Explosion by Killing and Maximum Principle in Symmetric Markov Processes

Keller and Lenz KL define a concept of stochastic completeness at infinity (SCI) for a regular symmetric Dirichlet form (, ). We show that (SCI) can be characterized probabilistically by using the predictable part ᵖ of the life time of the symmetric Markov process X= (Pₓ, Xₜ) generated by (, ), that is, (SCI) is equivalent to ₓ (=ᵖ1 implies a Liouville property that every bounded solution to ^ u=0 is zero quasi-everywhere and that the refined maximum principle in the sense of Berestycki-Nirenberg-Varadhan BNV holds for ^ if and only if () >1 (Theorem RMP).