The Path Dependent Heat Equation

The path dependent heat equation is the differential equation satisfied by a Brownian functional. These functionals arise naturally in different areas of stochastic calculus and mathematical finance, e.g., in the study of optimal stochastic control problems with delay and in the valuation problem of path-dependent financial derivatives. Path-dependent PDEs (PPDEs) have been recently introduced and there are still many open problems. In particular, we focus on the definition of weak solution (more precisely, viscosity solution) for PPDEs, which has been given in the paper On Viscosity Solutions of Path Dependent PDEs (to appear on Annals of Probability) by Ekren, Keller, Touzi, and Zhang. We present these results for the special case of the path dependent heat equation.