Abstract
Let (X
t
)
t⩾0 be a symmetric strong Markov process generated by non-local regular Dirichlet form 
as follows:
where J(x, y) is a strictly positive and symmetric measurable function on ℝd×ℝd. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup
In particular, we prove that for 
with α ∈ (0, 2) and V(x) = |x|λ with λ > 0, (T
V
t
)
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
V
t
)
t⩾0 is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ > 1, and (T
V
t
)
t⩾0 is intrinsically hypercontractive if and only if λ ⩾ 1. Besides, we also investigate intrinsic contractivity properties of (T
V
t
)
t⩾0 for the case that lim inf|x|→+∞
V(x) < +∞.
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Dedicated to Professor Mu-Fa Chen on the occasion of his 70th birthday
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Chen, X., Wang, J. Intrinsic contractivity properties of Feynman-Kac semigroups for symmetric jump processes with infinite range jumps. Front. Math. China 10, 753–776 (2015). https://doi.org/10.1007/s11464-015-0477-8
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DOI: https://doi.org/10.1007/s11464-015-0477-8