28 research outputs found
Pointwise Bounds for Steklov Eigenfunctions
Let (Ω,g) be a compact, real-analytic Riemannian manifold with real-analytic boundary âΩ. The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle SââΩ. These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations Pu=0 near the characteristic set {Ï(P)=0}
Simulation of Conditioned Diffusions on the Flat Torus
Diffusion processes are fundamental in modelling stochastic dynamics in
natural sciences. Recently, simulating such processes on complicated geometries
has found applications for example in biology, where toroidal data arises
naturally when studying the backbone of protein sequences, creating a demand
for efficient sampling methods. In this paper, we propose a method for
simulating diffusions on the flat torus, conditioned on hitting a terminal
point after a fixed time, by considering a diffusion process in R 2 which we
project onto the torus. We contribute a convergence result for this diffusion
process, translating into convergence of the projected process to the terminal
point on the torus. We also show that under a suitable change of measure, the
Euclidean diffusion is locally a Brownian motion.Comment: 10 pages, 6 figures, GSI Conferenc