#### 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
##### Abstract

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 ${\Omega }_{\lambda }\subseteq M$ almost fully contains a ball of radius $\sim 1∕\sqrt{{\lambda }_{1}\left({\Omega }_{\lambda }\right)}$, and such a ball can be centred at any point of maximum of the Dirichlet ground state ${\phi }_{{\lambda }_{1}\left({\Omega }_{\lambda }\right)}$. 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 $M$.

##### Keywords
Laplace eigenfunctions, nodal domains, Brownian motion
##### Mathematical Subject Classification 2010
Primary: 35P20, 53B20, 53Z05
Secondary: 35K05