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Kac's Moment Formula and the Feynman-Kac Formula for Additive Functionals of a Markov Process

P. J. Fitzsimmons and Jim Pitmangif

May 29, 1998


Mark Kac introduced a method for calculating the distribution of the integral tex2html_wrap_inline2800 for a function v of a Markov process tex2html_wrap_inline2804 and a suitable random time T, which yields the Feynman-Kac formula for the moment-generating function of tex2html_wrap_inline2808 . We review Kac's method, with emphasis on an aspect often overlooked. This is Kac's formula for moments of tex2html_wrap_inline2808 , which may be stated as follows. For any random time T such that the killed process tex2html_wrap_inline2814 is Markov with substochastic semi-group tex2html_wrap_inline2816 , any non-negative measurable function v, and any initial distribution tex2html_wrap_inline2820 , the tex2html_wrap_inline2822 moment of tex2html_wrap_inline2808 is tex2html_wrap_inline2826 where tex2html_wrap_inline2828 is the Green's operator of the killed process, tex2html_wrap_inline2830 is the operator of multiplication by v, and tex2html_wrap_inline2834 is the function that is identically 1.

Patrick Fitzsimmons
Wed May 17 08:50:36 PDT 2000