Questions tagged [brownian-motion]

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When does a correlated Brownian motion leave a square?

Let $B=(X,Y)$ be a correlated two-dimensional Brownian motion, that is, the components are standard Brownian motions and the covariance between $X_t$ and $Y_t$ is $t\rho$ for some constant $\rho \in [-...
Jochen Wengenroth's user avatar
15 votes
0 answers
463 views

Quantitative Skorokhod embedding

The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
Dor's user avatar
  • 723
13 votes
2 answers
1k views

How long for Brownian motion to "fill-out" a torus in d-dimensions?

I've been taken by the concise result1 that (roughly!), on a $2$-dimensional torus $\mathbb{T}^2$, the time it takes to visit nearly every point (within $\epsilon$, as $\epsilon \to 0$) is: $\frac{2}{\...
Joseph O'Rourke's user avatar
13 votes
1 answer
3k 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 ...
shu's user avatar
  • 1,091
12 votes
4 answers
1k 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 ...
Axel Boldt's user avatar
11 votes
2 answers
1k views

Local time of Brownian motion + Lipschitz continuous function

Let $\mathrm{ Lip} (M)$ denote the space of all functions on $[0,T]$ with Lipschitz constant and $L^\infty$ norm bounded by $M$. Let $(B_t)_t$ be a Brownian motion defined on the probability space $(\...
Nina Holden's user avatar
10 votes
2 answers
990 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 ...
Yue's user avatar
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10 votes
1 answer
537 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 ...
Ritwik's user avatar
  • 3,235
10 votes
2 answers
1k views

Covariance function of Brownian motion and the second derivative operator

I recently noticed something about the covariance function of a Brownian motion that I don't quite understand, and I was wondering if anyone could help me. Suppose $W$ is a Brownian motion, and we ...
Simon Lyons's user avatar
  • 1,646
10 votes
1 answer
350 views

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 ...
user57600's user avatar
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9 votes
3 answers
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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?
Hans's user avatar
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8 votes
2 answers
1k views

The Wiener measure of an open set

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 ...
Liviu Nicolaescu's user avatar
8 votes
2 answers
379 views

Regularity of translations for Brownian motion

Let $B_t$ be the classic Brownian motion. I understand that, if $s>1/2$, almost surely $B_t$ is nowhere $s$-Hölder continuous i.e. almost surely for no point $x$ it happens that $B_t\in C^s(x)$. ...
pipenauss's user avatar
  • 297
8 votes
2 answers
865 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 ...
John Jumper's user avatar
8 votes
2 answers
3k views

Finite time hitting probabilities for Brownian motion in the plane

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 2D). However, ...
Jeff Schenker's user avatar
8 votes
3 answers
1k 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 ...
Matthias Ludewig's user avatar
8 votes
3 answers
932 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.
kassandra's user avatar
  • 165
8 votes
1 answer
527 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 ...
ght's user avatar
  • 3,616
8 votes
1 answer
1k views

Location of maximum of Brownian motion with rough drift

I am interested in the distribution of the $\text{argmax}_{t \in [0,1]} \{B(t) + f(t)\}$, where $B$ is a Brownian motion (or Brownian bridge) and $f:[0,1] \to \mathbb{R}$ is a continuous function. ...
X. Wang's user avatar
  • 93
8 votes
0 answers
467 views

Basis for $L^2(\mathbb{R})$ that Solves the Heat Equation

This is a less-than-serious question that I asked on math.SE, but I suspect it is slightly more appropriate to ask it here. Consider the heat equation $$ u_t = \frac12 u_{xx} $$ On $\mathbb{T}$ with ...
Greg Zitelli's user avatar
7 votes
4 answers
947 views

Probability that planar Brownian motion doesn't "encircle" 0

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^2$ and $T = \text{inf}\{t : |B_t| = 1\}$. Let $E$ denote the event that $0$ is contained in the unbounded component of $\mathbb{R}^2 \...
user71299's user avatar
7 votes
2 answers
882 views

Brownian motion in $n$ dimensions

Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in \...
user82390's user avatar
7 votes
2 answers
4k 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 ...
wood's user avatar
  • 2,714
7 votes
2 answers
792 views

Reference for Feynman-Kac

I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's ‎An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...
waaat's user avatar
  • 73
7 votes
1 answer
3k views

Change of time variable in Wiener process

I'm following a solution of an SDE from here http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf Start with the SDE $$ dX_t = \delta dt + 2\sqrt{X_t} dW_t $$ consider a deterministic time change $...
chuse's user avatar
  • 173
7 votes
2 answers
388 views

Fractional Brownian motion of Riemann-Liouville type is not a semimartingale

Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...
El_mago's user avatar
  • 99
7 votes
1 answer
1k views

Moment bounds on exponential martingale

Consider the exponential martingale used in the Girsanov transformation of measure: $$Z(t) = \exp\Big(\int_0^tXdW - \frac{1}{2}\int_0^t|X|^2ds\Big)$$ so that $Z$ solves the sde $dZ = ZXdW$ where $W$ ...
user253775's user avatar
7 votes
1 answer
363 views

What happens when the diffusion term in an SDE becomes zero?

