Applied Analysis Reading Group

Hong Duong

A Hamiltonian-Gradient flow splitting approximation scheme for a Vlasov-Fokker-Planck equation

We combine the theory of gradient flows and Hamiltonian flows on the space of probability measures, which are developed separately by Jordan-Kinderlehrer-Otto [JKO98] and Ambrosio-Gangbo [AG08], in order to capture the dynamic underlying the initial value problem for a Vlasov-Fokker-Planck equation. We propose a new cost functional and a new two-step splitting scheme. The first step is a Hamiltonian flow and the second one is a gradient flow with respect to the cost functional on the space of probability measures. Convergence of the scheme is established.

References

[AG08] Luigi Ambrosio and Wilfred Gangbo. Hamiltonian ODEs in the Wasserstein space of probability measures. Comm. Pure Appl. Math., 61(1):18-53, 2008.

[JKO98] R. Jordan, D. Kinderlehrer, and F. Otto. The variational formulation of the fokker-planck equation. SIAM Journal on Mathematical Analysis, 29(1):1-17, 1998

(Joint work with Mark A. Peletier and Johannes Zimmer)

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