Large time asymptotics of Feynman-Kac functionals for symmetric stable processes (Q2836147)

Large-time behavior of heat kernels are derived for Feynman-Kac semigroups of \(\alpha\)-stable processes, which extends existing results for the Brownian motion. Let \(\alpha\in (0,2)\) and \(d>\alpha\), consider the Dirichlet form \((\mathcal E,\mathcal F)\) associated with \((-\Delta)^\alpha\) in \(L^2(\mathbb R^d;m)\), where \(m\) is the Lebesgue measure. For a positive Kato measure \(\mu\) with compact support, consider the Schrödinger form NEWLINE\[NEWLINE\mathcal E^\mu(f,g):= \mathcal E(f,g) -\mu(fg),\;\;f,g\in \mathcal F.NEWLINE\]NEWLINE NEWLINEAssume that \(\mu\) is critical such that the associated Schrödinger operator has a unique generalized ground state \(h_0\). Then the heat kernel \(p_t^\mu\) of the Schrödinger operator has the following long-time behavior: NEWLINE\[NEWLINE\int_{\mathbb R^d} p_t^\mu(x,y)d y\sim \begin{cases} \frac{\alpha \Gamma(d/2) \sin[(d/\alpha-1)\pi]}{2^{1-d}\pi^{1-d/2}\Gamma(d/\alpha)\mu(h_0)} h_0(x) t^{d/\alpha -1} &\text{if}\;d\in (\alpha, 2\alpha),\\ \frac{\Gamma(\alpha +1)h_0(x)t}{2^{1-d}\pi^{-d/2}\mu(h_0)\log t} &\text{if}\;d=2\alpha,\\ \frac{\mu(h_0)}{m(h_0^2)} h_0(x) t &\text{if}\;d>2\alpha.\end{cases}NEWLINE\]