Hamiltonian ODEs in the Wasserstein space of probability measures

By Luigi Ambrosio, Wilfrid Gangbo. 2008

163 citations recorded by the source corpus.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c010a1e15c28de1b3a4992f9d38a1f860f2566c0

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Citations in the corpus

Some Geometric Calculations on Wasserstein Space (2006, background): “As the partition of [0, 1 ] becomes finer, the term”
A Pontryagin Maximum Principle in Wasserstein spaces for constrained optimal control problems (2018, unknown): “[8, 9]), which incorporates natural Lipschitz and metric estimates in the smooth cases (see [45]).”
Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow (2011, background): “In the context of Hamiltonian systems, very general existence results, together with approximation schemes, have been obtained in [1]: however, uniqueness seems to be out of reach in such low regularity context.”
The Pontryagin Maximum Principle in the Wasserstein Space (2017, methodology): “notice that v[µ] depends on the whole measure µ and not only on its values at some points as it is usually the case for non-linear conservation laws. We now recall a theorem which was ﬁrst derived in [5] providing existence, uniqueness and representation formula for solutions of (4). We state here a”
Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework (2020, methodology): “Moreover, the functional approach to differential inclusions developed in this paper is very close in spirit to the formulation of gradient flows [10, Chapter 11] and Hamiltonian flows [8] in Wasserstein spaces, which correspond to special (but not less general) cases in which the sets of admissible”
On Properties of the Generalized Wasserstein Distance (2013, background): “Related to this problem, Wasserstein distances in the space of probability measures have revealed to be powerful tools, in particular for dealing with dynamics of measures (like the transport PDE, see e.g. [1, 2]).”
Generalized Wasserstein Distance and its Application to Transport Equations with Source (2012, background): “In particular, in [1] the authors prove existence and uniqueness of the solution under Lipschitzianity of v with respect to μ, when the space of measures (of given fixed mass) is endowed with the Wasserstein distance.”
Existence of a solution to an equation arising from the theory of Mean Field Games (2015, unknown): “2 of [3]) since we do not require that γ ∈ Γ0(μ, ν).”
Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems (2008, methodology): “We also study the geometric properties of this symplectic distribution and define the notion of Hamiltonian systems on M, thus providing a firm basis to the notion already introduced in [ 3 ].”
Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, background): “There are already wellposedness results in the first-order case, see [2, 25, 1, 26, 27, 28], even for equations adopting different notions of derivative with respect to the measure.”
Understanding MCMC Dynamics as Flows on the Wasserstein Space (2019, background): “(6) On the same topic, Ambrosio & Gangbo (2008) study the existence and simulation of the Hamiltonian flow on P(M) forM as a symplectic Euclidean space, and verify the conservation of Hamiltonian under certain conditions.”