  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.
