I'm hoping to analyze a certain kind of stochastic process, but I'm not sure what to call it/what to google to see if it's been studied already. Basically, in my problem there's a current state $x_t$, which at each time is updated in a finite number of directions $\{\xi ^i\}_{i=1}^N$, with a probability proportional to e.g. the similarity between $x_t$ and $\xi^i$. For example, it might be something like $p(\Delta x_t = \xi^i) = \frac{1}{Z}\exp(\beta \langle x_t ,\xi^i \rangle)$ with normalizer $Z$ and some parameter $\beta$.
The reason I'm interested, is that I want something like $p(\hat{x_{\infty}} | x_0)$, the distribution of the direction of $x$ at infinite time, given a particular initialization distribution. Intuitively, the state will probably eventually get caught into a basin of attraction and will keep walking in the same direction, but what I care about is how to determine which attractor an initial state will eventually fall into.
If I just need to simulate, so be it! But it seems like such a simple process, I was just wondering if someone's already studied it. Apologies for the very basic question, I've tried googling around and looking through various articles but just don't know what to call it!
Just playing around - suppose the direction vector is $\hat{x}(t)=(\cos[\theta(t)],\sin[\theta(t)])$.
Suppose the direction angle $\theta$ tends to relax exponentially toward a preferred direction $\bar{\theta}$ with a relaxation time $\tau$, but Gaussian white noise fluctuations $\xi(t)$ tend to disturb this relaxation. Then $$ \dot{\theta}(t) = \frac{\bar{\theta} - \theta}{\tau} + \xi(t). $$
If the correlation function of the Gaussian white noise is written $\langle \xi(t) \xi(s) \rangle = 2 D \delta(t-s)$, the Fokker-Planck equation for the angle is $$ \frac{\partial P}{\partial t}(\theta,t) = -\frac{\partial}{\partial \theta}\frac{\bar{\theta} - \theta}{\tau} P + D\frac{\partial^2}{\partial\theta^2} P.$$ This is an Ornstein-Uhlenbeck process. This Fokker-Planck equation is an advection-diffusion equation. It has the solution (didn't check, just guessing) $$ P(\theta,t) = \sqrt{\frac{1}{2 \pi D \tau\big(1-e^{-2t/\tau}\big)}}\exp\Big[-\frac{(\theta - \bar{\theta} - \theta_0 e^{-t/\tau})^2}{2 D\tau \big(1-e^{-2t/\tau}\big)}\Big], $$ where $P(\theta,0)=\delta(\theta-\theta_0)$ is the initial condition.
You could then write the distribution of the unit vector as $$ W(\hat{x},t) = \int_{-\infty}^\infty d\theta \delta^2\big(\hat{x}-\hat{x}(\theta)\big)P(\theta,t).$$ This is just a simplified model of the situation you describe.
For $t\gg \tau$ you get typical fluctuations around $\bar{\theta}$ of order $\delta\theta = 2 D \tau$. So the unit vector is symmetrically distributed with typical deviations between $\hat{x}(\bar{\theta}-\delta\theta)$ and $\hat{x}(\bar{\theta}+\delta\theta)$.
I vaguely recall something like this being in the book "Stochastic processes in physics and chemistry" by Van Kampen. If I remember he was discussing it in context of the alignment of dipoles with a magnetic field.
The key assumption in this approach is that the rate of change in the angle is linear in the deviation $\theta-\bar{\theta}$ from the preferred direction. If it is not linear, the problem gets very difficult.