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

Abstract

Let (X_t)_t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form (D, D(D)) as follows:

D(f,g)={\displaystyle {\int }_{{ℝ}^d}{\displaystyle {\int }_{{ℝ}^{\mathit{d}}}(f(x)-f(y))(g(x)-g(y))J(x,y)\mathrm{d}x\mathrm{d}y,\hspace{1em}f,g\in D(D),}}

where J(x, y) is a strictly positive and symmetric measurable function on {ℝ}^d\times {ℝ}^d. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup

T_t^V(f)(x)={\mathbb{E}}^x\left(\exp \left(-{\displaystyle {\int }_{\mathrm{0}}^{t}V(X_s)\mathrm{d}s}\right)f(X_t)\right),\hspace{1em}x\in {ℝ}^d,f\in L^{\mathrm{2}}({ℝ}^d;\mathrm{d}x).

In particular, we prove that for J\left(x,y\right)\approx {\left|x-y\right|}^{-d-a}{{l}}_{\left\{\left|x-y\right|\le 1\right\}}+{\mathrm{e}}^{-\left|x-y\right|}{{l}}_{\left\{\left|x-y\right|>1\right\}} with α ∈(0, 2) and V(x)={\left|x\right|}^{\lambda } with λ>0, {(T_t^V)}_{t\ge \mathrm{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)={\log }^{\lambda }(\mathrm{1}+\left|x\right|) with some λ>0, {(T_t^V)}_{t\ge \mathrm{0}} is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ>1, and {(T_t^V)}_{t\ge \mathrm{0}} is intrinsically hypercontractive if and only if \lambda \ge \mathrm{1}. Besides, we also investigate intrinsic contractivity properties of {(T_t^V)}_{t\ge \mathrm{0}} for the case that lim

Keywords

Symmetric jump process / Lévy process / Dirichlet form / Feynman- Kac semigroup / intrinsic contractivity

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