The Hardy inequality and the heat flow in curved wedges
We show that the polynomial decay rate of the heat semigroup of the Dirichlet Laplacian in curved planar wedges equals the sum of the usual dimensional decay rate and a multiple of the reciprocal value of the opening angle. To prove the result, we develop the method of self-similar variables for the associated heat equation and study the asymptotic behaviour of the transformed non-autonomous parabolic problem for large times. We also establish an improved Hardy inequality for the Dirichlet Laplacian in non-trivially curved wedges and state a conjecture about an improved decay rate in this case.
We study a random walk in a N dimensional hypercube.To each site of the hypercube we associate a random value according to a Bernoulli distribution.These values together with the temperature define the law of waiting times of the walk at each site.We exhibit a transition between the high and low temperatures regime by studying the distance between two trajectories starting from different initial conditions and subjected to the same noise.