Time reversibility of Stratonovich Diffusion: Reference Request - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:40:21Z http://mathoverflow.net/feeds/question/95196 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95196/time-reversibility-of-stratonovich-diffusion-reference-request Time reversibility of Stratonovich Diffusion: Reference Request Paul Tupper 2012-04-25T20:22:26Z 2012-04-30T22:31:30Z <p>Please consider the Stratonovich stochastic differential equation (SDE) $$dX = b(X)\circ dB$$ where $B$ is standard Brownian motion and $X(0)=X_0$. This corresponds to the Ito (SDE) $$dX = \frac{1}{2} b(X) b'(X) dt + b(X) dB.$$</p> <p>I would like a reference showing (or even just stating) that trajectories of this equation are time-reversible in the following sense: that for all $m\geq 1$ and $t_m > t_{m-1} > \ldots > t_1 >0$, the joint distribution of $$(X(t_1), \ldots, X(t_m) )$$ is identical to the joint distribution of $$(X(-t_1), \ldots, X(-t_m) ).$$</p> <p>Also, is there a particular term for this kind of time-reversibility? People also use time-reversibility to mean detailed balance for systems in equilibrium, which is different from this.</p> <p><strong>Motivation</strong> In a paper I am listing advantages of expressing diffusions in terms of the Stratonovich convention. I want to be able to briefly state that if the drift coefficient in a Stratonovich SDE is 0, then the equation is time-reversible in the sense I state above.</p> <p><strong>Edit: Further Explanation</strong> Here is a clarification of what I mean above, as well as a justification of my claim.</p> <p>Let $B(t)$ for $t \in \mathbb{R}$ be two-sided Brownian motion with $B(0)=0$. Let $X(t)$ solve the above Stratonovich SDE. Let $Y(t)=X(-t)$. Then $$dY(t) = dX(-t) = -b(X(-t)) \circ dB(-t) = b(Y(t)) \circ d\tilde{B}(t)$$ where $\tilde{B}(t) = -B(-t)$ is also a Brownian motion. So $Y$ solves the same equation as $X$ with a different Brownian motion. These formal manipulations can be justified by letting $B$ be approximated by smooth stochastic processes and then taking the limit using the Wong-Zakai result.</p> <p>Thanks for any help!</p>