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### Lévy's characterization of Brownian motion: right-continuous processes

Cool question. Proposition Let $(X_u)_{u}$ be right-continuous martingale with $X_0=0$, such that $(X^2_u-u)_u,(X_u^3-3uX_u)_u,(X_u^4-6uX_u^2+3u^2)_u$ are martingales. Then for every integer $M \ge 1$,...
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### prove $P\{\int_{0}^{\infty}f(W_s)ds=\infty\}=1$

We need to show that $$\int_0^T f(B_t) dt\to\infty, T\to\infty.$$ I will use the occupation density formula (UPD see a simpler argument below):  \int_0^T f(B_t) dt = \int_{\mathbb{R}} f(x) L^x_T(B)...
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