This is a follow-up on a previous question. Now the parabolic PDE of $P(t,x,v)$ has two spatial dimensions.

$$
\partial_t P = L^* P \tag1
$$
$$L^*P = \frac12\left(\kappa^2\frac{\partial^2}{\partial v^2}+2\rho\kappa\frac{\partial^2}{\partial x\partial v}+\frac{\partial^2}{\partial x^2}+\frac{\partial}{\partial x}\right)(vP)+\gamma\frac{\partial}{\partial v}((v-\theta)P)$$
where $\kappa,\rho,\theta$ are all positive constants and $\rho\in[0,1]$, with intial-boundary condition
$$P(t=0,x,v)= \delta(x)\delta(v-v_0),$$
$$P(t,x,v=0)=P(t,x,v=\infty)=P(t,x=-\infty,v)=P(t,x=\infty,v)=0.$$

Integrating $p$ over $x$ we get the parabolic equation in the previous question.

Without solving this equation explicitly, can we prove the solution approaches the stationary solution which is the elliptic PDE obtained from setting the time partial derivative of the original PDE to zero?

Currently it is a non-self-adjoint operator. Had it been a self-adjoint elliptic operator, its spectrum would be all discrete and non-negative (and approaching infinity). We can deduce easily that all positive eigenfunctions decay away as time approaches infinity leaving only the zero eigenfunction which is the stationary solution. Can we somehow transform Equation (1) into one with self-adjoint elliptic operator? More importantly, what is a general theory on the spectrum of this kind of elliptic operators?