Research Seminar in Mathematics - Solving PDEs by coin tossing

Speakers

Bair Budaev, University of California, Berkeley.

Abstract

Explicit solution of differential equations are always attractive because they permit computations of solutions at certain pints instead of dense meshes and because they provide qualitative information about the solutions, which is rarely available from numerical methods, such as finite differences or finite elements methods. Unfortunately, explicit solutions are usually restricted to problem in domains of a few simple shapes and with equations with special coefficients.

However, there is a class of explicit solutions of differential equations which is free of the above mentioned limitation. These are probabilistic solutions that represent unknown quantities as mathematical expectations of certain functionals computed along random trajectories, such as the Brownian motion, which can be easily simulated using generators or random numbers. Explicit solutions of this kind were discovered almost 100 years ago (Philips & Wiener, J.Math.Phys., 1923, v.2; Courant & Friedrich & Levy, Math.Ann., 1928, v.100), but for several decades they remained unused.

However, they remained practically unknown until the work of Feynman (Rev.Modern Phys, 1948, v.20, 367-387) who invented "path integrals" and of Kac (Trans.Amer.Math.Soc., 1949, v.65,1-13), who demonstrated the connection between Feynman's path integrals and the Brownian motion.
These studies resulted in the Feynman-Kac formula providing explicit solutions of the Laplace and the diffusion equations in domains of arbitrary shape, and very soon this approach was generalized to a broad class of elliptic and parabolic equations.

The advantages of probabilistic solutions of PDEs include but are not limited to: the versatility, no need in dense meshes, minimal restrictions on the shape of the domain and on the coefficients of the equation, transparent explanation of qualitative properties if the solution, as well as low requirements of computer memory and unlimited possibility of parallel computing.

Despite serious advantages, the area of applications of probabilistic solutions of PDEs for a long time remained limited to scalar second-order elliptic and parabolic equations. However, in recent years these methods were extended for the analysis of some systems of PDEs, non-linear equations, including Burgers and Navier-Stokes equations, of higher order equations, such as the bi-harmonic equation, the telegrapher and Helmholtz equations describing wave propagation, etc. Currently, this method is widely used in the Financial mathematics, especially for the analysis of the Black Scholes equation, which describes the price evolution of European options.

The presentation will discuss probabilistic solutions of PDEs represented by the Feynman-Kac formula as well as some of its recent extensions.