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I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory contains the vertices of a equilateral triangle.

Mathematical Formulation

Let $(W_t)_{t\in[0,1]}$ a standard Wiener process on the interval $[0,1]$. Does it exist $\ell>0$ and $(t_1,t_2,t_3)\in [0,1]^3$ such that: $$ \begin{cases} ||(t_1,W_{t_1})-(t_2,W_{t_2})|| = \ell\\ ||(t_3,W_{t_3})-(t_2,W_{t_2})|| = \ell\\ ||(t_1,W_{t_1})-(t_3,W_{t_3})|| = \ell\\ \end{cases} ~~~~\mathrm{?} $$

Illustration

The problem I mention is then to find a equilateral triangle of length $\ell$ as in the drawing.

I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory contains the vertices of a equilateral triangle.

Mathematical Formulation

Let $(W_t)_{t\in[0,1]}$ a standard Wiener process on the interval $[0,1]$. Does there exist $\ell>0$ and $(t_1,t_2,t_3)\in [0,1]^3$ such that: $$ \begin{cases} \|(t_1,W_{t_1})-(t_2,W_{t_2})\| = \ell\\ \|(t_3,W_{t_3})-(t_2,W_{t_2})\| = \ell\\ \|(t_1,W_{t_1})-(t_3,W_{t_3})\| = \ell\\ \end{cases} ~~~~\mathrm{?} $$

Illustration

The problem I mention is then to find a equilateral triangle of length $\ell$ as in the drawing.

I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory contains the vertices of a equilateral triangle.

Mathematical Formulation

Let $(W_t)_{t\in[0,1]}$ a standard Wiener process on the interval $[0,1]$. Does it exist $\ell>0$ and $(t_1,t_2,t_3)\in [0,1]^3$ such that: $$ \begin{cases} |W_{t_1}-W_{t_2}| = \ell\\ |W_{t_3}-W_{t_2}| = \ell\\ |W_{t_3}-W_{t_1}| = \ell\\ \end{cases} ~~~~\mathrm{?} $$

Illustration

The problem I mention is then to find a equilateral triangle of length $\ell$ as in the drawing.

I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory contains the vertices of a equilateral triangle.

Mathematical Formulation

Let $(W_t)_{t\in[0,1]}$ a standard Wiener process on the interval $[0,1]$. Does it exist $\ell>0$ and $(t_1,t_2,t_3)\in [0,1]^3$ such that: $$ \begin{cases} ||(t_1,W_{t_1})-(t_2,W_{t_2})|| = \ell\\ ||(t_3,W_{t_3})-(t_2,W_{t_2})|| = \ell\\ ||(t_1,W_{t_1})-(t_3,W_{t_3})|| = \ell\\ \end{cases} ~~~~\mathrm{?} $$

Illustration

The problem I mention is then to find a equilateral triangle of length $\ell$ as in the drawing.

I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory contains the vertices of a equilateral triangle.

Mathematical Formulation

Let $(W_t)_{t\in[0,1]}$ a standard Wiener process on the interval $[0,1]$. Does it exist $\ell>0$ and $(t_1,t_2,t_3)\in [0,1]^3$ such that: $$ \begin{cases} |W_{t_1}-W_{t_2}| = \ell\\ |W_{t_3}-W_{t_2}| = \ell\\ |W_{t_3}-W_{t_1}| = \ell\\ \end{cases} ~~~~\mathrm{?} $$

I am curious about the following simple problem but I couldn't do any progress on it. I would like to know whether it is possible to prove (with probabilistic proof) that a brownian trajectory contains the vertices of a equilateral triangle.

Mathematical Formulation

Let $(W_t)_{t\in[0,1]}$ a standard Wiener process on the interval $[0,1]$. Does it exist $\ell>0$ and $(t_1,t_2,t_3)\in [0,1]^3$ such that: $$ \begin{cases} |W_{t_1}-W_{t_2}| = \ell\\ |W_{t_3}-W_{t_2}| = \ell\\ |W_{t_3}-W_{t_1}| = \ell\\ \end{cases} ~~~~\mathrm{?} $$

Illustration

The problem I mention is then to find a equilateral triangle of length $\ell$ as in the drawing.