Brownian Motion, Levy Flight, and the Diffusion Equations

1 Brownian Motion, Levy Flight, and the Diffusion Equations 

We start with a very simple one-dimensional diffusion equation 

It should be noted that we integrate with respect to the forward variable y in (3.236). In this case, (3.236) has a very nice probabilistic interpretation. Consider the Brownian motion B t), which is a stochastic process with independent increments, such that B(t + s) - B(s) is normally distributed with zero mean and variance 2Dt. The corresponding transition probability density function p y, t x) is given by (3.237). Therefore the solution (3.236) has a probabilistic representation 

Of course (3.234) and (3.240) are identical in form, but only the forward equation (3.240) has the physical meaning of a transport equation for particles. We will discuss the difference between forward and backward equations in the next section. It turns out that, it is more convenient to deal with the backward equation (3.234). Let us give an example. The Brownian motion B t) starting at x can be rewritten in terms of the standard Wiener process W(t) as 

3 Random Walks and Mesoscopic Reaction-Transport Equations 

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