Given a stochastic PDE: $\partial_tf(x,t) = D\partial^2_{xx}f(x,t) -vf(x,t) + \lambda(x)$.
The textbook author suggests that the solution for $f(x,t)$ is $f(x,t) = \int_{-\infty}^{+\infty}\frac{dx'}{\sqrt{4\pi Dt}}f(x', t=0)e^{-\frac{(x'-x)^2}{4\pi Dt}-vt} + \int_{0}^{t}dt' \int_{-\infty}^{+\infty}\frac{dx'}{\sqrt{4\pi D(t'-t)}}\lambda(x')e^{-\frac{(x'-x)^2}{4\pi D(t-t')}-v(t-t')}$.
I am not quite sure how the author developed the closed-form solution and would like to learn more.
Does anyone know any good starting points in solving the above equation? In addition, are there any good references (or textbooks) that develop methods of finding closed form solutions for stochastic PDEs?
Thank you in advance for any help.