As an illustrative example, consider a modified O-U process $dX_t = -X_tdt + \exp(-t)dW_t$. It is not too hard to understand that after a while the behaviour is dominated by the deterministic component, which will drive the process to $0$ almost surely (for instance by showing that its asymptotic mean and variance converge to 0).

**My question is the following**: are there any results on proving such a convergence from the generator directly? Formally, I have a (as well-behaved as you wish) generator $\mathcal{A}_t f(x) = \lim_{s \to 0+}\frac{\mathbb{E}[f(X_{t+s}) \mid X_t=x] - f(x)}{s}$, and a known value $x^*$, for which I want to verify that $X_t \to x^*$ a.s.. Can this be done directly by checking some properties of $\mathcal{A}_t$.

My first idea was to simply find conditions under which $\mathcal{A}_t$ converges to a $\mathcal{A}_{\infty}$ for which the Dirac $\delta_{x^*}$ is invariant. This is of course insufficient: it may happen that the decay in the process is too fast to eventually reach the desired asymptotic (imagine adding a $\exp(-t)$ in the drift of the OU above).

For context, I am trying to understand the convergence of continuous-time approximations of stochastic gradient descent (see, e.g., https://jmlr.org/papers/v20/17-526.html) in terms of their generator.

Thanks!