269 research outputs found
The "hot spots" problem in planar domains with one hole
There exists a planar domain with piecewise smooth boundary and one hole such
that the second eigenfunction for the Laplacian with Neumann boundary
conditions attains its maximum and minimum inside the domain
Forward Brownian Motion
We consider processes which have the distribution of standard Brownian motion
(in the forward direction of time) starting from random points on the
trajectory which accumulate at . We show that these processes do not
have to have the distribution of standard Brownian motion in the backward
direction of time, no matter which random time we take as the origin. We study
the maximum and minimum rates of growth for these processes in the backward
direction. We also address the question of which extra assumptions make one of
these processes a two-sided Brownian motion.Comment: The latest version has an extra result (Theorem 5.2). The old Theorem
5.2 is now called Theorem 5.
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