Consider this time-homogeneous SDE, in the Ito sense: $$dX_t= -(X_t-a)\,dt+\sigma(X_t)\,dW_t,$$ where $W_t$ is standard Brownian motion, $a<b\in\mathbb{R}$, $X_0\leq b$ a.s., and $\sigma(b)=0$. ...
ColorfulLion's user avatar
7 votes
1 answer
271 views

A Converse of the Skorokhod Embedding Theorem

I am wondering whether the following "sort of converse" of Skorokhod's embedding theorem holds: Suppose that $\{D_t\}_{t \geq 0}$ is a stochastic process with continuous paths, $D_0 = 0$, and suppose ...
Probabilist's user avatar
7 votes
1 answer
727 views

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 ...
Timothy Budd's user avatar
  • 3,545
7 votes
2 answers
316 views

Probabilistic characterization of first Neumann eigenvalue

In this MO post, a question has been asked (and answered) about the probabilistic interpretation of the first Dirichlet eigenvalue of the Laplacian in terms of boundary hitting times. I wish to ask ...
SMS's user avatar
  • 1,293
6 votes
4 answers
603 views

Number of intervals needed to cross, Brownian motion

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...
user avatar
6 votes
2 answers
1k 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 ...
Goulifet's user avatar
  • 2,174
6 votes
2 answers
680 views

Reference for LIL for fractional Brownian motion

(Cross-posted to https://math.stackexchange.com/questions/2377810/law-of-iterated-logarithm-for-fractional-brownian-motion.) It seems strange but, even after consulting several books, and hours ...
Joe's user avatar
  • 759
6 votes
2 answers
374 views

Slight variation on law of the iterated logarithm

Let$$M_t = \max\{B_s : 0 \le s \le t\},\text{ }m_t = \min\{B_s : 0 \le s \le t\},$$where $B_t$ is a standard Brownian motion. My question is, does there exist $r$ such that with probability one,$$\...
user avatar
6 votes
1 answer
2k views

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 ...
nabla's user avatar
  • 205
6 votes
1 answer
543 views

Is this a Brownian motion?

I am building a 2D stochastic process as follows. I start with a point $P_0=(0,0)$. Then $P_k=(X_k,Y_k)$ is defined as follows, for $k>0$: \begin{align} X_k & =X_{k-1}+R_k \cos(2\pi\theta_k) \\ ...
Vincent Granville's user avatar
6 votes
1 answer
367 views

Convolution of two Brownian motions

Suppose $B_1(t)$ and $B_2(t)$ are two independent, standard Brownian motions. What is the distribution of \begin{align*} G(t) = \int_0^t B_1(\tau)B_2(t-\tau)d\tau \qquad \end{align*} Or, at least an ...
Tom Chen's user avatar
  • 229
6 votes
1 answer
2k views

Brownian motion and its maximum and its minimum

Let $W_u, 0\leq u \leq t$ be Brownian motion. Let $m_t= min_{0\leq u\leq t} W_u$ and $M_t = max_{0 \leq u \leq t} W_u$. The fact that $(M_t , W_t)$ is absolutely continuous with respect to Lebesgue ...
Seongqjini's user avatar
6 votes
1 answer
304 views

In the plane, does complement of Brownian path have infinitely many connected components?

Let $d = 2$. Do we have that with $P_x$—probability $1$, for every $T> 0$ the complement $W[0, T]^c$ of the Brownian path up to time $T$ has infinitely many connected components? I had seen this ...
Edward Hoenn's user avatar
6 votes
1 answer
351 views

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 $0$, ...
yilin wang's user avatar
6 votes
1 answer
452 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 ...
MassiveJack's user avatar
6 votes
1 answer
601 views

weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
Zhao Guohuan's user avatar
6 votes
0 answers
225 views

Running maximum/supremum of Brownian motion: add information to make it a Markov process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it ...
Ziv's user avatar
  • 121
6 votes
0 answers
219 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 ...
r_faszanatas's user avatar
5 votes
3 answers
898 views

"Practical" use of time-continuous stochastic processes like Wiener process or Poisson (point) process?

If one uses the Wiener process as an ingredient to model something, then for practical purposes one could just as well take a simple discrete random walk (with sufficiently fine scale). If one uses a ...
Mr H's user avatar
  • 59
5 votes
2 answers
1k views

Average Value of Area Closed by Brownian Motion

Two dimensional brownian motion will intersect its own path infinitly many times. What is the average value of area, closed by curve during an intersection in brownian motion?
Mohammad Ghiasi's user avatar
5 votes
2 answers
651 views

Endpoint of Brownian motion conditional on high maxima

Note: This question is closely related to an earlier question: A large noise limit. Let $W$ be a standard one dimensional Brownian motion. For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...
Nate River's user avatar
  • 4,822
5 votes
2 answers
176 views

Density near at $0$ for the integral of the positive part of the Brownian motion

This question was asked recently on MO and then deleted by the owner, user Aalon. I think the question deserves to be answered, which is what I will try to do here. Aalon was reading this paper, where ...
Iosif Pinelis's user avatar
5 votes
1 answer
502 views

Largeness of the set of zeroes of a Brownian motion

Definitions: A measurable subset $S$ of $\mathbb R$ is said to be mesoscopic if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero ...
Nate River's user avatar
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