Harmonic extension technique: probabilistic and analytic perspectives

Abstract

Consider a path of the reflected Brownian motion in the half-plane $\{y\ge 0\}$, and erase its part contained in the interior $\{y>0\}$. What is left is, in an appropriate sense, a path of a jump-type stochastic process on the line $\{y=0\}$ -- the boundary trace of the reflected Brownian motion. It is well known that this process is in fact the 1-stable Lévy process, also known as the Cauchy process. The PDE interpretation of the above fact is the following. Consider a bounded harmonic function $u$ in the half-plane $\{y>0\}$, with sufficiently smooth boundary values $f$. Let $g$ denote the normal derivative of $u$ at the boundary. The mapping $f\mapsto g$ is known as the Dirichlet-to-Neumann operator, and it is again well known that this operator coincides with the square root of the 1-D Laplace operator $-\hat{I}”$. Thus, the Dirichlet-to-Neumann operator coincides with the generator of the boundary trace process. Molchanov and Ostrovskii proved that isotropic stable Lévy processes are boundary traces of appropriate diffusions in half-spaces. Caffarelli and Silvestre gave a PDE counterpart of this result: the fractional Laplace operator is the Dirichlet-to-Neumann operator for an appropriate second-order elliptic equation in the half-space. Again, the Dirichlet-to-Neumann operator turns out to be the generator of the boundary trace process. During my talk I will discuss boundary trace processes and Dirichlet-to-Neumann operators in a more general context. My main goal will be to explain the connections between probabilistic and analytical results. Along the way, I will introduce the necessary machinery: Brownian local times and additive functionals, Krein's spectral theory of strings, and Fourier transform methods.

31 pages; lecture notes for a series of talks given at the 3rd Korean Croatian Summer Probability Camp in Å ibenik, Croatia, on June 26-28, 2024; v2 includes solutions to exercises