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72 views

Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots, S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin. Let $\tau_{N}$ be the first time $S_{n}$ exits ...
3
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1answer
77 views

Reflected Brownian Motion with random barrier?

I am looking for a way to say something about $$P\left(\max_{t\in[0,n]} W_t+|W_m|> x\right),$$ for $n>m$, where $W$ is a brownian motion with drift.
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0answers
117 views

Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is $$ P\left(B_t\in d y, L_t^0\in d v\right) = ...
1
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1answer
67 views

Is an arbitrary Brownian-motion path a viscosity solution of every differential equation?

Is an arbitrary Brownian path a viscosity solution of every differential equation? My intuition is that a path of Brownian motion is so ill-behaved that it not only does not have derivatives ...
4
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1answer
137 views

Area enclosed by Brownian motion (without winding number)

The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
4
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2answers
213 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?
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1answer
76 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 ...
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2answers
214 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 $\sigma$, ...
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0answers
76 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
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1answer
186 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 ...
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0answers
33 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
72 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
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2answers
137 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
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1answer
268 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 ...
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1answer
243 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 ...
6
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1answer
283 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
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1answer
102 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 ...
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1answer
128 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
71 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 ...
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0answers
28 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 ...
3
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0answers
57 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
399 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
19 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 ...
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2answers
77 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
98 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
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0answers
95 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" ...
5
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2answers
213 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
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1answer
172 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
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1answer
122 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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3answers
203 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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0answers
98 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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0answers
66 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
101 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
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1answer
101 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
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1answer
128 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
60 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
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1answer
167 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 ...
5
votes
1answer
291 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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0answers
192 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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1answer
739 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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0answers
114 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
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1answer
165 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
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1answer
415 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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2answers
729 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
votes
2answers
253 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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0answers
65 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
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1answer
239 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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0answers
365 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
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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 $\mathbb{R}^d$, ...