# Tag Info

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 ...
• 184k

### 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 ...
• 9,843

### 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]. ...
• 14.1k
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 ...
• 14.1k
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,...
• 122k

### 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 ...
• 122k

### "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 ...
• 4,509
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 $(0,0)$...
• 9,843

### 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 ...
• 3,187
Accepted

• 6,105
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 ...
• 184k
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 ...
• 8,041
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 ...