pr.probability - 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 $f in L^1(mu)$,

$lim_{trightarrow infty}frac{1}{2t}int_{-t}^t f(T_s x) ds = int f(x)dmu(x)$.

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 $trightarrow infty$? 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.

Thanks for your help.