Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\tag2$$ is a Gelfand triple, $(W_t)_{t\ge0}$ is a $Q$- or cylindrical Wiener process on a separable Hilbert space $U$ and $U_0:=Q^{\frac12}U$. If $V=H=\mathbb R^d$, it is well-known that $$\tilde X_t:=X_{T-t}\;\;\;\text{for } t\in[0,T]$$ is again the solution to a SDE; see, for example, Time Reveral of Diffusions.
In the general, infinite-dimensional, setting, I wasn't able to find much in the literature. I guess that one problem is that in order to show the result in the link we need to assume that the distribution of $X_t$ has a density $p_t$ wrt the Lebesgue measure and at the end the drift of $(\tilde X_t)_{t\in[0,\:T]}$ contains the gradient $\nabla\ln p_t$.
The paper Infinite Dimensional Diffusion Models contains a result in the desired direction, but the equation considered there is less general than $(2)$.
I think one possibly could apply a Galerkin approach, considering finite-dimensional SDEs, using the available result from that case, and go back to the actual infinite-dimensional space. I'm open for any references and suggestions. In the case I'm mainly interested in $V=H^1(D)$ and $H=L^2(D)$ for some bounded open $D\subseteq\mathbb R^d$. $b$ and $\sigma$ don't need to depend on time; something like $b(u)=\operatorname{div}(a(\nabla u) \, \nabla u)$ and $\sigma$ will be of similar shape.
