# How can we prove that a stochastic process converges to a deterministic value?

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!

Set initial condition $$X_{0}=x_{0}=x^{*}=0$$ for simplicity. We can identify its long-term behaviour by studying its density $$p(t,x)$$ and thus the corresponding Fokker-Plank equation (in terms of the adjoint of the generator $$A^{*}$$)
$$\partial_{t}p(t,x)=A^{*}p(t,x)=\partial_{x}(-xp(t,x))-\frac{1}{2}\partial_{xx}(e^{-2t}p(t,x))$$ with initial data $$p(x,0)=\delta_{x}$$.
I plan to return to try to solve this or at least use parabolic estimates to get limiting behaviour, but I am not sure how doable it is especially because of the $$e^{-2t}$$ factor (here they discuss the similar issue of time-dependent coefficients). Did you also want an exact solution?