Intrinsic contractivity properties of Feynman-Kac semigroups for symmetric jump processes with infinite range jumps.

Chen, Xin; Wang, Jian

Let ( X) be a symmetric strong Markov process generated by non-local regular Dirichlet form [InlineMediaObject not available: see fulltext.] as follows: [Figure not available: see fulltext.] where J( x, y) is a strictly positive and symmetric measurable function on ℝ×ℝ. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup [Figure not available: see fulltext.] In particular, we prove that for [InlineMediaObject not available: see fulltext.] with α ∈ (0, 2) and V( x) = | x| with λ > 0, ( T) is intrinsically ultracontractive if and only if λ > 1; and that for symmetric α-stable process ( X) with α ∈ (0, 2) and V( x) = log (1 | x|) with some λ > 0, ( T) is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ > 1, and ( T) is intrinsically hypercontractive if and only if λ ⩾ 1. Besides, we also investigate intrinsic contractivity properties of ( T) for the case that lim inf V( x) < ∞.

JUMP processes; DIRICHLET forms; MATHEMATICAL forms; CHARACTERS sums (Mathematics); MARKOV processes; SEMIGROUPS (Algebra)

Frontiers of Mathematics in China, 2015, Vol 10, Issue 4, p753

1673-3452

Academic Journal

10.1007/s11464-015-0477-8