Some Limit Theorems on Reversed Brownian Motion

Abstract

Let $B$ be the standard one-dimensional Brownian motion, $T_x = \inf\{s \mid s \geq 0; B(s) = x\}, x > 0, T = T_0 \wedge T_a$, \begin{equation*}\gamma_x(t) = \begin{cases} \sup\{s \mid s \leq t; B(s) = x\}\\ , \gamma(t) = \gamma_0(t) \vee \gamma_a(t)\\ 0 \text{if above set} = \phi\end{cases}\end{equation*} where $a > 0$ is fixed and $W(s) = B(T - s), 0 \leq s \leq T$. Let $Z(s)$ be the restriction of $B$ to the interval $\lbrack\gamma(t), t\rbrack$, that is, $Z(s) = B(\gamma(t) + s), 0 \leq s \leq L(t)$. In this paper we use a "time reversal" argument to study relations as $t \rightarrow \infty$ between the processes $Z$ and $W$ under $P^y(\cdot \mid B(t) = x)$ and to evaluate some limits related to $L(t)$.