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!



Time reversibility of Stratonovich Diffusion: Reference Request
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](https://en.wikipedia.org/wiki/Kolmogorov_backward_equations_(diffusion)),
$$-\frac{\partial}{\partial t}p(x,t)=\mu(x)\frac{\partial}{\partial x}p(x,t) + \tfrac{1}{2}\sigma^2(x)\frac{\partial^2}{\partial x^{2}}p(x,t)$$
$$\frac{\partial}{\partial t}p(x,t)=-\frac{\partial}{\partial x}[\mu(x)p(x,t)] + \tfrac{1}{2}\frac{\partial^2}{\partial x^2}[\sigma^2(x)p(x,t)]$$
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](https://old.i2m.univ-amu.fr/~pardoux/TRD.pdf) (1986) and in [On the drift of a reversed diffusion](https://link.springer.com/chapter/10.1007%2FBFb0005071?LI=true) (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.


