TY - JOUR
T1 - Wasserstein Hamiltonian Flow With Common Noise On Graph
AU - Cui, Jianbo
AU - Liu, Shu
AU - Zhou, Haomin
N1 - Funding Information:
*Received by the editors April 18, 2022; accepted for publication (in revised form) December 1, 2022; published electronically April 20, 2023. https://doi.org/10.1137/22M1490697 Funding: This research is partially supported by Georgia Tech Mathematics Application Portal (GT-MAP) and by research grants NSF DMS-1830225, and ONR N00014-21-1-2891. The research of the first author is partially supported by start-up funds (P0039016, P0041274) from Hong Kong Polytechnic University, the Hong Kong Research Grant Council ECS grant 25302822, and the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics. \dagger Corresponding author. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong ([email protected]). \ddagger School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (sliu459@ gatech.edu, [email protected]).
Publisher Copyright:
Copyright © by SIAM.
PY - 2023/4
Y1 - 2023/4
N2 - We study the Wasserstein Hamiltonian flow with a common noise on the density manifold of a finite graph. Under the framework of the stochastic variational principle, we first develop the formulation of stochastic Wasserstein Hamiltonian flow and show the local existence of a unique solution. We also establish a sufficient condition for the global existence of the solution. Consequently, we obtain the global well-posedness for the nonlinear Schrödinger equations with common noise on a graph. In addition, using Wong-Zakai approximation of common noise, we prove the existence of the minimizer for an optimal control problem with common noise. We show that its minimizer satisfies the stochastic Wasserstein Hamiltonian flow on a graph as well.
AB - We study the Wasserstein Hamiltonian flow with a common noise on the density manifold of a finite graph. Under the framework of the stochastic variational principle, we first develop the formulation of stochastic Wasserstein Hamiltonian flow and show the local existence of a unique solution. We also establish a sufficient condition for the global existence of the solution. Consequently, we obtain the global well-posedness for the nonlinear Schrödinger equations with common noise on a graph. In addition, using Wong-Zakai approximation of common noise, we prove the existence of the minimizer for an optimal control problem with common noise. We show that its minimizer satisfies the stochastic Wasserstein Hamiltonian flow on a graph as well.
KW - density manifold
KW - optimal transport
KW - stochastic Hamiltonian flow on graph
KW - Wong-Zakai approximation
UR - http://www.scopus.com/inward/record.url?scp=85152959800&partnerID=8YFLogxK
U2 - 10.1137/22M1490697
DO - 10.1137/22M1490697
M3 - Journal article
AN - SCOPUS:85152959800
SN - 0036-1399
VL - 83
SP - 484
EP - 509
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 2
ER -