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### Feynman-Kac for heat equation on a compact manifold with boundary

It is known that for any open $\Omega \subset \mathbb{R}^n$, given $f \in L^2(\Omega)$, $x \in \Omega$, one has $$e^{t\Delta}f(x) = \mathbb{E}_x(f(\omega(t))\psi_\Omega(\omega, t)),$$ where $\Delta$ ...
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### Winding number of a random walk on the square lattice before hitting the origin

Let us consider a simple random walk on $\mathbb{Z}^2$ started at $(x,0)$ and killed upon hitting the origin. Define the total winding number $w_x$ around the origin to be the (signed) number of ...
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### Short time asymptotics for Brownian motion on a compact manifold

Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in ...
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### Two dimensional Brownian motion moving from one point to other

Suppose $W_t= (X_t,Y_t)$ is a $2$d standard Brownian motion starting at $(-1,0)$. How do I show that there is a positive probability that $W_t$ moves from $(-1,0)$, to a neighborhood of $(1,0)$, say ...
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### Brownian hitting probability of a small body

Consider a Brownian motion $B(t)$ starting from the origin $0$ in $\mathbb{R}^n$. Consider the ball $B(0, r)$ and an open set $V \subset B(0, r)$. If it is known that the probability of the Brownian ...
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### Brownian motion in perturbed (asymptotically flat) metric

Let $g_{\mathbb{R}^n}$ denote the usual Euclidean metric on $\mathbb{R}^n$ and let $B_g(t)$ denote the Brownian motion associated to a complete metric $g$ on $\mathbb{R}^n$. Consider a Brownian motion ...
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### Probability Brownian motion lies between $2$ functions

Suppose $a_j \in \mathbb{R}$, $b_j \ge 0$, and $0 = t_0 < t_1 < \ldots < t_J$ are time points. Let $W_t$ be a standard Brownian motion. Is it possible to further simplify the expression \...
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### How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here. Say we have a stochastic process described by a ...
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### Normality criterion based on Brownian motion

Consider analytic family $\mathcal{F}$ btw domains $U,V\subset \mathbb{C}$. For any $f\in \mathcal{F}$ we have time-changed Brownian motion $f(B_{t})=\widetilde{B}_{\int_{0}^{t}|f(B_{s})|^{2}ds}$. So ...
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### BM hitting times with exponential killing process

Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point $p_s=(x_s,y_s,z_s)$ . BM starts at point $p_0=(x_0,y_0,z_0)$ and when it riches the ...
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### Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
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### Large deviation for Brownian path on $[0,\infty)$

It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path. If we equip the space of continuous function starting from $0$, ...
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### $X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]

What are the SDE's satisfied by the following processes? $X_t = B_t^q$ $X_t = (\sin B_t)^q$ $X_t = B_t^q (\sin B_t)^r$ Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the ...
### Distribution of Brownian local time at first hitting times of $1$ and of $\pm1$? [closed]
Here, $(B_t)$ is a standard Brownian motion, and $(L_t)$ its local time at $0$. Consider $$T=\inf\{t : B_t = 1\},\qquad\tau =\inf\{t : |B_t| = 1\}.$$ What is the distribution of $L_T$? What is the ...