Consider a Brownian bridge of length $r$ on the plane. What is the expected (non-signed) smallest area of the disc spanned by the loop? By "non-signed" I mean that if a loop goes a …

After an extensive unsuccessful search: I need a reference (preferably a book) for the Donsker's invariance principle for Riemannian manifolds. Thanks.

If $v(\omega,t) : \Omega \times [0,\infty) \to \mathbb{R}$ is a Standard Brownian motion, then for what values of $s,\omega$ does the Laplace transform $l(\omega,s) = \int_0^\infty …

Assume we have a reflected Brownian motion in a smooth bounded domain $D \subseteq \mathbb R^d$. It can have nonzero (but constant) drift, non-identity (but constant) covariance ma …

A Bessel Bridge is a Brownian Motion, conditioned such that $B(0) = B(1) = 0$ and $B([0, 1]) \ge 0$. A raised Bessel Bridge is a generalization of this: it's a Brownian Motion con …

Suppose $X$ and $Y$ are two Brownian motions such that $|X|$ and $|Y|$ are independent. Then it is easy to show that $\langle X,Y \rangle =0$ using the Tanaka formula, for example, …

Suppose X_1,X_2,...,X_n are n Brownian motions with respect to the same filtration such that X_1 is independent of X_j for all j=2,...,n. Is it true that X_1 is independent of (X_2 …

I have been trying to prove the following conjecture for a while, but so far to no avail. Would be very grateful for some tips! (I edited the entire thing to make it clearer) The …

Suppose $D$ is a bounded Lipschitz domain in $R^n$ and $X_1$, $X_2$ are two independent reflected Brownian motions (RBMs) in $D$. Is it true that $$P[X_1(t)\neq X_2(t) \text{ for …

from math.stackexchange I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have \begin{align} dX_t=b(X_t)dt+\sigm …

Preliminaries An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independe …

Hi, let ${B(t), t\in R}$ be a Brown motion, then $$ \varlimsup_{t\downarrow 0}\frac{B(t)}{\sqrt{2t\log\log(1/t)}} = 1 $$ almost surely in the sense of Wiener measure. I find the r …

Suppose X_0 = 0. Is there a simple solution that is similar with that of the 1-d case, a standard exercise for stoc calculus ? Anyone could give a hint ? Thanks :)

I am trying to show the following statement Let $D\subset \mathbb{R}^2$ be an open and bounded subset. $\Pi=(P^x : x \in D )$ a Family of standard Brownian Motions started at $ …

Consider a Brownian particle in the plane with a circular trap at the origin. If we give the particle enough time it falls into the trap (since Brownian motion is space filling in …