70 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:31C15 #msc:31C25 #msc:35Hxx #msc:35J70
paper · pdf · doi:10.48550/arxiv.0903.5479
arxiv created 2009/03/31 · arxiv updated 2009/12/01
Let \cal E be a Dirichlet form on L2(X) and Ω an open subset of X. Then one can define Dirichlet forms \cal ED, or \cal EN, corresponding to \cal E but with Dirichlet, or Neumann, boundary conditions imposed on the boundary ∂Ω of Ω. If S, SD and SN are the associated submarkovian semigroups we prove, under general assumptions of regularity and locality, that Stϕ= SDtϕ for all ϕ∈ L2(Ω) and t>0 if and only if the capacity \mathop\rm capΩ(∂Ω) of ∂ Ω relative to Ω is zero. Moreover, if S is conservative, i.e. stochastically complete, then \mathop\rm capΩ(∂Ω)=0 if and only if SD is conservative on L2(Ω). Under slightly more stringent assumptions we also prove that the vanishing of the relative capacity is equivalent to SDt ϕ= SNt ϕ for all ϕ∈ L2(Ω) and t>0.