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A spectral criterion for the finiteness or infiniteness of stopped Feynman-Kac functionals of diffusion processes. (English) Zbl 0611.60072

A criterion is given for the finiteness or infiniteness of the Feynman- Kac functionals \[ u(q,D;x)=E_ x\exp \{\int^{\tau_ D}_{0}q(X(s))ds\} \] of d-dimensional diffusion processes X(t), \(t\geq 0\), with generator L, where D is a bounded open region in \({\mathbb{R}}^ d\), \(\tau_ D\) is the first exit time from D and \(q\in C(\bar D)\). The conditions are formulated in terms of the top \(\lambda_{q,D}\) of the spectrum of the Schrödinger operator \(L_{q,D}\), which is an extension of \(L+q\) acting on smooth functions which vanish on \(\partial D\). An explicit variational formula for \(\lambda_{q,D}\) is obtained. The case of unbounded regions is also discussed.
Reviewer: B.Grigelionis

MSC:

60J60 Diffusion processes
60J57 Multiplicative functionals and Markov processes
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