The brownian-motion tag has no wiki summary.

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### Trapping a particle

A particle starts a brownian walk in the middle of a long tunnel in the plane, at one end of the tunnel is a region Y of given area A.
Does the shape of region Y affect average time for the particle ...

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**1**answer

86 views

### Fractional Brownian motion via Hilbert space

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:
Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...

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110 views

### Conformal invariance of Brownian motion in higher dimensions

We know for planar Brownian motion, that conformal maps composed with Brownian motion are also Brownian motion (preserve distribution).
Does it follow for higher dimensions?
I think it follows for ...

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64 views

### Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...

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41 views

### Reference for “Newtonian capacity estimates probability that A is hit by a Brownian motion”

I am looking for the following statement
"In fact, the Newtonian (logarithmic) capacity gives an estimate, up to a constant factor, the probability that A is hit by a Brownian motion started, say, ...

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73 views

### Probability that d-Brownian Motion ,d>3, avoids a set A

In other words, the probability that Brownian motion stays within $A^{c}$. So far I found that it is 1, for random cylinders and thorns (http://www.math.upenn.edu/~pemantle/papers/burdzy.pdf).
What ...

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**1**answer

85 views

### Order statistics of Brownian motions

I've been struggling with proving a conjecture concerning order statistics of Brownian motions for a while. The conjecture I'm looking to prove is the following: (I have run Monte Carlo simulations ...

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63 views

### Wiener measure of hitting sets A,B but not C (or easier hitting A but not C)

I am trying to formulate the measure of event
$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,
where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise ...

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37 views

### Probability of hitting a Borel set by transient Brownian motion ($d\geq 3$)

I am looking for references/progress made in estimating the hitting probability for Borel sets.
For spheres we have $P_{x}(T_{B_{r}(0)}<\infty)=(\frac{|r|}{|x|})^{d-2}$, where $x=B_{0}$ for ...

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**1**answer

87 views

### Branching Brownian Motion and the KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...

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34 views

### Time Brownian motion spends above a curve

Let $W(t)$ be Brownian motion on the positive real line. I am looking for the critical function $g(t)$ sucht that
$$\int_0^\infty Ind(W(t) > g(t)) dt = \infty$$
vs.
$$\int_0^\infty Ind(W(t) > ...

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265 views

### Properties of the algebraic self-difference set of Brownian motion zeros

As I was trying to exhibit new interesting(?) path transformations of Brownian motion, I became interested in
the (random) set of times $t$ such that $B(t)=B(t+1)=0$, where $B(t)$ denotes a standard ...

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124 views

### Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type
$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$
converges to zero (in probability/distribution) for $h ...

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**1**answer

258 views

### Properties of the time integral of Wiener process

Let $W_t$ be a Wiener process and consider the time integral
$$ X_T:= \int_0^T W_t dt $$
It is often mentionend in literature that $X_T$ is a Gaussian
with mean 0 and variance $T^3/6$.
I am ...

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98 views

### number of times Brownian motion hits boundaries

Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...

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134 views

### Escape Time of Fractional Brownian Motion

Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$.
Is the expected time known ...

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**1**answer

140 views

### Cross variation two not independent Brownian motions

How can I calculate the cross variation between a standard Brownian motion $(B_t)_{t\geq 0}$ and the process $(B^T_{t})_{t\geq T}$ given by $B^T_t= B_t-B_{t-T}$? Here $T$ is just a positive number.
I ...

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614 views

### Do Random Walks on the Hexagonal Lattice have a limit?

For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that
the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn
induces a tiling of ...

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183 views

### Area covered by Brownian motion of 2D disc

I would like to know the expected value for the area covered by a disc of radius $R$ whose center undergoes Brownian motion (diffusion).
Specifically, let $\mathbf{X}_t$ represent a two-dimensional ...

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60 views

### A cylinder leaking Brownian particles, cut in half by a mirror

This question is tangentially related to Probability a Brownian particle with an exponentially distributed lifetime hits a sphere before vanishing.
I have an infinitely long cylinder of some radius ...

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**1**answer

181 views

### Probability a Brownian particle with an exponentially distributed lifetime hits a sphere before vanishing

Imagine I have a point source $p_0 = (x_0,y_0,z_0)$ that releases a point-like Brownian particle with a lifetime given by an exponentially distributed rate parameter $\lambda$. When the particle's ...

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223 views

### What is the characteristic functional for Brownian motion on a sphere?

I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...

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84 views

### Question about infinite-dimensional BM

Suppose we are given an $L^2(\mathcal{D})$-valued Brownian motion $W_t$ defined by
$$W_t:=\sum_{k=1}^{\infty}\sqrt{\sigma_k}W_t^k\phi_k(x),$$
where $\mathcal{D}$ is bounded domain in $\mathbb{R}^d$, ...

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155 views

### Law of the $L^2$ norm of a Brownian motion and related

Let $B_t$ be a Brownian motion with variance 1. We know that $\int_0^1 B(t) \mathrm{d} t \sim \mathcal{N}(0,1/3)$. I am interested to know what we can say about the law of the two random variables
$X ...

