2014/03/14 by Xin Chen, Jian Wang, Chen, Xin +1 · 3 citations
Economics, Econometrics and Finance · Mathematics · #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications #advanced mathematical theories #math.PR
paper · pdf · doi:10.48550/arxiv.1403.3486
31 pages
arxiv created 2015/01/24 · arxiv updated 2015/01/27
Consider the symmetric non-local Dirichlet form (D,\D(D)) given by D(f,f)=∫\Rd∫\Rd(f(x)-f(y))2 J(x,y) dx dywith \D(D) the closure of the set of C1 functions on \Rd with compact support under the norm √(D1(f,f)), where D1(f,f):=D(f,f)+∫ f2(x) dx and J(x,y) is a nonnegative symmetric measurable function on \Rd× \Rd. Suppose that there is a Hunt process (Xt)t≥ 0 on \Rd corresponding to (D,\D(D)), and that (L,\D(L)) is its infinitesimal generator. We study the intrinsic ultracontractivity for the Feynman-Kac semigroup (TtV)t≥ 0 generated by LV:=L-V, where V≥ 0 is a non-negative locally bounded measurable function such that Lebesgue measure of the set \x∈ \Rd: V(x)≤ r\ is finite for every r>0. By using intrinsic super Poincaré inequalities and establishing an explicit lower bound estimate for the ground state, we present general criteria for the intrinsic ultracontractivity of (TtV)t≥ 0. In particular, if J(x,y)\asymp|x-y|-d-α\I_\|x-y|≤ 1\+e-|x-y|γ\I_\|x-y|> 1\ for some α∈ (0,2) and γ∈(1,∞], and the potential function V(x)=|x|θ for some θ>0, then (TtV)t≥ 0 is intrinsically ultracontractive if and only if θ>1. When θ>1, we have the following explicit estimates for the ground state ϕ1 c1exp(-c2 θ^\fracγ-1γ|x| log^\fracγ-1γ(1+|x|)) ≤ ϕ1(x) ≤ c3exp(-c4 θ^\fracγ-1γ|x| log^\fracγ-1γ(1+|x|)) , where ci>0 (i=1,2,3,4) are constants. We stress that, our method efficiently applies to the Hunt process (Xt)t ≥ 0 with finite range jumps, and some irregular potential function V such that lim|x| → ∞V(x)≠∞.