On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equations

By Wilfrid Gangbo, Adrian Tudorascu. Journal des Mathématiques Pures et Appliquées, 2019

125 citations recorded by the source corpus.

Abstract In this paper we elucidate the connection between various notions of differentiability in the Wasserstein space: some have been introduced intrinsically (in the Wasserstein space, by using typical objects from the theory of Optimal Transport) and used by various authors to study gradient flows, Hamiltonian flows, and Hamilton–Jacobi equations in this context. Another notion is extrinsic and arises from the identification of the Wasserstein space with the Hilbert space of square-integrable random variables on a non-atomic probability space. As a consequence, the classical theory of well-posedness for viscosity solutions for Hamilton–Jacobi equations in infinite-dimensional Hilbert spaces is brought to bear on well-posedness for Hamilton–Jacobi equations in the Wasserstein space.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e6e2f6c04a27fff351e2999207c0c5797587f40a

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Citations in the corpus

Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, methodology): “Regarding the relation between partial differential equations adopting the derivatives introduced by Lions (as in the present paper) and equations using notions of differentiability as those adopted in optimal transport theory, we mention results in this direction in the first-order case in [28] and”
Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity (2021, unknown): “The theory developed in [20, 31, 36, 40] shows that ∂μU(μ, ·) can be characterized by the property U(Lξ+η)− U(μ) = E [ 〈∂μU(μ, ξ), η〉 ] + o(‖η‖2), ∀ ξ, η, with Lξ = μ.”
Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence (2020, background): “Let us just briefly mention some contributions in this direction: a general analysis in metric setting can be found in [2, 36], see also [37] for the particular choice of the Wasserstein space.”
Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games (2020, background): “[8] [25]) ♯(Sn) = ♯(S −1 n ) = LΩ, lim n→∞ ‖y − x ◦ Sn‖ = 0.”
Zero-sum stochastic differential games of generalized McKean–Vlasov type (2018, background): “in [16], [17], but concern first-order equations, and to the best of our knowledge, the proof of a comparison principle for second-order equations as in (5.”
Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls (2018, methodology): “Assume F is continuously Fréchet differentiable, Lions [29] showed that the Fréchet derivative DF takes the following form: for some deterministic function h : P2(Rd)× Rd → Rd, DF (ξ) = h(Pξ, ξ), (2.2) see also Cardaliaguet [7], Gangbo & Tudorascu [23], and Wu & Zhang [36].”