Hopf-Cole transformation via generalized Schr\"odinger bridge problem

We study generalized Hopf-Cole transformations motivated by the Schr"odinger 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.