Hopf type lemmas for subsolutions of integro-differential equations

Abstract

In the paper we prove a lower bound for subsolutions of the integro-differential equation: −Au+cu=0 in a domain D. It states that there exists a Borel function ψ, strictly positive on D, depending only on the coefficients of the operator A, c and D such that for any subsolution u(⋅), that satisfies supy∈DSu(y)≥0, one can find a constant a>0 (that in general depends on u), for which supy∈DSu(y)−u(x)≥aψ(x), x∈D. The bound is valid for a wide class of Lévy type integro-differential operators A, non-negative, bounded and measurable function c and a quite general domain D⊂Rd. Here DS is a certain set containing the closure of D and determined by the support of the Levy jump measure associated with A. In some cases a non-negative eigenfunction corresponding to the operator in D can be admitted as the function ψ. In particular, this occurs when the transition probability semigroup associated with A is ultracontractive. The main assumptions made about A are: there exists a strong Markov solution to the martingale problem associated with the operator and its resolvent satisfies some minorization condition. This type of a result we call the generalized Hopf lemma.