Weak Solutions to a Fractional Fokker-Planck Equation via Splitting and Wasserstein Gradient Flow

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Bowles_Malcolm_MSc_2014.pdf (426.38 KB)

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2014-08-22

Authors

Bowles, Malcolm

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Abstract

In this thesis, we study a linear fractional Fokker-Planck equation that models non-local (`fractional') diffusion in the presence of a potential field. The non-locality is due to the appearance of the `fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular (Gaussian) diffusion. Motivated by the observation that, in contrast to the classical Fokker-Planck equation (describing regular diffusion in the presence of a potential field), there is no natural gradient flow formulation for its fractional counterpart, we prove existence of weak solutions to this fractional Fokker-Planck equation by combining a splitting technique together with a Wasserstein gradient flow formulation. An explicit iterative construction is given, which we prove weakly converges to a weak solution of this PDE.

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Keywords

splitting, Fractional Laplacian, Wasserstein Gradient Flow

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http://hdl.handle.net/1828/5591

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ETD (Electronic Theses and Dissertations)
Theses (Mathematics and Statistics)
 
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