Time reversibility of Stratonovich Diffusion: Reference Request

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) $$dX = \frac{1}{2} b(X) b'(X) dt + b(X) dB.$$

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 $$(X(-t_1), \ldots, X(-t_m) ).$$

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.

Motivation 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.

Edit: Further Explanation Here is a clarification of what I mean above, as well as a justification of my claim.

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.

Thanks for any help!

• Yes, my further explanation proves the result. But what I was originally looking for was a reference to cite for it. It seems like such a simple and basic property that it must have at least been mentioned before somewhere. It's kind of like time reversibility for Hamiltonian systems but I've never seen the connection made explicit. – Paul Tupper Jun 28 '17 at 16:58
• In particular, the property itself doesn't have anything to do with diffusions, or even with the Markov property necessarily. – Paul Tupper Jun 28 '17 at 17:32

To understand what this "time reversibility" means, it is instructive to replace the stochastic differential equations by the corresponding equations for the time dependence of the probability distribution $p(x,t)$. This is sufficient for the comparison of two-time correlations, in this case between $t=0$ and $t'>0$ (forward in time) or $t'<0$ (backward in time).

"Time-reversibility" from the OP then amounts to the statement that the right-hand-side of the Kolmogorov backward and forward equations, $$-\frac{\partial}{\partial t}p(x,t)=\mu(x)\frac{\partial}{\partial x}p(x,t) + \tfrac{1}{2}\sigma^2(x)\frac{\partial^2}{\partial x^{2}}p(x,t)$$ $$\frac{\partial}{\partial t}p(x,t)=-\frac{\partial}{\partial x}[\mu(x)p(x,t)] + \tfrac{1}{2}\frac{\partial^2}{\partial x^2}[\sigma^2(x)p(x,t)]$$ becomes identical if the drift $\mu$ is related to the diffusion $D=\tfrac{1}{2}\sigma^2$ by $$\mu=\frac{\partial D}{\partial x},$$ which is indeed the case, as one can readily verify by substitution. The forward and backward Kolmogorov equations then have the same form $$\pm\frac{\partial}{\partial t}p(x,t)=\frac{\partial}{\partial x}\left[D(x)\frac{\partial}{\partial x}p(x,t)\right]$$

In response to the questions in the OP:

• Concerning a reference to the literature: The general relation between forward and backward diffusion is discussed in Time reversal of diffusions (1986) and in On the drift of a reversed diffusion (2005). Neither of these two papers makes the observation that there is a special case such that the drift for the forward and backward diffusion becomes the same, so as far as I can tell the observation in the OP is novel.

• Concerning a term to refer to this special case of equivalence of diffusion and reverse diffusion, I would suggest self-adjoint diffusion, since the corresponding Kolmogorov operator $\nabla D\nabla$ is self-adjoint.

• As mentioned in the OP, none of this has any bearing on the physics notions of "time-reversibility" or "detailed balance", which refer to a more general relation between drift and diffusion, involving the steady-state equilibrium density. I might add that from a physics point of view, the identity of drift and derivative of diffusion required for self-adjoint diffusion is quite unnatural. This may explain why the special case of a self-adjoint Kolmogorov equation has not appeared before in the literature.

• I am a bit confused that how can you justify replacing stochastic differentials by ordinary ones... – Henry.L Jun 26 '17 at 12:21
• @Henry.L -- are you referring to the equivalence of Langevin dynamics (SDE in the sense of Ito or Stratonovich) and the Fokker-Planck or Kolmogorov PDE ? I thought these are fully equivalent descriptions of a stochastic process. – Carlo Beenakker Jun 26 '17 at 12:48
• Actually no. I am well aware that Ito SDE and FP PDE are equivalent, but from what OP described, I thought he just wanted a ref. about the special situation (for simplicity let's assume we are discussing about $L^2$ processes) where the trajectories form a symmetric space w.r.t. to $t$. Such a situation is studied, AFAIK, using path analysis in late 1990s by D.Stroock. And it is not quite about "time-reversal" but "symmetric processes" – Henry.L Jun 26 '17 at 13:03
• And the HAUSSMANN-PARDOUX paper you cited did not specify that the reversed process share the identical finite dimensional distribution families with the original process as the OP discussed. So your answer(which is great!) actually explained a slightly different situation from what OP described as I felt...But restate it in terms of PDE does not reveal too much more about the situation in OP. – Henry.L Jun 26 '17 at 13:06
• @Henry.L --- thanks for the clarification, I have added the restriction to two-time correlations implied by the PDE representation of the SDE. – Carlo Beenakker Jun 27 '17 at 15:47