12
votes

Accepted

### Probability Brownian motion lies between $2$ functions

This is the problem of Brownian motion between two moving absorbing boundaries. For a linear time dependence some analytical progress can be made, but for arbitrary time dependence no closed-form ...

11
votes

### White noise vs. black noise

Percolation 'noise' is generated by a perfectly good family of random variables, the 'quad-crossing' events. Basically, for every diffeomorphic image of the unit square, this random variable is $1$ if ...

10
votes

### The Wiener measure of an open set

This is known as the support theorem for Brownian motion. Besides the proof in the answer of Iosif Pinelis and the proof in Exercise 1.8 of [1], there is also a proof on page 59 of [2]. ...

9
votes

Accepted

### Largeness of the set of zeroes of a Brownian motion

Yes, the local time (at zero) maps the zero set of Brownian motion to an interval. See e.g. Lemma 6.9 page 159 in [1] for continuity.
[1] Brownian motion, by Peter Mörters and Yuval Peres. Cambridge ...

9
votes

Accepted

### The Wiener measure of an open set

$\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\om}{\omega}$Let $g:=f_0$. There is some real $\de>0$ such that
\begin{equation*}
\om(g,\de):=\max\{|g(y)-g(x)|\colon x,y\in[0,...

9
votes

### Regularity of translations for Brownian motion

If $f$ is absolutely continuous with $\int_0^1 f'(t)^2\,dt<\infty$, then, by Girsanov's formula (see e.g. Theorem 5.1. in [1]), the process $(B_t+f(t))_{t\in[0,1]}$ is a standard Brownian motion ...

9
votes

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

You answered your own question, I think. Many physics/economics models involve partial differential equations which often are studied using Feynman-Kac (among many other methods), which involves ...

9
votes

Accepted

### Equilateral triangle in a Brownian path

The answer is yes, here is a sketch of proof. Take a decreasing sequence $\{t_n\}_{n \ge 0}$ such that $W_{t_n} = 0$ and such that $\lim_{n \to \infty}t_n = 0$. Consider the triangles formed by $(0,0)$...

8
votes

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

It looks like the $d$-dimensional case is easier generally than the $d=2$ case according to the excerpt below from this paper (see page three).
...the two-dimensional model is also more difficult ...

8
votes

Accepted

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

A very general answer, in dimension $d\geq 3$,
is in the following paper of Dembo, Peres and Rosen
https://projecteuclid.org/euclid.ejp/1464037588:
for compact $d$-dimensional manifolds,
$$C_\...

8
votes

Accepted

### SDE driven by fractional Brownian motion

When $H > 1/2$, one can interpret the integral as a Riemann-Stieltjes integral and one gets solutions by classical Picard iteration in $C^\alpha$ for $1/2 < \alpha < H$. It's not hard to see ...

7
votes

Accepted

### Endpoint of Brownian motion conditional on high maxima

$\newcommand{\ep}{\varepsilon}\newcommand{\vpi}{\varphi}\newcommand{\de}{\delta}$Yes, this is true:
By the reflection principle (see e.g. Proposition 2, for $M:=\max_{0\le t\le1}W_t$,
\begin{equation}
...

7
votes

Accepted

### Quadratic variation of supremum of brownian motion

The quadratic variation is identically $0$, i.e.
$$\langle S, S \rangle_t = 0$$
for all $t$, almost surely.
To see this, note that $S$ is almost surely increasing, hence has bounded variation almost ...

6
votes

### Reference for LIL for fractional Brownian motion

There is formula here (item 3):
https://math.stackexchange.com/a/2417816/21498
and how about that:
Séminaire de Probabilités XLI pp 161-179
Part of the Lecture Notes in Mathematics book series (LNM,...

6
votes

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

I consulted Igor Borisov, who is an expert on these (and other) matters and, in particular, wrote a paper (Theory Probab. Appl. 25 (1980), no. 3, 454--465 (1981)) on the existence of a bounded density ...

6
votes

Accepted

### Orthonormal frame bundles on a manifold

On the orthonormal frame bundle we have soldering forms $\omega_i$ and connection forms $\omega_{ij}$. A lift is horizontal just when $\omega_{ij}=0$ on it. So the velocity can be described by its $\...

