How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial differential equation?

$$\frac{\partial}{\partial t}p = \frac{k^2}2\frac{\partial^2}{\partial v^2}(vp)+\frac{\partial}{\partial v}(\gamma(v-\theta)p) \tag1$$ with initial and boundary conditions $$P(t=0,v) = \delta(v-v_0),$$ $$P(t,v=0)=P(t,v=\infty)=0,$$ and all variables being real, $k, \gamma, \theta>0$. We can start with $k,\gamma,\theta$ being constant then graduating to they being functions of only $v$ then finally to being functions of $t$ and $v$.

We can solve this equation via Fourier transform and express the solution with special functions. Then we can see the solution approaches a Gamma distribution at time infinity. My question is 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?

I am thinking about obtaining the properties of spectrum (eigenvalues) of the elliptic operator on the right hand side of Equation (1). Currently it is a non-self-adjoint operator. Had it been a self-adjoint elliptic operator, its spectrum are 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?

There is a sequel to this question.