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.


Thanks for your help.



Birkhoff ergodic theorem for dynamical systems driven by a Wiener process
Not a stupid question, but I think the answer is no. 


The paper Random Ergodic Theorems with Universally Representative Sequences
by Lacey, Petersen, Wierdl and Rudolph gives a counterexample in the case where the system is being driven by a simple symmetric random walk (based on an application of Strassen's functional law of the iterated logarithm). I'm pretty sure the same technique would give a counterexample here.


The paper can be found online at: <http://www.numdam.org/item?id=AIHPB_1994__30_3_353_0>


