We study generalized Hopf–Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf–Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf–Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.
Zhongjian WangDepartment of Mathematics, The University of Hong KongXue LuoSchool of Mathematical Sciences, Beihang University (Shahe campus)Stephen S.-T. YauDepartment of Mathematical Sciences, Tsinghua UniversityZhiwen ZhangDepartment of Mathematics, The University of Hong Kong
Numerical Analysis and Scientific ComputingOptimization and ControlProbabilitymathscidoc:2004.25001