Advances in Applied Probability

Fractional diffusion and fractional heat equation

J. M. Angulo, M. D. Ruiz-Medina, V. V. Anh, and W. Grecksch

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This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is formulated in terms of the Fourier transform of its spatially and temporally homogeneous Green function. The spectral density of the resulting solution is then obtained explicitly. The result implies that the solution of the fractional heat equation may possess spatial long-range dependence asymptotically.

Article information

Adv. in Appl. Probab. Volume 32, Number 4 (2000), 1077-1099.

First available: 12 February 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60G60: Random fields

Diffusion processes stochastic heat equation Bessel potential Riesz potential


Angulo, J. M.; Ruiz-Medina, M. D.; Anh, V. V.; Grecksch, W. Fractional diffusion and fractional heat equation. Advances in Applied Probability 32 (2000), no. 4, 1077--1099. doi:10.1239/aap/1013540349.

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