Questions tagged [brownian-motion]
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404
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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 [-...
15
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0
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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 ...
13
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2
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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}{\...
13
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1
answer
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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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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 ...
11
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2
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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 $(\...
10
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2
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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 ...
10
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1
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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 ...
10
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2
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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 ...
10
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1
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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 ...
9
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3
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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?
8
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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 ...
8
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2
answers
379
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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 $B_t\in C^s(x)$.
...
8
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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 ...
8
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2
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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, ...
8
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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 ...
8
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3
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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.
8
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1
answer
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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 ...
8
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1
answer
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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. ...
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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 ...
7
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4
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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 \...
7
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2
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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 \...
7
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2
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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 ...
7
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2
answers
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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 ...
7
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1
answer
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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
$...
7
votes
2
answers
388
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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 $...
7
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1
answer
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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 $W$ ...
7
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1
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363
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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 $\sigma(b)=0$. ...
7
votes
1
answer
271
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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 ...
7
votes
1
answer
727
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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 ...
7
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2
answers
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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 ...
6
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4
answers
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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$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...
6
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2
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Law of the $L^2$ norm of a Brownian motion and related
Let $B_t$ be a Brownian motion with variance 1. We know that $\int_0^1 B(t) \mathrm{d} t \sim \mathcal{N}(0,1/3)$. I am interested to know what we can say about the law of the two random variables
$X ...
6
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2
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Reference for LIL for fractional Brownian motion
(Cross-posted to https://math.stackexchange.com/questions/2377810/law-of-iterated-logarithm-for-fractional-brownian-motion.)
It seems strange but, even after consulting several books, and hours ...
6
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2
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Slight variation on law of the iterated logarithm
Let$$M_t = \max\{B_s : 0 \le s \le t\},\text{ }m_t = \min\{B_s : 0 \le s \le t\},$$where $B_t$ is a standard Brownian motion. My question is, does there exist $r$ such that with probability one,$$\...
6
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1
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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 ...
6
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1
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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) \\
...
6
votes
1
answer
367
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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 ...
6
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1
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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 ...
6
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1
answer
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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 ...
6
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1
answer
351
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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 $0$, ...
6
votes
1
answer
452
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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 ...
6
votes
1
answer
601
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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 $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...
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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 ...
6
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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 ...
5
votes
3
answers
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"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 ...
5
votes
2
answers
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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?
5
votes
2
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651
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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 ...
5
votes
2
answers
176
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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 ...
5
votes
1
answer
502
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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 ...