All Questions
12 questions
2
votes
1
answer
159
views
Measurability of two hitting times at the stopped $\sigma$-algebra
Let $\mathcal{F}=(\mathcal{F}_t)_{t\ge 0}$ be the complete filtration generated by the Brownian motion $B $, and let $a<0<b$. Define the stopping times
$\tau_a=\inf\{t\ge 0\mid B_t=a\}$ and $\...
- 503
0
votes
0
answers
95
views
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 ...
- 11
1
vote
1
answer
168
views
Resources to understand Lebesgue measure of Brownian motion's path [closed]
[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47]
Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
- 11
1
vote
0
answers
103
views
Continuity of Wiener measure on open balls
Let $\mu$ be the Wiener measure on $C_0 [0, T]$, the space of continuous functions starting at $0$, under the sup norm.
Question: Is it true that the function $r \mapsto \mu(B_r(x))$ is continuous in $...
- 6,155
5
votes
1
answer
548
views
Largeness of the set of zeroes of a Brownian motion
Definitions:
A measurable subset $S$ of $\mathbb R$ is said to be mesoscopic if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero ...
- 6,155
1
vote
0
answers
72
views
Local time as a measurable map from Wiener space
Let $B$ be a Brownian motion on $[0,1]$. The local time of $B$, which I will denote by $L$, is defined as the process on $\mathbb R$ such that
$$\int_0^1 F(B_t)~dt=\int_\mathbb R F(x)L(x)~dx,\qquad\...
- 891
0
votes
0
answers
274
views
Constructing uncountably many independent random variables with same distribution from Brownian motion?
It is well known one cannot construct uncountable many independent random variables on $([0, 1], \mathcal{B}[0, 1], \lambda)$. ($\lambda$ Lebesgue measure.)
Also, one can clearly construct infinitely ...
user152718
3
votes
3
answers
1k
views
Continuity of Brownian motion constructed from Kolmogorov extension theorem?
I'm trying to construct Brownian motion using the Kolmogorov extension theorem.
I am happy with the construction of a process with the required FDDs as (the canonical process associated with) a ...
user152718
2
votes
0
answers
56
views
What is the Wiener measure of the set of curves with given Hölder constant on a Riemannian manifold?
Let $M$ be a connected Riemannian manifold and $x_0 \in M$. For $0 < \alpha < \frac 1 2$, let
$$H = \{ c : [0,1] \to M \mid c(0) = x_0 \text{ and } \exists C>0 \text { s.t. } d(c(s), c(t)) \...
- 5,407
2
votes
1
answer
326
views
What is the Wiener measure of the curves with Hölder index $\frac 1 2$?
One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index ...
- 5,407
3
votes
1
answer
156
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 ...
- 5,474
2
votes
2
answers
571
views
Family of Brownian Motions
I am trying to show the following statement
Let $D\subset \mathbb{R}^2$ be an open and bounded subset. $\Pi=(P^x : x \in D )$ a Family of standard Brownian Motions started at $x \in D$. Then $\Pi$ ...
- 279