Intrinsic contractivity properties of Feynman-Kac semigroups for symmetric jump processes with infinite range jumps
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Let (X t ) t⩾0 be a symmetric strong Markov process generated by non-local regular Dirichlet form Open image in new window as follows: where J(x, y) is a strictly positive and symmetric measurable function on ℝ d ×ℝ d . We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup In particular, we prove that for Open image in new window with α ∈ (0, 2) and V(x) = |x|λ with λ > 0, (T t V ) t⩾0 is intrinsically ultracontractive if and only if λ > 1; and that for symmetric α-stable process (X t ) t⩾0 with α ∈ (0, 2) and V(x) = logλ (1+|x|) with some λ > 0, (T t V ) t⩾0 is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ > 1, and (T t V ) t⩾0 is intrinsically hypercontractive if and only if λ ⩾ 1. Besides, we also investigate intrinsic contractivity properties of (T t V ) t⩾0 for the case that lim inf|x|→+∞ V(x) < +∞.
KeywordsSymmetric jump process Lévy process Dirichlet form Feynman-Kac semigroup intrinsic contractivity
MSC60G51 60G52 60J25 60J75
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