177 research outputs found
Green's and Dirichlet spaces associated with fine Markov processes
AbstractThis is the second paper in a series devoted to Green's and Dirichlet spaces. In the first paper, we have investigated Green's space K and the Dirichlet space H associated with a symmetric Markov transition function pt(x, B). Now we assume that p is a transition function of a fine Markov process X and we prove that: (a) the space H can be built from functions which are right continuous along almost all paths; (b) the positive cone K+ in K can be identified with a cone M of measures on the state space; (c) the positive cone H+ in H can be interpreted as the cone of Green's potentials of measures μ ϵ M. To every measurable set B in the state space E there correspond a subspace K(B) of K and a subspace H(B) of H. The orthogonal projections of K onto K and of H onto H(B) can be expressed in terms of the hitting probabilities of B by the Markov process X. As the main tool, we use additive functionals of X corresponding to measures μ ϵ M
Positive solutions of Schr\"odinger equations and fine regularity of boundary points
Given a Lipschitz domain in and a nonnegative
potential in such that is bounded
in we study the fine regularity of boundary points with respect to
the Schr\"odinger operator in . Using potential
theoretic methods, several conditions equivalent to the fine regularity of are established. The main result is a simple (explicit if
is smooth) necessary and sufficient condition involving the size of
for to be finely regular. An essential intermediate result consists in
a majorization of for
positive harmonic in and . Conditions for
almost everywhere regularity in a subset of are also
given as well as an extension of the main results to a notion of fine
-regularity, if , being two potentials, with and a second order elliptic operator.Comment: version 1. 23 pages version 3. 28 pages. Mainly a typo in Theorem 1.1
is correcte
Bessel capacities on compact manifolds and their relation to Poisson capacities
AbstractA motivation for this paper comes from the role of Choquet capacities in the study of semilinear elliptic partial differential equations. In particular, the recent progress in the classification of all positive solutions of Lu=uα in a bounded smooth domain E⊂Rd was achieved by using, as a tool, capacities on a smooth manifold ∂E. Either the Poisson capacities (associated with the Poisson kernel in E) or the Bessel capacities (related to the Bessel kernel) have been used. In this and many other applications there is no advantage in choosing any special member in a class of equivalent capacities. (Two capacities are called equivalent if their ratio is bounded away from 0 and ∞.) In the literature Bessel capacities are considered mostly in the space Rd. We introduce two versions of Bessel capacities on a compact N-dimensional manifold. A class Capℓ,p of equivalent capacities is defined, for ℓp⩽N, on every compact Lipschitz manifold. Another class CBℓ,p is defined (for all ℓ>0, p>1) in terms of a diffusion process on a C2-manifold. These classes coincide when both are defined. If the manifold is the boundary of a bounded C2-domain E⊂Rd, then both versions of the Bessel capacities are equivalent to the Poisson capacities
The backbone decomposition for spatially dependent supercritical superprocesses
Consider any supercritical Galton-Watson process which may become extinct
with positive probability. It is a well-understood and intuitively obvious
phenomenon that, on the survival set, the process may be pathwise decomposed
into a stochastically `thinner' Galton-Watson process, which almost surely
survives and which is decorated with immigrants, at every time step, initiating
independent copies of the original Galton-Watson process conditioned to become
extinct. The thinner process is known as the backbone and characterizes the
genealogical lines of descent of prolific individuals in the original process.
Here, prolific means individuals who have at least one descendant in every
subsequent generation to their own.
Starting with Evans and O'Connell, there exists a cluster of literature
describing the analogue of this decomposition (the so-called backbone
decomposition) for a variety of different classes of superprocesses and
continuous-state branching processes. Note that the latter family of stochastic
processes may be seen as the total mass process of superprocesses with
non-spatially dependent branching mechanism.
In this article we consolidate the aforementioned collection of results
concerning backbone decompositions and describe a result for a general class of
supercritical superprocesses with spatially dependent branching mechanisms. Our
approach exposes the commonality and robustness of many of the existing
arguments in the literature
On the spatial Markov property of soups of unoriented and oriented loops
We describe simple properties of some soups of unoriented Markov loops and of
some soups of oriented Markov loops that can be interpreted as a spatial Markov
property of these loop-soups. This property of the latter soup is related to
well-known features of the uniform spanning trees (such as Wilson's algorithm)
while the Markov property of the former soup is related to the Gaussian Free
Field and to identities used in the foundational papers of Symanzik, Nelson,
and of Brydges, Fr\"ohlich and Spencer or Dynkin, or more recently by Le Jan
Buyback Problem - Approximate matroid intersection with cancellation costs
In the buyback problem, an algorithm observes a sequence of bids and must
decide whether to accept each bid at the moment it arrives, subject to some
constraints on the set of accepted bids. Decisions to reject bids are
irrevocable, whereas decisions to accept bids may be canceled at a cost that is
a fixed fraction of the bid value. Previous to our work, deterministic and
randomized algorithms were known when the constraint is a matroid constraint.
