28 research outputs found
Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
We explore the connection between fractional order partial differential
equations in two or more spatial dimensions with boundary integral operators to
develop techniques that enable one to efficiently tackle the integral
fractional Laplacian. In particular, we develop techniques for the treatment of
the dense stiffness matrix including the computation of the entries, the
efficient assembly and storage of a sparse approximation and the efficient
solution of the resulting equations. The main idea consists of generalising
proven techniques for the treatment of boundary integral equations to general
fractional orders. Importantly, the approximation does not make any strong
assumptions on the shape of the underlying domain and does not rely on any
special structure of the matrix that could be exploited by fast transforms. We
demonstrate the flexibility and performance of this approach in a couple of
two-dimensional numerical examples
Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces
We find necessary and sufficient conditions for a finite K–bi–invariant
measure on a compact Gelfand pair (G, K) to have a square–integrable
density. For convolution semigroups, this is equivalent to having a
continuous density in positive time. When (G, K) is a compact Riemannian
symmetric pair, we study the induced transition density for
G–invariant Feller processes on the symmetric space X = G/K. These
are obtained as projections of K–bi–invariant L´evy processes on G,
whose laws form a convolution semigroup. We obtain a Fourier series
expansion for the density, in terms of spherical functions, where the
spectrum is described by Gangolli’s L´evy–Khintchine formula. The
density of returns to any given point on X is given by the trace of
the transition semigroup, and for subordinated Brownian motion, we
can calculate the short time asymptotics of this quantity using recent
work of BaËśnuelos and Baudoin. In the case of the sphere, there is an
interesting connection with the Funk–Hecke theorem