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

Question

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!

Answer 1

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?

If you also want an SDE perspective on more general drifts see "An SDE perspective on stochastic convex optimization" where they study invariant measure limits.