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A Stochastic Differential Equation Model with Jumps for Fractional Advection and Dispersion

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Abstract

The path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation (SDE) driven by Brownian noise. This model may not be adequate for highly heterogeneous media. This paper extends the model to a general SDE driven by a Lévy noise. Particle paths follow a Markov process with long jumps. Their transition probability density solves a forward equation derived here via pseudo-differential operator theory and Fourier analysis. In particular, the SDE with stable driving noise has a space-fractional advection-dispersion equation (fADE) with variable coefficients as the forward equation. This result provides a stochastic solution to anomalous diffusion models, and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations.

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  1. Department of Statistics and Probability, Michigan State University, East Lansing, MI, USA

    Paramita Chakraborty

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  1. Paramita Chakraborty
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Correspondence to Paramita Chakraborty.

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Chakraborty, P. A Stochastic Differential Equation Model with Jumps for Fractional Advection and Dispersion. J Stat Phys 136, 527–551 (2009). https://doi.org/10.1007/s10955-009-9794-1

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  • DOI: https://doi.org/10.1007/s10955-009-9794-1

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