Question

In this problem, you will demonstrate that the Gaussian PDF in Equation ( 9.64) is in fact the solution to the diffusion Equation ( 9.63). To do this, we will use frequency domain
methods. Define

time- varying characteristic function of the random process X (t).
(a) Starting from the diffusion Equation (9.63), show that the characteristic function must satisfy

Also, determine the appropriate initial condition for this differential equation. That is, find Φ (ω, t).
(b) Solve the first- order differential equation in part (a) and show that the characteristic function is of the form

(c) From the characteristic function, find the resulting PDF given by Equation (9.62).

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