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1answer
38 views

Defining a brownian bridge indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
1
vote
2answers
190 views

$\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?

I am trying to prove the following Lemma, which seems intuitive, but I still have doubts: Lemma Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two bounded processes, $\mu$ and $\sigma$, ...
0
votes
0answers
71 views

Rolling map as a diffeomorphism?

Let $M$ be a (compact) Riemannian manifold and $x \in M$. For a piecewise smooth path $\gamma: [0, T] \longrightarrow M$, we can define Cartan's development map (or rolling map) $$(\Phi\gamma)(t) = ...
1
vote
1answer
124 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 ...
0
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0answers
29 views

References for symmetric α-stable process (SSP) for $a>2$

Many properties of Brownian motion have been extended to SSP's for $0\leq \alpha\leq 2$ and so it is quite easy to find literature on them. However, I am currently studying the SSP for $\alpha>2$ ...
4
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0answers
65 views

“One sided” fast times of Brownian motion

Let $B_t$, $t \in [0,1]$ be a standard Brownian motion. We call a time $t$ fast up if $$ \limsup_{h \searrow 0} \frac{B(t+h) - B(t)}{\sqrt{2 h \ln(1/h)}} =1. $$ (Note the absence of absolute value ...
5
votes
2answers
111 views

Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds

I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a ...
5
votes
1answer
262 views

Can a Brownian motion be fast at its extrema?

After pondering this MO question > Location of maximum of Brownian motion with rough drift <, I wonder whether a Brownian motion can be fast (i.e. beats the law of the iterated logarithm) at its ...
0
votes
1answer
217 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
5
votes
1answer
218 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. ...
1
vote
1answer
91 views

Deriving Newtonian capacity of sphere from Brownian motion

We have the following result by Spitzer (see (1) or Port) $lim_{t\to \infty}\frac{1}{t}\int_{\mathbb{R}^{n}/B_{r_{0}}}P_{x}(T_{B_{r_{0}}}<t)dx=Cap(B_{r_{0}})=\frac{r_{0}}{4\pi}$ By Chuancun and ...
2
votes
1answer
112 views

Maximizer of random walk with very small drift

This is an extended question based on Large deviations for maximizer of random walk with drift. Let $$S_k = X_1 + \ldots + X_k,$$ where $X_i$ are i.i.d. with mean $-\mu < 0$ and unit variance. ...
2
votes
1answer
60 views

Hitting probability of semiball

For fixed x and hemisphere H of radius r and centered at the origin, I wonder what is $P_{x}(T_{H}<\infty)$. Attempt Firstly, I wonder if there is any relation between $P_{x}(T_{H}<N)$ and ...
0
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0answers
26 views

Density of $\int_{B}\frac{|1-|B_{T}|^{2}|}{|y-B_{T}|^{3}}dS(y)$

For $B\subset \partial B(0,1)))$ and random variable $B_{T}\in Int(B(0,1))$ with density $p_{T}$, is there a density for $\int_{B}\frac{|1-|B_{T}|^{2}|}{|y-B_{T}|^{3}}dS(y)$? Context The original ...
2
votes
0answers
48 views

What is the probability of B.M. hitting two disjoint spheres $(d\geq 3)$?

The hitting probability for spheres centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$, where $|x|>r$. 1)So I was wondering how can one compute ...
8
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1answer
369 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 ...
0
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1answer
18 views

$P_{x}(T_{A}<\infty)<P_{x}(T_{B}<\infty)$ imply $Cap_{N}(A)<Cap_{N}(B)$, where $Cap_{N}$ is Newtonian capacity

We start a Brownian motion at $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a large enough ball that contains compact sets A B. In other words, the B.M. starts on the exterior of A and B. Then if the ...
0
votes
2answers
64 views

Newtonian capacity of sphere equals its hitting probability by Brownian motion?

Do we have $Cap_{N}(S(0,r))=P_{x}(T_{S(0,r)}<\infty)$ for $x\in [B(0,r)]^{c}$, where $B(0,r)$ is a ball centered at the origin ? I know for $x=0$, they are both equal to 1. How can I go about ...
0
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2answers
97 views

Asymptotics for Hitting the sphere from the Outside

The problem is: consider A a solid ball centered at 0 and the exterior starting point $x\in A^{c}$, what is the behavior of $P_{x}(T_{B_{r}(0)}>t)$ for $d\geq 3$ as $t\to \infty$,where ...
2
votes
0answers
81 views

(Reference) Asymptotics of hitting probability by Brownian motion

The problem is: Given compact set A with positive finite volume (eg. ball,cube), what happens to $P_{x}(T_{A}>t)$ as $t\to \infty$, where $T_{A}=inf_{t>0}(B_{t}\in A)$ and x is in the "exterior" ...
4
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2answers
206 views

Brownian motion and hitting a Quadrilateral

I want to compute the hitting probability of a bounded plane by a Brownian motion starting at the origin. In other words, given the coordinates of a quadrilateral A , can we compute ...
1
vote
1answer
160 views

Reference question: Brownian motion and surface area

I am doing research on the hitting probability of various sets (eg. 3D convex) and specifically how changes in perimeter/surface area change the hitting probability. By hitting probability I mean ...
10
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1answer
286 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 ...
2
votes
1answer
116 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, ...
2
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3answers
174 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 ...
4
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0answers
93 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 ...
0
votes
0answers
58 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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0answers
93 views

Probability that d-Brownian Motion ,$d\geq 3$, avoids a fixed set A

In other words, the probability that Brownian motion stays within $A^{c}$. What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...
0
votes
1answer
96 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 ...
3
votes
1answer
126 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 ...
2
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0answers
59 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 ...
1
vote
1answer
134 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 ...
0
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0answers
37 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) > ...
5
votes
1answer
289 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 ...
1
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0answers
175 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 ...
6
votes
1answer
588 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 ...
1
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0answers
111 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 ...
5
votes
1answer
156 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 ...
2
votes
1answer
348 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 ...
4
votes
2answers
690 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 ...
8
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2answers
233 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 ...
2
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0answers
62 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 ...
2
votes
1answer
224 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 ...
1
vote
0answers
307 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 ...
2
votes
1answer
91 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 $\mathbb{R}^d$, ...
5
votes
2answers
182 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 ...
4
votes
3answers
413 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?
2
votes
1answer
180 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 ...
0
votes
0answers
77 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 ...
4
votes
2answers
212 views

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 ...