Evolution of bivariate copulas in discrete processes

A copula function makes a bridge between multivariate joint distributions and univariate marginal distributions, and provides a flexible way of describing nonlinear dependence among random circumstances. We introduce a new family of bivariate copulas which evolves according to the discrete process of heat equation. We prove the convergence of solutions as well as the measure of dependence. Numerical experiments are also performed, which shows that our procedure works substantially well.