From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE

By Viorel Barbu, Michael Röckner. Annals of Probability, 2018

134 citations recorded by the source corpus.

We construct weak solutions to a class of distribution dependent SDE, of type $dX(t)=b\left( X(t), \displaystyle\frac{d\mathcal{L}_{X(t)}}{dx}(X(t))\right) dt+\sigma\left( X(t),\displaystyle\frac{d\mathcal{L}_{X(t)}}{dt}(X(t))\right) dW(t)$ for possibly degenerate diffusion matrices $\sigma$ with $X(0)$ having a given law, which has a density with respect to Lebesgue measure, $dx$. Here ${\mathcal{L}}_{X(t)}$ denotes the law of $X(t)$. Our approach is to first solve the corresponding nonlinear Fokker-Planck equations and then use the well known superposition principle to obtain weak solutions of the above SDE.

Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/0ea00f4489db5e6f502790009331f94635f840b9

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Citations in the corpus

Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “Moreover, superposition principles for controlled SDEs and two classes of (particularly singular) McKean-Vlasov SDEs can be found in [50] and [7, 6], respectively.”
Weak quantitative propagation of chaos via differential calculus on the space of measures (2019, background): “[34] or, for more up-to-date panorama on research on existence and uniqueness, see [32, 20, 1, 12]).”
Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance (2020, background): “This type of SDEs are also called McKean-Vlasov SDEs and mean field SDEs, and have been intensively investigated due to its wide applications, see for instance [1, 2, 5, 8, 10, 11, 12, 21, 23] and references within.”
Large and Moderate Deviation Principles for McKean-Vlasov SDEs with Jumps (2020, background): “of Gaussian noise (see [12, 29]), nonlinear and nonlocal integral Fokker-Planck PDEs can be related to MVSDEs with Lévy noise (see [26, 34, 37]).”