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A Fleming-Viat particle representation of Dirichlet Laplacian
(Springer-Verlag GmbH, 2000-11)
We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D ...
A counterexample to the "hot spots" conjecture
(Princeton University and Institute for Advanced Study, 1999-01)
We construct a counterexample to the "hot spots" conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point ...