2015/01/25 by Xin Chen, Jian Wang, Chen, Xin +1
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #advanced mathematical theories #math.PR
paper · pdf · doi:10.48550/arxiv.1501.06128
openalex publication_date 2015/01/25 · arxiv created 2015/04/24 · arxiv updated 2015/04/27 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Let (Xt)t≥ 0 be a symmetric strong Markov process generated by non-local regular Dirichlet form (D,\D(D)) as follows \beginsplit D(f,g)=∫\Rd∫\Rd(f(x)-f(y))(g(x)-g(y)) J(x,y) dx dy, f,g∈ \D(D) \endsplit 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 TVt(f)(x)=\Eex(exp(-∫0tV(Xs) ds)f(Xt)), x∈\Rd, f∈ L2(\Rd;dx). In particular, we prove that for J(x,y)\asymp|x-y|-d-α\I_\|x-y|≤ 1\+e-|x-y|\I_\|x-y|> 1\ with α∈ (0,2) and V(x)=|x|λ with λ>0, (TtV)t≥ 0 is intrinsically ultracontractive if and only if λ>1; and that for symmetric α-stable process (Xt)t≥0 with α∈ (0,2) and V(x)=logλ(1+|x|) with some λ>0, (TtV)t≥ 0 is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ>1, and (TtV)t≥ 0 is intrinsically hypercontractive if and only if λ≥1. Besides, we also investigate intrinsic contractivity properties of (TtV)t ≥ 0 for the case that \liminf|x| → ∞V(x)<∞.