Geodesic
Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations with Applications to Cell Biology
Fractional calculus and applied analysis, Tome 9 (2006) no. 4, pp. 351-369

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In this paper the multi-dimensional Monte-Carlo random walk simulation models governed by distributed fractional order differential equations (DODEs) and multi-term fractional order differential equations are constructed. The construction is based on the discretization leading to a generalized difference scheme (containing a finite number of terms in the time step and infinite number of terms in the space step) of the Cauchy problem for DODE. The scaling limits of the constructed random walks to a diffusion process in the sense of distributions is proved.
Keywords: Random Walk, Anomalous Diffusion, Confined Diffusion, Distributed Order Differential Equation, Monte-Carlo Simulation, 65C05, 60G50, 39A10, 92C37 
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     author = {Andries, Erik and Umarov, Sabir and Steinberg, Stanly},
     title = {Monte {Carlo} {Random} {Walk} {Simulations} {Based} on {Distributed} {Order} {Differential} {Equations} with {Applications} to {Cell} {Biology}},
     journal = {Fractional calculus and applied analysis},
     pages = {351--369},
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Andries, Erik; Umarov, Sabir; Steinberg, Stanly. Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations with Applications to Cell Biology. Fractional calculus and applied analysis, Tome 9 (2006) no. 4, pp. 351-369. http://geodesic.mathdoc.fr/item/FCAA_2006_9_4_a2/