Brownian Motion, 2nd Edition

8 The PDE connection

We want to discuss some relations between partial differential equations (PDEs) and Brownian motion. For many classical PDE problems probability theory yields concrete representation formulae for the solutions in the form of expected values of a Brownian functional. These formulae can be used to get generalized solutions of PDEs (which require less smoothness of the initial/boundary data or the boundary itself) and they are amenable to Monte–Carlo simulations. Purely probabilistic existence proofs for classical PDE problems are, however, rare: Classical solutions require smoothness, which does usually not follow from martingale methods.⁵ This explains the role of Proposition 7.3 g) and Proposition 8.10. Let us point out ...