Let $W_u, 0\leq u \leq t$ be brownian motion. Let $m_t= min_{0\leq u\leq t} W_u$ and $M_t = max_{0 \leq u \leq t} W_u$.
The fact that $(M_t , W_t)$ is absolutely continuous to Lebesgue measure on $\mathbb{R}^2$ is known in some stochastic calculus book.
By symmetry of Brownian motion, $(m_t,W_t)$ is absolutely continous to Lebesgue measure on $\mathbb{R}^2$.
I want to know whether $(m_t, M_t , W_t)$ is absolutely continuous to Lebesgue measure on $\mathbb{R}^3$.