Xt is a weak solution to the SDE with dXt = ( −αXt + γ )dt + β dBt , ∀t ≥ 0 X0 = x0. α, β , and γ constants, and Bt is a brownina motion.
need to find the PDE for the transition density of X at time t, pt(x0,.), and solve it
the expectation is E[Xt]=γ /α +exp(-αt)(x0-γ/α) and the variance V[Xt]=b^2/(2α)(1-exp(-2αt).
and i know that Xt follows a normal distribution and that it has a stationary distribution. so p(t,x,y)=N~(E[Xt],V[Xt])
but how to find the PDE of the transition density/ probability distribution?