Characteristic Function and Transport Equation for the Particle Density

3 Characteristic Function and Transport Equation for the Particle Density 

Characteristic functions are very useful tools for studying random processes. It turns out that reaction-transport equations can also be effectively handled by using the characteristic function of the underlying random walks. In what follows, we will see how this function helps to define the transport operator, a pseudo-differential 

For illustrative purposes we begin with the transport of particles that follow the path of the compound Poisson process (3.71), X t) = Z,. The corresponding 

The function V (fe) plays a very important role in defining a transport operator. It follows from (3.87) that the function p k, t) satisfies the equation 

Applying the inverse Fourier transform to (3.89) with (3.88) and the standard convolution theorem, we obtain the Kolmogorov-Feller equation (3.74). Thus the particle density p(x, t) can be interpreted as the inverse Fourier transform of the characteristic function p k, t) = E(e ). Since p k, 0) = 1, the initial particle density is p(x, 0) = 5(x). The integral operator on the RHS of the Kolmogorov-Feller equation (3.74) can be considered as a pseudo-differential operator with symbol (3.88). Recall that a pseudo-differential operator L. acting on the variable x is defined by its Fourier transform as T L p x, t)] = ir k)p k, t), where fik) is referred to as the symbol of (see, for example, [15]). 

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