# Questions tagged [brownian-motion]

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### Comparison theorem for SDEs driven by a continuous martingale

Consider the well-known comparison theorem for SDEs, versions of which appear in several textbooks, e.g., Karatzas and Shreve, Proposition 5.2.18, or Revuz and Yor, Theorem IX.3.7. The result states ...
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### Equilateral triangle in a Brownian path

I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory ...
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### Are the paths of the Brownian motion contained in a suitable RKHS?

Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$. But is ...
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### Conditioned random walk over a graph

I want to solve for a conditioned random walk over a graph. I have a directed graph $G$. The random walkers start at a fixed node, Source. They all need to end up at fixed node, Sink. So the random ...
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### On the convergence of a martingale

Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by : $$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$ and for $t\geq 0$, we ...
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### Polar form of 2D Brownian motion

Consider two independent unidimensional Brownian motion $w_1$ and $w_2$. What is the polar form of $(w_1,w_2)$? If $r(t)$ and $\phi(t)$ satisfy $(w_1,w_2) = r(t)(\cos(\phi(t)),\sin(\phi(t)))$, how to ...
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### Macroscopic sets - a notion of largeness for Lebesgue null sets

Let $E$ be a measurable subset of $\mathbb R$. We say $E$ is $\alpha$-macroscopic, for $0 \leq \alpha \leq 1$, if there exists an $\alpha$-Holder continuous function $f: \mathbb R \to \mathbb R$ such ...
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### Stochastic braids

I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
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### Reflecting Brownian motion in disk

What is the transition density function of a reflecting Brownian motion in $\mathbb D \overset{\mathrm{def}}= \{z \in \mathbb C : \lvert z\rvert < 1\}$ and how to compute it? The transition density ...
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### From large deviations to finite time probability tails

Cross-Post from Math.SE Let $(B_t)$ be a standard $d$-dimensional Brownian motion. It is well-known that $$\mathbb P(\sup_{s\in[0,t]}|B_s|\ge \alpha) \le 4de^{-\alpha^2/2dt}.$$ One possibility to ...
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### Fractional Brownian motion covariance with a twist

Let $H \in (0, 1)$, $D \in \mathbb{R}$ and assume that the following function $$r ( t, s ) = \frac{1}{2} \, \Big[ t^{2H} + s^{2H} - | t - s |^{2H} \Big] + D \, t^H s^H, \quad t, \, s \geq 0$$ is ...
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### Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$

I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion. I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
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### Joint tail for Brownian motion $P[B_{t_1}>g_1,...,B_{t_n}>g_n]$

Maybe not surprisingly there seems to be a lack of in-depth study of sharp estimates for the joint tail of Brownian motion over different times $$P[B_{t_1}>g_1,...,B_{t_n}>g_n]$$ for strictly ...
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### Converse Cameron-Martin theorem for shifts by adapted processes

Let $W$ be a standard one dimensional Brownian motion, $\mathcal F_t$ its natural filtration, and $\mathbb P$ be the induced Wiener measure on $\Omega := C[0, 1]$. Given a $C[0, 1]$ valued random ...
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### Derive the solution of the diffusion equation from the solution of a random walk

Summary The probability distribution (pdf) of a random walk in 1 dimension is represented by a Bessel function. On the other hand, the pdf of a Brownian motion in free space is represented by a ...
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### Brownian motion hitting open set starting from its boundary

Let $\{W(t),\,t \in [0,1]\}$ be a standard Brownian motion in $\mathbb{R}^d$, starting from $0$. Let $U$ be a non-empty open set such that $0 \in \partial U$. Which conditions on $U$ are necessary and ...
The Wiener sausage of a standard Brownian motion $\{W(t),0 \leq t \leq T\}$ in $\mathbb{R}^2$ is the set $S(T,R)=\bigcup_{0 \leq t \leq T} W(t)+B(0,R)$, where $B(x,r)$ denotes a ball in $\mathbb{R}^2$ ...
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 $B_t\in C^s(x)$. ...