## From Feynman-Kac formula to Feynman integrals via analytic continuation.(English)Zbl 0812.60052

Summary: By using a calculus based on Brownian bridge measures, it is shown that under mild assumptions on $$V$$ (e.g. $$V$$ is in the Kato class) the fundamental solution (FS) $$q(t,x,y)$$ for the heat equation $$\partial_ tu = ({1 \over 2} \Delta - V)u$$ can be represented by the Feynman-Kac formula. Furthermore, it has an analytic continuation in $$t$$ over $$\mathbb{C}_ +$$, where $$\mathbb{C}_ + = \{z \in \mathbb{C}$$, $$\text{Re} z>0\}$$, and $$q(\varepsilon + it$$, $$x,y)$$ can be expressed via Wiener path integrals. For small $$\varepsilon>0$$ it can be considered as an approximation of the FS for the Schrödinger equation $$\partial_ t \psi = i ({1 \over 2} \Delta - V) \psi$$. We also give an estimate of $$q(t,x,y)$$ for $$t \in \mathbb{C}_ +$$.

### MSC:

 60H30 Applications of stochastic analysis (to PDEs, etc.) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry 60J65 Brownian motion 60J35 Transition functions, generators and resolvents 81S40 Path integrals in quantum mechanics
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### References:

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