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 $u(\cdot )$, that satisfies ${sup}_{y\in D_S}u(y)\ge 0$, one can find a constant $a>0$ (that in general depends on u), for which ${sup}_{y\in D_S}u(y)-u(x)\ge a\mathit{\psi }(x)$, $x\in D$. 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 $D\subset {\mathbb{R}}^d$. Here $D_S$ 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.