CTRWs

Continuous-Time Random Walks (CTRWs) are stochastic models in which a particle makes a sequence of jumps separated by random waiting times. Each step comprises a waiting time τ drawn from a waiting-time distribution ψ(τ), followed by a spatial jump Δx drawn from a jump distribution λ(Δx). The position after n jumps is X_n = sum of the Δx_i, and the total time is T_n = sum of the τ_i. The motion X(t) is the position after N(t) jumps have occurred, with N(t) = max{n: T_n ≤ t}. If ψ is exponential, the process is memoryless and reduces to a classical Markovian walk with ordinary diffusion; if ψ has a heavy tail, the CTRW can exhibit anomalous diffusion.

In Fourier-Laplace space the process is described by the Montroll-Weiss equation: P(k,s) = [1 − ψ̃(s)] / [s (1 − ψ̃(s) λ̃(k))], where P(k,s) is the Fourier-Laplace transform of the probability density P(x,t), and ψ̃ and λ̃ denote the Laplace and Fourier transforms of ψ and λ, respectively. The inverse transforms yield P(x,t). This formulation captures how waiting times and jump lengths combine to shape the spread of the particle.

Limiting behavior: heavy-tailed waiting times with tail ~ τ^{−1−α} (0<α<1) lead to subdiffusion, with mean-squared displacement ⟨x^2(t)⟩ ∼ t^α, and, in the continuum limit, to a time-fractional diffusion equation ∂^α_t P = D ∂^2_x P (Caputo derivative). If the jump distribution λ has infinite variance (Lévy tails), space-fractional diffusion arises. When both effects are present, one obtains fractional diffusion with both time and space derivatives.

Variants and applications: CTRWs admit subordination representations X(t) = B(S(t)) with a Brownian motion B and a random time process S. They are used to model transport in disordered media, amorphous materials, and intracellular transport, as well as certain financial processes, providing a flexible framework for non-Markovian dynamics and anomalous transport phenomena.