On divergence of expectations of the Feynman-Kac type with singular potentials. (English) Zbl 1356.60125

This paper aims to give sufficient conditions for the divergence of the expectations \[ E_x [u_0(B_t) \exp{(\int_0^t V(B_s)ds)}], \] where the initial datum \(u_0\) belongs to \(C_0(\mathbb{R}^N)\), \(V\) is a singular potential on \(\mathbb{R}^N\) that explodes at the origin and \(B\) is either a Brownian motion starting at \(x\) or a rotationally invariant \(\alpha\)-stable process. An analogous problem is also discussed in the half-space for a Brownian motion and a singular potential on the boundary of \(\mathbb{R}^{N-1}\times (0,\infty)\). A key ingredient in the proofs is an estimate derived from the eigenvalue expansions for hitting distributions of Bessel processes.


60J65 Brownian motion
60G52 Stable stochastic processes
60F99 Limit theorems in probability theory
60J55 Local time and additive functionals
35K05 Heat equation
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