Birkhoff ergodic theorem for dynamical systems driven by a Wiener process - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:50:31Z http://mathoverflow.net/feeds/question/29011 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29011/birkhoff-ergodic-theorem-for-dynamical-systems-driven-by-a-wiener-process Birkhoff ergodic theorem for dynamical systems driven by a Wiener process RadonNikodym 2010-06-21T23:41:38Z 2010-11-25T07:54:48Z <p>At the risk of asking a stupid question I have the following problem.</p> <p>Suppose I have a measure preserving dynamical system <code>$(X, \mathcal{F}, \mu, T_s)$</code>, where </p> <ul> <li><code>$X$</code> is a set </li> <li>$\mathcal{F}$ is a sigma-algebra on $X$,</li> <li>$\mu$ is a probability measure on $X$,</li> <li>$T_s:X \rightarrow X$, is a group of measure preserving transformations parametrized by $s \in \mathbb{R}$.</li> </ul> <p>Suppose that this dynamical system is ergodic, so that for any $f \in L^1(\mu)$,</p> <p><code>$\lim_{t\rightarrow \infty}\frac{1}{2t}\int_{-t}^t f(T_s x) ds = \int f(x)d\mu(x)$</code>.</p> <p>Now let $B_s$ be a real valued Wiener process such that $B_0 = 0$, then I can define the following process:</p> <p><code>$\frac{1}{t}\int_{0}^t f(T_{B_s} x) ds$</code></p> <p>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.</p> <p>Thanks for your help.</p> http://mathoverflow.net/questions/29011/birkhoff-ergodic-theorem-for-dynamical-systems-driven-by-a-wiener-process/47307#47307 Answer by Anthony Quas for Birkhoff ergodic theorem for dynamical systems driven by a Wiener process Anthony Quas 2010-11-25T07:53:57Z 2010-11-25T07:53:57Z <p>Not a stupid question, but I think the answer is no. </p> <p>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.</p> <p>The paper can be found online at: <a href="http://www.numdam.org/item?id=AIHPB_1994__30_3_353_0" rel="nofollow">http://www.numdam.org/item?id=AIHPB_1994__30_3_353_0</a></p>