Vol. 11, No. 1, 2018

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Nodal geometry, heat diffusion and Brownian motion

Bogdan Georgiev and Mayukh Mukherjee

Vol. 11 (2018), No. 1, 133–148

We use tools from n-dimensional Brownian motion in conjunction with the Feynman–Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold M. On one hand we extend a theorem of Lieb (1983) and prove that any Laplace nodal domain Ωλ M almost fully contains a ball of radius  1λ1 (Ωλ ), and such a ball can be centred at any point of maximum of the Dirichlet ground state φλ1(Ωλ). This also gives a slight refinement of a result by Mangoubi (2008) concerning the inradius of nodal domains. On the other hand, we also prove that no nodal domain can be contained in a reasonably narrow tubular neighbourhood of unions of finitely many submanifolds inside M.

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Laplace eigenfunctions, nodal domains, Brownian motion
Mathematical Subject Classification 2010
Primary: 35P20, 53B20, 53Z05
Secondary: 35K05
Received: 11 April 2016
Revised: 20 June 2017
Accepted: 10 August 2017
Published: 17 September 2017
Bogdan Georgiev
Max Planck Institute for Mathematics
Mayukh Mukherjee
Max Planck Institute for Mathematics