Consider the local time of a standard Brownian motion $L_t^a$ for $(t,a)\in\mathbb{R}_+\times\mathbb{R}$. Is the probability of the event $$A:=\big\{ \text{There exists a neighborhood } N \text{ of } (0,0) \text{ such that } L_t^a>0 \text{ for } (t,a)\in N \text{ with } t>0\big\}$$ greater than zero? I haven't been able to characterize the behavior of $L_t^a$ in $a$ and $t$ jointly around the origin...
2026-04-30 00:09:27.1777507767
Local time of Brownian motion around the origin
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