Exercise 12.4 (Fokker–Planck Equation) For a Brownian motion, prove that the transition probability function P(x, y, t) as well as the density function p(x, y, t) satisfies the PDE

This PDE is called the Fokker–Planck equation or the Kolmogorov forward equation by an obvious reason; cf. (7.7).

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t 2 2 -P+ 2 2 P=-lay P.