Submitted Paper
Inserted: 7 dec 2005
Year: 2005
Abstract:
In this paper we consider a Hamiltonian $H$ on ${\cal P}_2(*R*^{2d})$, the set of probability measures with finite quadratic moments on the phase space $*R*^{2d}$, which is a metric space when endowed with the Wasserstein distance $W_2.$ We study the initial value problem $d\mu_t/dt+\nabla \cdot (J_d v_t\mu_t)=0,$ where $J_d$ is the canonical symplectic matrix, $\mu_0$ is prescribed, $v_t$ is a tangent vector to ${\cal P}_2(*R*^{2d})$ at $\mu_t$, and belongs to $\partial H(\mu_t)$, the subdifferential of $H$ at $\mu_t.$
Concerning existence of solutions, two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where $\mu_0$ is absolutely continuous. It ensures that $\mu_t$ remains absolutely continuous and $v_t=\nabla H(\mu_t)$ is the element of minimal norm in $\partial H(\mu_t).$ The second method handles any initial measure $\mu_0$. If we furthermore assume that $H$ is $\lambda$--convex, proper and lower semicontinuous on ${\cal P}_2(*R*^{2d})$, we prove that the Hamiltonian is preserved along any solution of our evolutive system.
Keywords: Optimal transport, Wasserstein distance, Hamiltonian ODE's
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