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305 views

### When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?

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162 views

### Proving that Brownian motion has no points of increase

I am reading Burdzy's paper on the points of increase of Brownian motion:
Burdzy's Paper
He is proving that, almost surely, a Brownian motion, has no points of increase. What he actually proves is ...

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69 views

### The supreme distribution of Brownian motion increment

Let $W_t$ be an one-dimensional standard Brownian motion, and $\theta_s$ is the shift such that $\theta_s( W_t)=W_{t+s}-W_s$, then are there any reference available regarding the distribution of the ...

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### Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...

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153 views

### Quadratic variation for discrete Martingale

Is there any analogue of continuous martingale quadratic variation for the discrete case? If so, are there any theorems which characterize simple random walk using quadratic variation - similar to ...

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286 views

### Expectation of the time t standard brownian motion stopped at itself's square

I have a one dimensional standard brownian motion $W$ defined under a stochastic basis with probability $\mathbf{Q}$ and filtration $\left(\mathscr{F}\right)_{t\in{\mathbf{R}}_{+}}$, and I want to ...

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145 views

### Distribution of last time Brownian motion crosses a line

Is the distribution of the last time Brownian motion crosses a line y=a*x known? (Equivalently, the distribution of the last time a Brownian motion with downwards drift hits 0.) It's not hard to give ...

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385 views

### Does the strong law of Large Number hold for an infinite dimensional Brownian motion?

For finite-dimensional Brownian motion $W_t$, it is well known that
\begin{equation}
\lim_{t\to \infty}\frac{W_t}{t}=0,\text{ a.s. }\ \ \ \ \hspace{1cm} \langle 1\rangle
\end{equation}
Now suppose we ...

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218 views

### Argmax of random walk vs of Brownian motion

Consider a random walk on $\mathbb{Z}$ with triangular drift and jumps that are standard normals. That is,
$$
\begin{cases}
RW_{t+1} = RW_t - d + \epsilon_t, \quad t \geq 0,\\
RW_{t-1} = RW_t - d + ...

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507 views

### random walk and Brownian motion on Riemannian manifold

As we know, the random walk on $\mathbb{Z}/n$ will converge(in some sense) to the Brownian motion on $\mathbb{R}$ when $n\to\infty$. I would like to know is there some higher dimensional analogy ...

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294 views

### Can one use Brownian motion to prove that two manifolds are not conformally equivalent?

Let me start by a very simple example; consider the following question:
"Let D1 be a square and D2 a rectangle (boundary included). View them
as subsets of the complex plane. Does there exist a ...

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173 views

### Malliavin calculus w.r.t $G$-Brownian motion

I wonder if it is possible to define a Malliavin calculus w.r.t $G$-Brownian motion defined on a Sublinear Expectation Space available on this link.
Gâ€“Brownian motion has a very rich and interesting ...

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309 views

### Brownian motion on Metric spaces

Is there a generalization of Brownian motion to general metric spaces (which should probably be length spaces)?
This should be a process satisfying
$$d(B_t, B_s) \sim \mathcal{N}(0, t-s)$$
and such ...

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217 views

### Ito Diffusions with low regularity?

I would like to have an ItĂ´ Diffusion
$$ X_t = \int_0^t b(s) \mathrm{d}s + \int_0^t \sigma(s) \mathrm{d}B_s.$$
where the (vector- and matrix-valued, respectively) functions $b$ and $\sigma$ have lower ...

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372 views

### Constructing Riemann maps using Brownian motion?

There's a relation between two-dimensional Brownian motion and conformal maps, see e.g. Thurston's answer to this question. Given two non-empty simply-connected domains $U$ and $V$ in the complex ...

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104 views

### Brownian flow and flow of metric

Suppose we have a Brownian flow of diffeomorphisms on R^n and we wish to represent it as a stochastic process on the metric i.e - Given a point x, the metric transforms as F*(t)(G), the pullback of G.
...

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186 views

### Brownian motion on Homogeneous spaces

Suppose we have a Brownian motion(or transition density) on a Lie group G and a Riemmanian manifold H on which G acts transitively and isometrically. Can we construct a Brownian motion( or transition ...

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282 views

### Measurability of subspace of set of all functions

Set $X=\mathbb{R}^n$ and let $X^{I}$, the space of maps from the (bounded or unbounded) interval $I$ to $X$, be endowed with the locally convex topology of pointwise convergence.
Is it true that the ...

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216 views

### Area of a Brownian bridge on the plane

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 around a unit square ...

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59 views

### Tail for the integral of a diffusion process

I would like to compute the following tail,
$$
\mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right),
$$
assuming
$$
\mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x),
$$
and $X$ is a diffusion ...

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287 views

### Reference needed: Donsker's Invariance Principle for Riemannian Manifolds

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

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### Exponential Ergodicity for Reflected Brownian Motion in a Bounded Domain

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 matrix, and oblique ...

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123 views

### Can we express a one-dimensional raised Bessel Bridge as a function of a single Brownian Motion?

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 conditioned such that ...

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206 views

### Independence using reflecting brownian motion

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, and thus $X$ and ...

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103 views

### Family of Brownian motions

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,...,X_n)?

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### Integrating a Bessel Bridge

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 independent Brownian Motion. ...