Consider the long time behavior for time in-homogeneous diffusion such as $$dX_t = dB_t - \nabla V(X_t)\,dt + b_t(X_t)dt,$$ where $V(x)$ is a smooth convex function and $b_t(x)$ is a time-dependent drift. I believe there're results which roughly say that

"If $b_t$ converges to $b$ as $t\to\infty$ and $V$ is sufficiently convex relative to $b$, then the law of $X_t$ converges (exponentially fast)",

but I couldn't find a close enough reference of such extensions (to time in-homogeneous case) for either the Bakry-Emery theory or the free-energy approach. Any suggestion would be appreciated.

Consider the long time behavior for a time in-homogeneous diffusion such as $$dX_t = dB_t - \nabla V(X_t)\,dt + b_t(X_t)dt,$$ where $V(x)$ is a smooth convex function and $b_t(x)$ is a time-dependent drift. I believe there're results which roughly say that

``If $b_t$ converges to $b$ as $t\to\infty$ and $V$ is sufficiently convex relative to $b$, then the law of $X_t$ converges (exponentially fast)",

but I couldn't find a close enough reference of such extensions (to time in-homogeneous case) for either the Bakry-Emery theory or the free-energy approach. Any suggestion would be appreciated.