How to perform Laplace transform to a Fokker-Planck equation?

Question

I've been stuck at point where Laplace transform is needed to get the solution. This is the FP equation I'm trying to solve $$ \frac{\partial}{\partial t} P = \gamma \frac{\partial}{\partial v} [(v-\theta)P] + \frac{1}{2} \frac{\partial}{\partial x} (vP) + \rho \kappa \frac{\partial^{2}}{\partial x \partial v} (vP) + \frac{1}{2} \frac{\partial^{2}}{\partial x^{2}} (vP) + \frac{\kappa^{2}}{2} \frac{\partial^{2}}{\partial v^{2}} (vP) $$ The initial condition for the above equation is a product of two delta functions $$ P_{t=0}(x,v\mid v_{i}) = \delta (x) \delta (v-v_{i}) $$ Now, since x appears in the derivative operator first Fourier transform can be taken. $$ P_{t} (x,v\mid v_{i}) = \int_{-\infty}^{+\infty} \frac{dp_{x}}{2\pi} e^{ip_{x}x} \bar{P}_{t,p_{x}}(v\mid v_{i}) $$ the solution after FT looks like this: $$ \frac{\partial}{\partial t} \bar{P} = \gamma \frac{\partial}{\partial v} [(v-\theta)\bar{P}] -\left[ \frac{p_{x}^{2} - ip_{x}} {2} v - i\rho p_{x} \kappa \frac{\partial}{ \partial v} v - \frac{\kappa^{2}}{2} \frac{\partial^{2}}{\partial x^{2}} v \right] \bar{P} $$ after FT the equation is linear in v and quadratic in $\partial / \partial v$, it can be simplified by taking the Laplace transform over v $$ \tilde{P}_{t,p_{x}} (p_{v},v\mid v_{i}) = \int_{0}^{+\infty}dv e^{-p_{v}v} \bar{P}_{t,p_{x}}(v\mid v_{i}) $$ the final solution is $$ \left[\frac{\partial}{\partial t} + \left(\Gamma p_{v} + \frac{\kappa^{2}}{2}p_{v}^{2} - \frac{p_{x}^{2} - ip_{x}}{2}\right) \frac{\partial}{\partial p_{v}} \right] \tilde{P} = -\gamma \theta p_{v} \tilde{P} $$ where $$ \Gamma = \gamma + i\rho \kappa p_{x} $$ So my question is how they perform the Laplace transform and arrive at the final solution? I've tried so many times but definitely I'm missing something. Could someone show the interim steps? Thanks!