Eigenvalue problems arise in mathematical physics problems (e.g., heat equation, vibrating membrane problem) when one tries to find a particular solution for which the time and space variable separate. The solution is then obtained as a series of the sum of the particular solutions found in this fashion.
Probabilistic techniques can be used in the study of the eigenvalue problems. The technique of coupling of diffusions is a useful tool for various estimates in probability and analysis. In particular, coupling of reflecting Brownian motions can be used for obtaining various estimates related to the eigenfunctions and eigenvalues of the Laplacian in smooth bounded domains.
We introduce a new type of coupling of reflecting Brownian motions in smooth bounded domains, called scaling coupling. As an application we prove a strong maximum principle for antisymmetric second Neumann eigenfunctions of smooth planar bounded convex domains with one line of symmetry (the hot spots conjecture).
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