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In mathematics, a continuous-time random walk (CTRW) is a generalization of random walks in which wandering particles wait a random amount of time between jumps. This is a random jump process, and the jump length and waiting time are distributed randomly. More generally, it can be seen as a special case of the Markov renewal process.
CTRW was introduced by Montroll and Weiss as an extension of the physical diffusion process to effectively describe anomalous diffusion, that is, super- and sub-diffusive cases. The equivalent formula for CTRW is given by the generalized master equation. A link between CTRW and diffusion equations with fractional time derivatives has been established. Similarly, the spatiotemporal fractional diffusion equation can be considered as a continuous approximation of CTRW with continuous distribution jumps or CTRW on a lattice.
1. Expression
Consider a random process X ( t ) X(t) defined byX(t)：
X ( t ) = X 0 + ∑ i = 1 N ( t ) Δ X i X(t)=X_{0}+\sum _{i=1}^{N(t)}\Delta X_{i} X(t)=X0+i=1∑N(t)ΔX _i
The amount of increase Δ X i \Delta X_{i}ΔX _iIs iid ( Independent and identically distributed random variables ) random variable, the value range is Ω \OmegaΩ， N ( t ) N(t) N ( t ) is the interval( 0 , t ) (0,t)(0,The number of jumps within t ) . at timettt takes valueXXThe probability of the process of X is given by
P ( X , t ) = ∑ n = 0 ∞ P ( n , t ) P n ( X ) P(X,t)=\sum _{n=0}^{\infty }P(n,t)P_{n}(X) P(X,t)=n=0∑∞P(n,t)Pn(X)
Here P n ( X ) P_{n}(X)Pn( X ) is the process atnnTake the value XX after n jumpsProbability of X , and P ( n , t ) P(n,t)P(n,t ) is at timettnnafter tProbability of n jumps.
2. Montroll-Weiss formula
We use τ \tauτ meansN ( t ) N(t)The waiting time between two jumps of N ( t ) , with ψ ( τ ) \psi (\tau)ψ ( τ ) represents its distribution. ψ ( τ ) \psi (\tau)The Laplace transform of ψ ( τ ) is defined as:
ψ ~ ( s ) = ∫ 0 ∞ d τ e − τ s ψ ( τ ) {\tilde {\psi }}(s)=\int _{0}^{\infty }\mathrm{d}\tau\ ,e^{-\tau s}\psi (\tau )p~(s)=∫0∞dτe− τ s ψ(τ)
Similarly, the jump distribution f ( Δ X ) f(\Delta X)The characteristic function of f ( ΔX ) is given by its Fourier transform:
f ^ ( k ) = ∫ Ω d ( Δ X ) eik Δ X f ( Δ X ) {\hat {f}}(k)=\int _{\Omega }\mathrm{d}(\Delta X)\ ,e^{ik\Delta X}f(\Delta X)f^(k)=∫Ohd(ΔX)eikΔXf(ΔX)
It can be shown that the probability P ( X , t ) P(X,t)P(X,The Laplace-Fourier transform of t ) is given by:
P ~ ^ ( k , s ) = 1 − ψ ~ ( s ) s 1 1 − ψ ~ ( s ) f ^ ( k ) {\hat {\tilde {P}}}(k,s)={\frac {1-{\tilde {\psi }}(s)}{s}}{\frac {1}{1 -{\tilde {\psi }}(s){\hat {f}}(k)} }P~^(k,s)=s1−p~(s)1−p~(s)f^(k)1
The above formula is called the Montroll-Weiss formula.
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