Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line
F Cacciafesta, AS De Suzzoni - Communications in Contemporary …, 2020 - World Scientific
F Cacciafesta, AS De Suzzoni
Communications in Contemporary Mathematics, 2020•World ScientificWe prove that the Gibbs measures ρ for a class of Hamiltonian equations written as (*)∂ tu=
J (−△ u+ V′(| u| 2) u) on the real line are invariant under the flow of (∗) in the sense that
there exist random variables X (t) whose laws are ρ (thus independent from t) and such that
t↦ X (t) is a solution to (∗). Besides, for all t, X (t) is almost surely not in L 2 which provides as
a direct consequence the existence of global weak solutions for initial data not in L 2. The
proof uses Prokhorov's theorem, Skorohod's theorem, as in the strategy in [N. Burq, L …
J (−△ u+ V′(| u| 2) u) on the real line are invariant under the flow of (∗) in the sense that
there exist random variables X (t) whose laws are ρ (thus independent from t) and such that
t↦ X (t) is a solution to (∗). Besides, for all t, X (t) is almost surely not in L 2 which provides as
a direct consequence the existence of global weak solutions for initial data not in L 2. The
proof uses Prokhorov's theorem, Skorohod's theorem, as in the strategy in [N. Burq, L …
We prove that the Gibbs measures for a class of Hamiltonian equations written as on the real line are invariant under the flow of in the sense that there exist random variables whose laws are (thus independent from ) and such that is a solution to . Besides, for all , is almost surely not in which provides as a direct consequence the existence of global weak solutions for initial data not in . The proof uses Prokhorov’s theorem, Skorohod’s theorem, as in the strategy in [N. Burq, L. Thomann and N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, preprint (2014); arXiv:1412.7499v1 [math.AP]] and Feynman–Kac’s integrals.
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