Since interested in convergence to zero or $x_{*}$, one approach is phrasing it as a gradient descent SDE [An SDE perspective on stochastic convex optimization][1]

Here are some other tools to help you study the growth of the process. The most comprehensive tool is Feller's test (Shreve-Karatzas theorem 5.29).
Secondly, one can also use Lyapunov functions as done in ["Stochastic Stability of Differential Equations"][2]. Also see "Solutions of Stochastic Differential Equations obeying the Law of the Iterated Logarithm, with Applications to Financial Markets∗".

Thirdly, if one is looking for invariant measures, you can use the stationary Fokker-Plank eg. https://math.stackexchange.com/questions/683775/invariant-measures-for-stochastic-processes and ["Long-time dynamics
of stochastic differential equations"][3].


  [2]: https://link.springer.com/book/10.1007/978-3-642-23280-0
[3]:https://arxiv.org/pdf/2106.12998.pdf


  [1]: https://arxiv.org/pdf/2207.02750.pdf