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October, 1986 A Spectral Criterion for the Finiteness or Infiniteness of Stopped Feynman-Kac Functionals of Diffusion Processes
Ross Pinsky
Ann. Probab. 14(4): 1180-1187 (October, 1986). DOI: 10.1214/aop/1176992361

Abstract

Consider the Feynman-Kac functional $u(q, D; x) = E_x\exp(\int^{\tau_D}_0 q(x(s)) ds),$ where $D$ is a bounded open region in $R^d, \tau_D$ is the first exit time from $D, q \in C(\bar{D})$, and $x(s)$ is a diffusion process on $R^d$ with generator $L$. We give a criterion for the finiteness or infiniteness of $u(q, D; x)$ in terms of the top of the spectrum of the Schrodinger operator $L_{q, D}$, an extension of $L + q$ acting on smooth functions which vanish on $\partial D$. As we also have a variational formula for the top of the spectrum, we thus obtain a criterion explicitly in terms of a variational formula.

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Ross Pinsky. "A Spectral Criterion for the Finiteness or Infiniteness of Stopped Feynman-Kac Functionals of Diffusion Processes." Ann. Probab. 14 (4) 1180 - 1187, October, 1986. https://doi.org/10.1214/aop/1176992361

Information

Published: October, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0611.60072
MathSciNet: MR866341
Digital Object Identifier: 10.1214/aop/1176992361

Subjects:
Primary: 60J60

Keywords: Diffusion processes , Feynman-Kac functionals , large deviations

Rights: Copyright © 1986 Institute of Mathematical Statistics

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Vol.14 • No. 4 • October, 1986
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