6
votes

Accepted

### Use stochastic process to express solution to Laplace equation in the whole space

If $f(x) / (1 + |x|)$ is integrable, then the solution $u$ is equal to the Newtonian potential of $f$:
$$ -u(x) = \frac{1}{4\pi} \int_{\mathbb R^3} \frac{f(y)}{|x - y|} \, dy . $$
And the Newtonian ...

6
votes

Accepted

### Is this a Brownian motion?

I vote for Mateusz Kwaśnicki. The condition for whether the random walk you generate this way scales towards Brownian motion under taking long times and rescaling is whether or not the variance is ...

6
votes

### Endpoint of Brownian motion conditional on high maxima

Iosif Pinelis has already posted an answer, but here is an alternate answer communicated to me by Yuval Peres on a different website. Any typoes/mistakes are most definitely mine.
Write
$$\tau = \text{...

6
votes

Accepted

### Derive the solution of the diffusion equation from the solution of a random walk

To carry out the limit, it helps to start from an integral representation of the Bessel function,
$$P_n(T)=e^{-T}I_n(T)=\frac{1}{2\pi}\int_{-\pi}^\pi \exp [i k n+T \cos k-T]\,dk.$$
For $T\gg 1$ this ...

5
votes

Accepted

### Reference for Feynman-Kac

Please see p.282 of the following ref, it does not call it Feynman-Kac Formula but the whole section 5 is discussing it. It formalized the diffusion using a tensor field over a manifold which is the ...

5
votes

Accepted

### Moment bounds on exponential martingale

There are a number of ways to bound moments of $Z$ in terms of exponential moments of $X$. For some sharp results, see Theorem 1.5 of Kazamaki's book, "Continuous exponential martingales and BMO," as ...

5
votes

### Moments of the Hölder norm of Brownian process

It looks like it is finite for all $p > 1$. Here is a neat "bootstrap" argument (I still cannot believe it works, but I fail to find an error).
Denote $$S_T = \sup_{0\le s<t\le T} \dfrac{|B(t) -...

5
votes

### Moments of the Hölder norm of Brownian process

Much more is true: we have
$$\mathbb{E} [\exp(\epsilon \|B\|_{0,\alpha}^2)] < \infty$$
for some $\epsilon > 0$. Here $\|\omega\|_{0,\alpha} = \sup_{0 \le s < t \le T} \frac{|\omega(t)-\omega(...

5
votes

Accepted

### Is there Brownian motion on Alexandrov spaces?

Yes, there is a natural Brownian motion on an Alexandrov space.
In the following paper:
Kuwae, Kazuhiro; Machigashira, Yoshiroh; Shioya, Takashi, Sobolev spaces, Laplacian, and heat kernel on ...

5
votes

Accepted

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

Thank you for your work on this question! I am one of the authors of the cited paper and, yes, we stated a wrong claim (1) in the appendix of this paper (let me remark that the results in the main ...

5
votes

Accepted

### Pathwise stochastic integral as a linear operator on continuous functions

You can do the same for less regular functions, for example for functions with finite $p$-variation for some fixed $p \in [1,2)$ or functions in $\mathcal{C}^\alpha$ for some fixed $\alpha > 1/2$. ...

5
votes

Accepted

### What is the distribution of $2M_1-B_1$ where $M_t$ is the maximum process of the the Brownian motion $B_t$

Yes, the pdf of this distribution is
\begin{equation}
u\mapsto 2u^2 f(u)\,1(u>0) \tag{1}
\end{equation}
where $f$ is the standard normal pdf.
Indeed, by Proposition 2,
\begin{equation}
G(m,b):=...

5
votes

Accepted

### Each diffusion SDE is associated to a *unique* family of transition kernels

there are unique corresponding forward/backward equations (Fokker Plank) to an SDE, and unique solutions for them that correspond to transition kernels. See the nice notes here Lecture 10: Forward and ...

5
votes

Accepted

### Macroscopic sets - a notion of largeness for Lebesgue null sets

By Frostman's lemma, if $E$ is a compact set of positive $\alpha$-Hausdorff content, then there exists a probability Borel measure $\mu$ supported in $E$ such that $\mu(I) \leq c |I|^\alpha $ for ...

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