Connection between transition probability and SDE? Can someone highlight what is the connection between the transition probability of a continuous time stochastic process $X_t$, i.e. $p(x,t\vert x_0,0)$ and the stochastic differential equation of the evolution of a trajectory of the process $X_t$, i.e. $dX_t=bdt+\sigma dB_t$. How to get from $p$ to the SDE and what assumptions does one need to make?

From Wikipedia: 
https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation#One_dimension
The stochastic differential equation
$$
dX_t = \mu(X_t, t)dt+ \sigma(X_t,t) dW_t
$$
has the probability density function $p(x,t)$ satisfying 
$$
\frac{\partial}{\partial t}P(x,t)= -\frac{\partial}{\partial x}[\mu(x,t)p(x,t)]+\frac12\frac{\partial^2}{\partial x^2}[\sigma^2(x,t)p(x,t)]
$$
For the transition probability, you want to impose the following initial condition on the above PDE:
$$
p(x, 0) = \delta(x-x_0).
$$
Then $p(x,t)$ with this initial condition is $p(x,t|x_0,0)$.
