ABSTRACT
This chapter explores the relations between stochastic processes and partial differential equations (PDE): namely, the solution to a PDE is written as an expectation of a functional of the process, which refers to probabilistic representations for partial differential equations, or Feynman-Kac formulas. Historically, the first studied example is the heat equation, which is connected to Brownian motion. This random process, whose trajectories are very erratic, was described first by Robert Brown in 1827, then studied in other contexts by Louis Bachelier in 1900 and Albert Einstein in 1905. With the spectacular development of stochastic tools during the second half of the 20th century, this connection has been extended to more stochastic processes and more partial differential equations, under the generic name of the Feynman-Kac formulas.
