# Birkhoff ergodic theorem for dynamical systems driven by a Wiener process

At the risk of asking a stupid question I have the following problem.

Suppose I have a measure preserving dynamical system $(X, \mathcal{F}, \mu, T_s)$, where

• $X$ is a set
• $\mathcal{F}$ is a sigma-algebra on $X$,
• $\mu$ is a probability measure on $X$,
• $T_s:X \rightarrow X$, is a group of measure preserving transformations parametrized by $s \in \mathbb{R}$.

Suppose that this dynamical system is ergodic, so that for any $f \in L^1(\mu)$,

$\lim_{t\rightarrow \infty}\frac{1}{2t}\int_{-t}^t f(T_s x) ds = \int f(x)d\mu(x)$.

Now let $B_s$ be a real valued Wiener process such that $B_0 = 0$, then I can define the following process:

$\frac{1}{t}\int_{0}^t f(T_{B_s} x) ds$

Does anybody know how this process would behave as $t\rightarrow \infty$? Intuitively I would expect it to converge to a similar constant for a.e realisation of the brownian motion, but I can't find a convincing argument.