Let $X(t)$ be a Brownian motion in $\mathbb{R}^n$, stopped at some fixed time $T$. Is there a notion of Green's function for such a Brownian motion? I am guessing that there is, and $G(x, y) : = \int_0^T p(t, x, y)dt$, where $p(t, x, y)$ is the heat kernel in $\mathbb{R}^n$. Is this correct? Also, where can I find a proof that such fixed time stopped Brownian motions have the strong Markov property?
I am also requesting references where such fixed time stopped motions are discussed. I apologize if these questions are really basic, I am just beginning to learn Brownian motion, and the books I have access to do not discuss such fixed time stopped motions explicitly.