We extend this and give a deterministic algorithm for the case when the
constraint is an intersection of matroid constraints. We further prove a
matching lower bound on the competitive ratio for this problem and extend our
results to arbitrary downward closed set systems. This problem has applications
to banner advertisement, semi-streaming, routing, load balancing and other
problems where preemption or cancellation of previous allocations is allowed
Online Independent Set Beyond the Worst-Case: Secretaries, Prophets, and Periods
We investigate online algorithms for maximum (weight) independent set on
graph classes with bounded inductive independence number like, e.g., interval
and disk graphs with applications to, e.g., task scheduling and spectrum
allocation. In the online setting, it is assumed that nodes of an unknown graph
arrive one by one over time. An online algorithm has to decide whether an
arriving node should be included into the independent set. Unfortunately, this
natural and practically relevant online problem cannot be studied in a
meaningful way within a classical competitive analysis as the competitive ratio
on worst-case input sequences is lower bounded by .
As a worst-case analysis is pointless, we study online independent set in a
stochastic analysis. Instead of focussing on a particular stochastic input
model, we present a generic sampling approach that enables us to devise online
algorithms achieving performance guarantees for a variety of input models. In
particular, our analysis covers stochastic input models like the secretary
model, in which an adversarial graph is presented in random order, and the
prophet-inequality model, in which a randomly generated graph is presented in
adversarial order. Our sampling approach bridges thus between stochastic input
models of quite different nature. In addition, we show that our approach can be
applied to a practically motivated admission control setting.
Our sampling approach yields an online algorithm for maximum independent set
with competitive ratio with respect to all of the mentioned
stochastic input models. for graph classes with inductive independence number
. The approach can be extended towards maximum-weight independent set by
losing only a factor of in the competitive ratio with denoting
the (expected) number of nodes
Optimally defined Racah-Casimir operators for su(n) and their eigenvalues for various classes of representations
This paper deals with the striking fact that there is an essentially
canonical path from the -th Lie algebra cohomology cocycle, ,
of a simple compact Lie algebra \g of rank to the definition of its
primitive Casimir operators of order . Thus one obtains a
complete set of Racah-Casimir operators for each \g and nothing
else. The paper then goes on to develop a general formula for the eigenvalue
of each valid for any representation of \g, and thereby
to relate to a suitably defined generalised Dynkin index. The form of
the formula for for is known sufficiently explicitly to make
clear some interesting and important features. For the purposes of
illustration, detailed results are displayed for some classes of representation
of , including all the fundamental ones and the adjoint representation.Comment: Latex, 16 page
Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, with
near-optimal results known under a variety of constraints when the submodular
function is monotone. The case of non-monotone submodular maximization is less
understood: the first approximation algorithms even for the unconstrainted
setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC
'09, APPROX '09) show how to approximately maximize non-monotone submodular
functions when the constraints are given by the intersection of p matroid
constraints; their algorithm is based on local-search procedures that consider
p-swaps, and hence the running time may be n^Omega(p), implying their algorithm
is polynomial-time only for constantly many matroids. In this paper, we give
algorithms that work for p-independence systems (which generalize constraints
given by the intersection of p matroids), where the running time is poly(n,p).
Our algorithm essentially reduces the non-monotone maximization problem to
multiple runs of the greedy algorithm previously used in the monotone case.
Our idea of using existing algorithms for monotone functions to solve the
non-monotone case also works for maximizing a submodular function with respect
to a knapsack constraint: we get a simple greedy-based constant-factor
approximation for this problem.
With these simpler algorithms, we are able to adapt our approach to
constrained non-monotone submodular maximization to the (online) secretary
setting, where elements arrive one at a time in random order, and the algorithm
must make irrevocable decisions about whether or not to select each element as
it arrives. We give constant approximations in this secretary setting when the
algorithm is constrained subject to a uniform matroid or a partition matroid,
and give an O(log k) approximation when it is constrained by a general matroid
of rank k.Comment: In the Proceedings of WINE 201
Invariant four-forms and symmetric pairs
We give criteria for real, complex and quaternionic representations to define
s-representations, focusing on exceptional Lie algebras defined by spin
representations. As applications, we obtain the classification of complex
representations whose second exterior power is irreducible or has an
irreducible summand of co-dimension one, and we give a conceptual
computation-free argument for the construction of the exceptional Lie algebras
of compact type.Comment: 16 pages [v2: references added, last section expanded
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