Let $X_t$ be a geometric brownian motion with drift $\mu x$ and volatility $\sigma x$. There is a barrier at $x^*$ such that when $X_t$ reaches $x^*$, it is reset to some initial level $x_0 > x^*$. There is no symmetric upper barrier. What is the stationary distribution of such a process?
My background is in economics, so I'm unsure of the terminology I ought to be using. However, I was unable to find reference to this either via Wikipedia or a number of stochastic calculus textbooks.
I thought about how the forward equation would look for this stationary distribution $f$ $$ 0 = 1_{x = x_0} g + \frac{\mathrm{d}}{\mathrm{d}x}\left[\mu\left(x\right) f\left(x\right)\right] + \frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{d}x^2}\left[\sigma\left(x\right)^2 f\left(x\right)\right] $$ where $g$ denotes the flux through the barrier, but am unfamiliar with how I ought to proceed. Part of my confusion comes from the fact that $g$ is dependent upon the stationary distribution. The unconditional probability of hitting the boundary is dependent upon the distribution.