11 research outputs found
Skew-Unfolding the Skorokhod Reflection of a Continuous Semimartingale
The Skorokhod reflection of a continuous semimartingale is unfolded, in a
possibly skewed manner, into another continuous semimartingale on an enlarged
probability space according to the excursion-theoretic methodology of Prokaj
(2009). This is done in terms of a skew version of the Tanaka equation, whose
properties are studied in some detail. The result is used to construct a system
of two diffusive particles with rank-based characteristics and skew-elastic
collisions. Unfoldings of conventional reflections are also discussed, as are
examples involving skew Brownian Motions and skew Bessel processes.Comment: 20 pages. typos corrected, added a remark after Proposition 2.3,
simplified the last part of Example 2.
Diffusions with rank-based characteristics and values in the nonnegative quadrant
We construct diffusions with values in the nonnegative orthant, normal reflection along each of the axes, and two pairs of local drift/variance characteristics assigned according to rank; one of the variances is allowed to vanish, but not both. The construction involves solving a system of coupled Skorokhod reflection equations, then "unfolding" the Skorokhod reflection of a suitable semimartingale in the manner of Prokaj (Statist. Probab. Lett. 79 (2009) 534-536). Questions of pathwise uniqueness and strength are also addressed, for systems of stochastic differential equations with reflection that realize these diffusions. When the variance of the laggard is at least as large as that of the leader, it is shown that the corner of the quadrant is never visited
A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains
In the first (and abstract) part of this survey we prove the unitary
equivalence of the inverse of the Krein--von Neumann extension (on the
orthogonal complement of its kernel) of a densely defined, closed, strictly
positive operator, for some in a Hilbert space to an abstract buckling problem operator.
This establishes the Krein extension as a natural object in elasticity theory
(in analogy to the Friedrichs extension, which found natural applications in
quantum mechanics, elasticity, etc.).
In the second, and principal part of this survey, we study spectral
properties for , the Krein--von Neumann extension of the
perturbed Laplacian (in short, the perturbed Krein Laplacian)
defined on , where is measurable, bounded and
nonnegative, in a bounded open set belonging to a
class of nonsmooth domains which contains all convex domains, along with all
domains of class , .Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144