Continuous-time random walk

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. 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. 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. A simple formulation of a CTRW is to consider the stochastic process X ( t ) {displaystyle X(t)} defined by whose increments Δ X i {displaystyle Delta X_{i}} are iid random variables taking values in a domain Ω {displaystyle Omega } and N ( t ) {displaystyle N(t)} is the number of jumps in the interval ( 0 , t ) {displaystyle (0,t)} . The probability for the process taking the value X {displaystyle X} at time t {displaystyle t} is then given by Here P n ( X ) {displaystyle P_{n}(X)} is the probability for the process taking the value X {displaystyle X} after n {displaystyle n} jumps, and P ( n , t ) {displaystyle P(n,t)} is the probability of having n {displaystyle n} jumps after time t {displaystyle t} . We denote by τ {displaystyle au } the waiting time in between two jumps of N ( t ) {displaystyle N(t)} and by ψ ( τ ) {displaystyle psi ( au )} its distribution. The Laplace transform of ψ ( τ ) {displaystyle psi ( au )} is defined by Similarly, the characteristic function of the jump distribution f ( Δ X ) {displaystyle f(Delta X)} is given by its Fourier transform: One can show that the Laplace-Fourier transform of the probability P ( X , t ) {displaystyle P(X,t)} is given by The above is called Montroll-Weiss formula.