I am interested in obtaining a closed form solution for the stationary distribution $q(x,y)$ to the following Kolmogorov Forward Equation
$$ \frac{\partial}{\partial t} q(x,y,t) = - \frac{\partial}{\partial x} q(x,y,t) a(x,y) $$ $$+ \lambda_1 \Big( \delta(y)\int_{-\infty}^{\infty}q(x,z,t)dz - q(x,y,t) \Big) + \lambda_2 \Big( \frac{\partial}{\partial y}\big(F(y) \int_0^y q(x,z,t)dz \big) - q(x,y,t) \Big) $$
Where $q(x,y)$ is a joint density and $F(x)$ is a CDF with positive support. $\delta(x)$ is the Dirac Delta function. I was able to solve for the marginal density of $y$ in closed form (its CDF is $\frac{\lambda_1}{\lambda_1+\lambda_2(1-F(y))}$ over $y>0$) but that is all. What tools would you recommend? I am not too familiar with calculations involving Laplace transform in this context. Thank you