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I am struggling with finding a solution to the following Fokker Planck equation with linear potential:

$$\partial_{t}P(x,t)=k\partial_{x}P(x,t)+D\partial_{x}^{2}P(x,t)$$

Can anyone help me please?

P.s. I just found the equilibrium solution, which is the following (renormalizable for $x$ positive):

$P_{eq}(x)=\frac{k}{D}e^{-\frac{k}{D}x}$

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    $\begingroup$ Linear potential for all x has no stable solution. The solution you gave is for potential V(x) ~ x for positive x, and infinity for negative x. Your equation has no damping too. Please rephrase the question - searching for method of solving? steady state solution? what are the boundary conditions? Solution to linear potential appears in Garadiner's Stochastic methods book (p.122, 4th ed). Risken's 'Fokker-Planck' book is very helpful too (ch 5). $\endgroup$ – Alexander Feb 22 '20 at 11:38
  • $\begingroup$ Yes, you are right. I am looking for the solution of the above Fokker Planck solution for any t. $\endgroup$ – Umberto Tomasini Feb 22 '20 at 11:48
  • $\begingroup$ Thank you for your suggestions! $\endgroup$ – Umberto Tomasini Feb 22 '20 at 11:48
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The corresponding SDE is $$\mathrm{d}x_t = -k\mathrm{d}t + \sigma\mathrm{d}W_t$$ where $D = \frac{1}{2}\sigma^2$. This obviously is just the Gaussian with variance $\sigma^2t$ centered at $-kt$, i.e. your solution is (assuming Dirac at 0 as the initial condition) $$P(x, t) = \frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{(x + kt)^2}{4Dt}\right)$$

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