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I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's ‎An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...

In other words, the probability that Brownian motion stays within $A^{c}$. What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...