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In the theorem below $P_D$ means the heat kernel in the open $D \subset \mathbb{R}^m$ and $P_m$ is the heat kernel in whole $\mathbb{R}^m.$ I know absolutely nothing about what Brownian bridges are, ...

There is so much written about the Brownian motion and I suspect the answers to the questions below are hidden in somewhere in the literature but I cannot find them Denote by $E$ the Banach space ...

Let $V_t$ and $W_t$ be independent standard Wiener processes ($t\ge 0$, $W_t,V_t\in\mathbb R$). Let $C$ be the event that there is a continuous function $f$ such that for all $s$, $t$, $$ W_t=W_s\iff ...

How far away is $$\max_{x: x \in \{0, \ldots, N\}} \left|W\left(\frac{x}{N}\right)\right|$$ from $$\max_{0 \leq t \leq 1} |W(t)|$$ In other words, if you simulate a Wiener process over a finite ...

Let $W(t)$ be a standard brownian motion in $E \triangleq \mathbb{R}^d$. The transition probability from a state $x \in E$ at time $t$ to a state $y \in E$ at time $T$ is $$ p(x,t;y,T) = \frac{1}{\...