Suppose that the chaoticmotion of a particle along a line is depicted by a Markov process Y

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

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

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