2021/02/17 by Klimsiak, Tomasz, Komorowski, Tomasz
#35J08 #FOS: Mathematics #Primary 35B50 #Probability (math.PR) #Secondary 35J15
paper · doi:10.48550/arxiv.2102.08890
In the paper we prove a generalization of the Hopf lemma for weak subsolutions of the equation: -Au+cu=0 in D, for a wide class of Lévy type integro-differential operators A, bounded and measurable function c:D→[0,+∞) and domain D⊂ \BRd. More precisely, we prove that if \em the strong maximum principle (SMP) holds for A, then there exists a Borel function ψ:D→(0,+∞), depending only on the coefficients of the operator A, c and D such that for any subsolution u(⋅) one can find a constant a>0 (that in general depends on u), for which sup_y∈ \rm clD∪ \cal S(D)u(y)-u(x)≥ aψ(x), x∈ D. Here \rm clD is the closure of D. The set \cal S(D) - called the range of non-locality of A over D - is determined by the support of the Levy jump measure associated with A. This type of a result we call the \em generalized Hopf lemma. It turns out that the irreducibility property of the resolvent of A implies the SMP. The converse also holds, provided we assume some additional (rather weak) assumption on the resolvent. For some classes of operators we can admit the constant a to be equal to sup_y∈ \rm clD∪ \cal S(D)u(y). We call this type of a result a \em quantitative version of the Hopf lemma. Finally, we formulate a necessary and sufficient condition on A - expressed in terms of ergodic properties of its resolvent - which ensures that the lower bound ψ∈ \rm span(φD), where φD is a non-negative eigenfunction of A in D. We show that the aforementioned ergodic property is implied by the intrinsic ultracontractivity of the semigroup associated with A.