Suppose that the “chaotic”motion of a particle along a line is depicted by a Markov process Y_t with continuous trajectories. Let p(y|x, t) be the density of Y_t under the initial-time condition Y₀ = x. A. Einstein has shown that the function p(y|x, t) should satisfy the equation

which is called the diffusion equation with a diffusion coefficient σ². Show that the density of the r.v. Y_t = x+w_t satisfies (3.1) for σ = 1. Modify Yt to make this true for σ ≠ 1.