The Feynman–Kac Formula and Excursion Theory
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We provide a proof of the Feynman–Kac formula for Brownian motion, using excursion theory up to an independent exponential time θ. Call g(θ) the last zero before θ. The independence of the pre-g(θ) process and the post-g(θ) process and the representation of their laws in terms of the integrals of Wiener measure up to inverse local time, or first hitting times allow to recover a formulation of the Feynman–Kac formula via excursion theory.
KeywordsExcursion Theory Independent Exponential Time Inverse Local Time Wiener Measure Brownian Motion
- 3.M. Kac, On some connections between probability theory and differential and integral equations, in Proc. Second Berkeley Symp. Math. Stat. Prob., ed. by J. Neyman (University of California Press, Berkeley, 1951), pp. 189–215Google Scholar