102 research outputs found
Reflection principles for functions of Neumann and Dirichlet Laplacians on open reflection invariant subsets of
For an open subset of , symmetric with respect to a
hyperplane and with positive part , we consider the Neumann/Dirichlet
Laplacians and . Given a Borel
function on we apply the spectral functional calculus and
consider the pairs of operators and
, or and
. We prove relations between the integral kernels
for the operators in these pairs, which in particular cases of
and , , , were known as reflection principles for the Neumann/Dirichlet
heat kernels. These relations are then generalized to the context of symmetry
with respect to a finite number of mutually orthogonal hyperplanes.Comment: 25 page
Density behaviour related to Lévy processes
Let p t (x), f t (x) and q * t (x) be the densities at time t of a real Lévy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of p t (x), f t (x) and q * t (x), when t is small and x is large. Then for large x, these asymptotic behaviours are compared to this of the density of the Lévy measure. We show in particular that, under mild conditions, if p t (x) is comparable to tν(x), as t → 0 and x → ∞, then so is f t (x)
Spectral properties of the Cauchy process
We study the spectral properties of the transition semigroup of the killed
one-dimensional Cauchy process on the half-line (0,infty) and the interval
(-1,1). This process is related to the square root of one-dimensional Laplacian
A = -sqrt(-d^2/dx^2) with a Dirichlet exterior condition (on a complement of a
domain), and to a mixed Steklov problem in the half-plane. For the half-line,
an explicit formula for generalized eigenfunctions psi_lambda of A is derived,
and then used to construct spectral representation of A. Explicit formulas for
the transition density of the killed Cauchy process in the half-line (or the
heat kernel of A in (0,infty)), and for the distribution of the first exit time
from the half-line follow. The formula for psi_lambda is also used to construct
approximations to eigenfunctions of A in the interval. For the eigenvalues
lambda_n of A in the interval the asymptotic formula lambda_n = n pi/2 - pi/8 +
O(1/n) is derived, and all eigenvalues lambda_n are proved to be simple.
Finally, efficient numerical methods of estimation of eigenvalues lambda_n are
applied to obtain lower and upper numerical bounds for the first few
eigenvalues up to 9th decimal point.Comment: 37 pages, 1 figur
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