Chapter 10 is an introduction to the connections between probability theory and partial differential equations. At the beginning of §10.1, I show that martingales provide a link between probability theory and partial differential equations. More precisely, I show how to represent in terms of Wiener integrals solutions to parabolic and elliptic partial differential equations in which the Laplacian is the principal part. In the second part of §10.1, I derive the Feynman–Kac formula and use it to calculate various Wiener integrals. In §10.2 I introduce the Markov property of Wiener measure and show how it not only allows one to evaluate other Wiener integrals in terms of solutions to elliptic partial differential equations but also enables one to prove interesting facts about solutions to such equations as a consequence of their representation in terms of Wiener integrals. Continuing in the same spirit, I show in §10.2 how to represent solutions to the Dirichlet problem in terms of Wiener integrals, and in §10.3 I use Wiener measure to construct and discuss heat kernels related to the Laplacian and discuss ground states (a.k.a. stationary measures) for them.