45 research outputs found
Spectral analysis and zeta determinant on the deformed spheres
We consider a class of singular Riemannian manifolds, the deformed spheres
, defined as the classical spheres with a one parameter family of
singular Riemannian structures, that reduces for to the classical metric.
After giving explicit formulas for the eigenvalues and eigenfunctions of the
metric Laplacian , we study the associated zeta functions
. We introduce a general method to deal with some
classes of simple and double abstract zeta functions, generalizing the ones
appearing in . An application of this method allows to
obtain the main zeta invariants for these zeta functions in all dimensions, and
in particular and . We give
explicit formulas for the zeta regularized determinant in the low dimensional
cases, , thus generalizing a result of Dowker \cite{Dow1}, and we
compute the first coefficients in the expansion of these determinants in powers
of the deformation parameter .Comment: 1 figur
Reflecting diffusions and hyperbolic Brownian motions in multidimensional spheres
Diffusion processes moving inside
spheres and reflecting orthogonally on their
surfaces are considered. The stochastic differential equations
governing the reflecting diffusions are presented and their kernels and
distributions explicitly derived. Reflection is obtained by means of the
inversion with respect to the sphere . The particular cases of
Ornstein-Uhlenbeck process and Brownian motion are examined in detail.
The hyperbolic Brownian motion on the Poincar\`e half-space is
examined in the last part of the paper and its reflecting counterpart within
hyperbolic spheres is studied. Finally a section is devoted to reflecting
hyperbolic Brownian motion in the Poincar\`e disc within spheres concentric
with