All Questions

I am looking for a (preferably some monograph) reference on the following fact: $$ u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \} $$ satisfies the heat equation $$ \frac{\...

Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...

I am trying to learn the connection between Brownian motion and heat equation (in the spirit of Feynman-Kac, for example, here). I read (Michael E. Taylor's PDE book, Volume II, Chapter 11, ...

I have asked this question on MathSE, but I got no replies, so I thought of trying here. Consider the Gaussian distribution on $\mathbb{R}$ with mean $m$ and variance $t=\sigma^2$. This has the ...