A kernel formula for regularized Wasserstein proximal operators

We study a class of regularized proximal operators in Wasserstein-2 space. We derive their solutions by kernel integration formulas. We obtain the Wasserstein proximal operator using a pair of forward-backward partial differential equations consisting of a continuity equation and a Hamilton-Jacobi equation with a terminal time potential function and an initial time density function. We regularize the PDE pair by adding forward and backward Laplacian operators. We apply Hopf-Cole type transformations to rewrite these regularized PDE pairs into forward-backward heat equations. We then use the fundamental solution of the heat equation to represent the regularized Wasserstein proximal with kernel integral formulas. Numerical examples show the effectiveness of kernel formulas in approximating the Wasserstein proximal operator.