Multidimensional symmetric stable processes

This paper surveys recent remarkable progress in the study of potential theory for symmetric stable processes. It also contains new results on the two-sided estimates for Green functions, Poisson kernels and Martin kernels of discontinuous symmetric α-stable process in boundedC ^1,1 open sets. The new results give explicit information on how the comparing constants depend on parameter α and consequently recover the Green function and Poisson kernel estimates for Brownian motion by passing α ↑ 2. In addition to these new estimates, this paper surveys recent progress in the study of notions of harmonicity, integral representation of harmonic functions, boundary Harnack inequality, conditional gauge and intrinsic ultracontractivity for symmetric stable processes. Here is a table of contents.

1. Introduction
2. Green function and Poisson kernel estimates

1. Estimates on balls
2. Estimates on boundedC ^1,1 domains
3. Estimates on boundedC ^1,1 open sets

1. Harmonic functions and integral representation
2. Two notions of harmonicity
3. Martin kernel and Martin boundary
4. Integral representation and uniqueness
5. Boundary Harnack principle
6. Conditional process and its limiting behavior
7. Conditional gauge and intrinsic ultracontractivity

     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

On optimal realizations of finite metric spaces by graphs

Graph realizations of finite metric spaces have widespread applications, for example, in biology, economics, and information theory. The main results of this paper are:

1. Finding optimal realizations of integral metrics (which means all distances are integral) is NP-complete.
2. There exist metric spaces with a continuum of optimal realizations.

Furthermore, two conditions necessary for a weighted graph to be an optimal realization are given and an extremal problem arising in connection with the realization problem is investigated.     

Nonhomogeneous Markov processes

We present the foundations of the theory of nonhomogeneous Markov processes in general state spaces and we give a survey of the fundamental papers in this topic. We consider the following questions:

1. The existence of transition functions for a Markov process.
2. The construction of regularization of processes.
3. The properties of right and left processes: the strict Markov property, the behavior of excessive functions, etc.
4. The relation of right and left processes with dual homogeneous processes and the application of the results of the nonhomogeneous theory to dual homogeneous processes, etc.

     

Additive representations of non-additive measures and the choquet integral

This paper studies some new properties of set functions (and, in particular, “non-additive probabilities” or “capacities”) and the Choquet integral with respect to such functions, in the case of a finite domain. We use an isomorphism between non-additive measures on the original space (of states of the world) and additive ones on a larger space (of events), and embed the space of real-valued functions on the former in the corresponding space on the latter. This embedding gives rise to the following results:

the Choquet integral with respect to any totally monotone capacity is an average over minima of the integrand;

the Choquet integral with respect to any capacity is the difference between minima of regular integrals over sets of additive measures;

under fairly general conditions one may define a “Radon-Nikodym derivative” of one capacity with respect to another;

the “optimistic” pseudo-Bayesian update of a non-additive measure follows from the Bayesian update of the corresponding additive measure on the larger space.

We also discuss the interpretation of these results and the new light they shed on the theory of expected utility maximization with respect to non-additive measures.     

Platitude d'une adherence schematique et lemme de Hironaka generalise

The main aim of this article is to prove the following:Theorem (Generalized Hironaka's lemma). Let X→Y be a morphism of schemes, locally of finite presentation, x a point of X and y=f(x). Assume that the following conditions are satisfied:

1. O _Y,y is reduced.
2. f is universally open at the generic points of the components of X_y which contain x.
3. For every maximal generisation y′ of y in Y and every maximal generisation x′ of x in X which belongs to X_y, we have dim_x, (X_y')=dim_x(X_y)=d.
4. X_y is reduced at the generic points of the components of X_y which contain x and (X_y)_red is geometrically normal over K(y) in x.

Then there exist an open neighbourhood U of x in X and a subscheme U₀ of U which have the same underlying space as U such that f₀:U₀\arY is normal (i.e. f₀ is a flat morphism whose geometric fibers are normal).     

The shortest augmenting path method for solving assignment problems — Motivation and computational experience

In this paper we discuss the shortest augmenting path method for solving assignment problems in the following respect:

we introduce this basic concept using matching theory

we present several efficient labeling techniques for constructing shortest augmenting paths

we show the relationship of this approach to several classical assignment algorithms

we present extensive computational experience for complete problems, and

we show how postoptimal analysis can be performed using this approach and naturally leads to a new, highly efficient hybrid approach for solving large-scale dense assignment problems

     

On the completion of excellent rings in characteristicp>0

Si considera il seguente problema posto da Grothendieck (E.G.A.): SeA è un anello eccellente edm un ideale diA, (A, m) ^{^}=m-adico completamento diA è eccellente? Si mostra che la risposta è positiva nei seguenti casi:

1. A=algebra di tipo finito su un DVR completo di caratteristicap>0;
2. A=algebra di tipo finito su un DVRC contenente un corpok di caratteristicap>0 e finito suk [C ^p ] oppure tale che:

1. per ogni sottocampok′ dik contenentek ^p tale che [k:k′]<∞, il modulo universale finito dei differenzialiD _{k′ (C)} esiste;
2. il corpo residuoK diC soddisfa rank _KK ? _K/k <∞
3. C ha una Der-base.

