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

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.