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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.

MSC:

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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