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 ζᵖ of the life time ζ of the symmetric Markov process X=( Pₓ,Xₜ) generated by (,), that is, (SCI) is equivalent to ₓ(ζ=ζᵖ<∞)=0. We define a concept, { explosion by killing} (EK), by ₓ(ζ=ζⁱ<∞)=1. Here ζⁱ is the totally inaccessible part of ζ. We see that (EK) is equivalent to (SCI) and ₓ(ζ=∞)=1. Let Xʳᵉˢ be the { resurrected process} generated by the { resurrected form}, a regular Dirichlet form constructed by removing the killing part from (, ). Extending work of Masamune and Schmidt ({MS}), we show that (EK) is also equivalent to the ordinary conservation property of time changed process of Xʳᵉˢ by Aᵏₜ, where the Aᵏₜ is the positive continuous additive functional in the Revuz correspondence to the killing measure k in the Beurling-Deny formula (Theorem {ma-sh}). We consider the maximum principle for Schr\"odinger-type operator ^μ=-μ. Here is the self-adjoint operator associated with (,) %with non-local part and μ is a Green-tight Kato measure. Let λ(μ) be the principal eigenvalue of the trace of (,) relative to μ. We prove that if (EK) holds, then λ(μ)>1 implies a Liouville property that every bounded solution to ^μ u=0 is zero quasi-everywhere and that the { refined maximum principle} in the sense of Berestycki-Nirenberg-Varadhan {BNV} holds for ^μ if and only if λ(μ)>1 (Theorem {RMP}).