# Questions tagged [brownian-motion]

Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

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### What is the typical probability space when we talk about a Brownian motion?

What is the typical probability space $(\Omega, \mathcal{F}, P)$ when we talk about a Brownian motion $(B_t)$? I found some papers used $\Omega = C[0,\infty), \mathcal{F}=\mathcal{B}(C[0,\infty))$, ...
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### Questions in proving $\mathbb{P}\left(T_a<\infty\right)=1$ with $T_a:=\inf \{t>0: B_t \ge a\}$

Let $\left(B_t, t \geq 0\right)$ be a one-dimensional Brownian motion starting from the origin (i.e, $\left.B_0=0\right)$. Let $\mathcal{F}_t:=\sigma\left(B_s: s \leq t\right)$ be the filtration ...
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### What is the transition semigroup of toroidally wrapped Brownian motion?

More generally, let $(B_t)_{t\ge0}$ be an $\mathbb R^d$-valued Lévy process and $W:=\iota(B)$, where $$\iota:\mathbb R^d\to[0,1)^d\;,\;\;\;x\mapsto x-\lfloor x\rfloor$$ (the floor function is applied ...
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### What is the probability that Brownian motion hits $\mathbb{Q}^d$?

Let $B$ be a standard $d$-dimensional Brownian motion. What is $\mathbb{P}(\exists t > 0 : B_t \in \mathbb{Q}^d)$? For $d = 1$, this is clearly $1$. However, intuitively, it seems that it should be ...
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