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Results tagged with brownian-motion
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user 81488
0
votes
Arcsine law for Brownian motion with drift
I don't know if there is a nice closed-form expression for this, but you can try to work it out writing
$$
P[Z\leq y] = P[X_t\neq 0 \text{ for }x\in (y,1]] = \int_{-\infty}^{+\infty} f_{y}(v) P_v[X_t\ …
- 1,897
4
votes
In the plane, does complement of Brownian path have infinitely many connected components?
Yes. Just observe that
(1) on any fixed time interval the Brownian path intersects itself with positive probability (easy to see);
(2) but the above implies that on any time interval the Brownian pa …
- 1,897
4
votes
Slight variation on law of the iterated logarithm
You can use the fact that $B_t-m_t$ is a reflected Brownian motion (see e.g. Revuz-Yor, Chapter VI, Theorem 2.3). I think it shouldn't be difficult to show that
$$
\limsup_{t\to\infty} \frac{M_t-m_t}{ …
- 1,897
9
votes
Accepted
Brownian motion in $n$ dimensions
The process $\|B(t)\|$ is called $n$-dimensional Bessel process (or Bessel process with parameter $\nu=\frac{n}{2}-1$). I think formula $\bf 4$.1.1.4 of Borodin-Salminen "Handbook of Brownian Motion - …
- 1,897
5
votes
Accepted
Brownian motion in $\mathbb{R}^n$, probability of hitting a set
It's not that simple. See about polar/nonpolar points/sets e.g. in http://wiki.math.toronto.edu/TorontoMathWiki/index.php/Brownian_Motion_and_Harmonic_functions
If I remember correctly, a set is not …
- 1,897