Jacek Mucha | Searcharxiv

arXiv2019

Spectral theory for one-dimensional (non-symmetric) stable processes killed upon hitting the origin

We obtain an integral formula for the distribution of the first hitting time of the origin for one-dimensional $α$-stable processes $X_t$, where $α\in(1,2)$. We also find a spectral-type integral formula for the transition operators $P_0^t$ of $X_t$ killed upon hitting the origin. Both expressions involve exponentially growing oscillating functions, which play a role of generalised eigenfunctions for $P_0^t$.

math.PR↗

arXiv2017

Extension technique for complete Bernstein functions of the Laplace operator

We discuss representation of certain functions of the Laplace operator $Δ$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $(-Δ)^{1/2}$, the square root of the $d$-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the $(d + 1)$-dimensional Laplace operator $Δ_{t,x}$ in $(0, \infty) \times \mathbf{R}^d$. Caffarelli and Silvestre extended this to fractional powers $(-Δ)^{α/2}$, which correspond to operators $\nabla_{t,x} (t^{1 - α} \nabla_{t,x})$. We provide an analogous result for all complete Bernstein functions of $-Δ$ using Krein's spectral theory of strings. Two sample applications are provided: a Courant--Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schrödinger operators $ψ(-Δ) + V(x)$, as well as an upper bound for the eigenvalues of these operators. Here $ψ$ is a complete Bernstein function and $V$ is a confining potential.

math.AP↗