Harmonic extension technique: probabilistic and analytic perspectives

TL;DR

This paper explores the deep connections between probabilistic boundary trace processes of reflected Brownian motion and their PDE counterparts, focusing on Dirichlet-to-Neumann operators and fractional Laplacians, through a blend of probabilistic and analytical perspectives.

Contribution

It provides a comprehensive analysis linking boundary trace processes with Dirichlet-to-Neumann operators, extending known results to more general contexts and introducing key analytical tools.

Findings

Boundary trace processes correspond to stable Lévy processes.

Fractional Laplacian is the Dirichlet-to-Neumann operator for elliptic equations.

Probabilistic and analytical methods are unified to study boundary phenomena.

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 L\'evy 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 $-\Delta$. Thus, the Dirichlet-to-Neumann operator coincides with the generator of the boundary trace process. Molchanov and Ostrovskii proved that isotropic stable L\'evy 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.