Rahul Sharma (Editor)

Continuous time random walk

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In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times. More generally it can be seen to be a special case of a Markov renewal process.

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Motivation

CTRW was introduced by Montroll and Weiss as a generalization of physical diffusion process to effectively describe anomalous diffusion, i.e., the super- and sub-diffusive cases. An equivalent formulation of the CTRW is given by generalized master equations. A connection between CTRWs and diffusion equations with fractional time derivatives has been established. Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices.

Formulation

A simple formulation of a CTRW is to consider the stochastic process X ( t ) defined by

X ( t ) = X 0 + ∑ i = 1 N ( t ) Δ X i ,

whose increments Δ X i are iid random variables taking values in a domain Ω and N ( t ) is the number of jumps in the interval ( 0 , t ) . The probability for the process taking the value X at time t is then given by

P ( X , t ) = ∑ n = 0 ∞ P ( n , t ) P n ( X ) .

Here P n ( X ) is the probability for the process taking the value X after n jumps, and P ( n , t ) is the probability of having n jumps after time t .

Montroll-Weiss formula

We denote by τ the waiting time in between two jumps of N ( t ) and by ψ ( τ ) its distribution. The Laplace transform of ψ ( τ ) is defined by

ψ ~ ( s ) = ∫ 0 ∞ d τ e − τ s ψ ( τ ) .

Similarly, the characteristic function of the jump distribution f ( Δ X ) is given by its Fourier transform:

f ^ ( k ) = ∫ Ω d ( Δ X ) e i k Δ X f ( Δ X ) .

One can show that the Laplace-Fourier transform of the probability P ( X , t ) is given by

P ~ ^ ( k , s ) = 1 − ψ ~ ( s ) s 1 1 − ψ ~ ( s ) f ^ ( k ) .

The above is called Montroll-Weiss formula.

Examples

The Wiener process is the standard example of a continuous time random walk in which the waiting times are exponential and the jumps are continuous and normally distributed.

References

Continuous-time random walk Wikipedia
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