     

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$ . Our main results are:

• A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d).
• The equivalence of the heat flow in $L^{2}(X,\mathfrak {m})$ generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional $\mathrm {Ent}_{\mathfrak {m}}$ in the space of probability measures .
• The proof of density in energy of Lipschitz functions in the Sobolev space $W^{1,2}(X,\mathsf {d},\mathfrak {m})$ .
• A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper.

Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality.     

The quasi-isometry classification of rank one lattices

Let X be a symmetric space—other than the hyperbolic plane—of strictly negative sectional curvature. Let G be the isometry group of X. We show that any quasi-isometry between non-uniform lattices in G is equivalent to (the restriction of) a group element of G which commensurates one lattice to the other. This result has the following corollaries:

1. Two non-uniform lattices in G are quasi-isometric if and only if they are commensurable.
2. Let Γ be a finitely generated group which is quasi-isometric to a non-uniform lattice in G. Then Γ is a finite extension of a non-uniform lattice in G.
3. A non-uniform lattice in G is arithmetic if and only if it has infinite index in its quasi-isometry group.

     

Symmetric differentials, Kähler groups and ball quotients

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients:

• Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kähler manifold X are controlled by the global symmetric differentials on X.
• An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).

     

A sub-Riemannian curvature-dimension inequality,volume doubling property and the Poincaré inequality

Let $\mathbb M $ be a smooth connected manifold endowed with a smooth measure $\mu $ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$ , and which is symmetric with respect to $\mu $ . We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold:

• The volume doubling property;
• The Poincaré inequality;
• The parabolic Harnack inequality.

The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.     

Weak isomorphism of measure-preserving diffeomorphisms

On the two-dimensional torus we construct twoC{∞}-diffeomorphismsT ₁,T ₂ satisfying:

1. T ₁,T ₂ preserve Lebesgue measure and are ergodic with respect to it,
2. T ₁,T ₂ are measurable factors of each other,
3. T ₁,T ₂ are not measure-theoretically isomorphic.

     

Convex rearrangements of stable processes

The asymptotic behavior of convex rearrangements for smooth approximations of random processes is considered. The main results are.

- the relations between the convergence of convex rearrangements of absolutely continuous on [0, 1] functions and the weak convergence of its derivatives considered as random variables on the probability space {[0, 1], ß_{[0, 1]}, λ} are established:

- a strong law of large numbers for convex rearrangements of polygonal approximations of stable processes with the exponent α, 1<α≦2, is proved:

- the relations with the results by M. Wshebor (see references) on oscillations of the Wiener process and with the results by Yu. Davydov and A. M. Vershik (see references) on convex rearrangements of random walks are discussed.

     

Δ2 degrees without Σ1 induction

In this paper, we study the structure of Turing degrees below 0′ in the theory that is a fragment of Peano arithmetic without Σ₁ induction, with special focus on proper d-r.e. degrees and non-r.e. degrees. We prove:

1. P ^? + BΣ₁ + Exp ? There is a proper d-r.e. degree.
2. P ^? +BΣ₁+ Exp ? IΣ₁ ? There is a proper d-r.e. degree below 0′.
3. P ^? + BΣ₁ + Exp ? There is a non-r.e. degree below 0′.

     

Monotonicity and error bounds for networks of Erlang loss queues

Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are

1. classical circuit switch telephone networks (loss networks) and
2. present-day wireless mobile networks.

Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to

• upper bounds for loss probabilities and
• analytic error bounds for the accuracy of the approximation for various performance measures.

The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:

• pure loss networks as under (i)
• GSM networks with fixed channel allocation as under (ii).

The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning.     

Die Stoppverteilungen eines Markoff-Prozesses mit lokalendlichem Potential

Consider a Markovian standard semigroup P_t, t≥o, with potential kernel U=∫P_tdt on a locally compact space E. Let μ be a finite measure on E with locally finite potential μ^U and X_t, t≥O, the process having (P_t) as transition semigroup and μ as initial law. Then for a measure ν on E the following two statements are equivalent:

1. μ^U≥νU;
2. there exists a “randomized” stopping time T such that X_T is distributed according to ν.

     

Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain:

• Geometric conditions ensuring the compactness of the underlying manifold (Bonnet–Myers type results);
• Volume estimates of metric balls;
• Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian;
• Spectral gap estimates.

     

Immersionen mit paralleler zweiter Fundamentalform: Beispiele und Nicht-Beispiele

1. We show that symmetric R-spaces can. be imbedded in euclidean space with parallel second fundamental tensor (Dα=0).
2. We give a restrictive necessary condition for totally geodesic submanifolds of the Grassmannian to be the Gauss image of an immersion with Dα=0, c.f. [9].

     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $