Stationary Symmetric <Emphasis Type=Italic>α</Emphasis>-Stable Discrete Parameter Random Fields

We establish a connection between the structure of a stationary symmetric α-stable random field (0<α<2) and ergodic theory of non-singular group actions, elaborating on a previous work by Rosiński (Ann. Probab. 28:1797–1813, 2000). With the help of this connection, we study the extreme values of the field over increasing boxes. Depending on the ergodic theoretical and group theoretical structures of the underlying action, we observe different kinds of asymptotic behavior of this sequence of extreme values. Supported in part by NSF grant DMS-0303493, NSA grant MSPF-05G-049 and NSF training grant “Graduate and Postdoctoral Training in Probability and Its Applications” at Cornell University.     

State-dependent Foster–Lyapunov criteria for subgeometric convergence of Markov chains

We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster–Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in Connor and Kendall (2007) [2] concerning the existence of so-called ‘tame’ Markov chains. Furthermore, we show that our ‘subsampled drift condition’ implies the existence of finite moments for the return time to a small set.     

Random orderings of the integers and card shuffling

In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.     

Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise

We construct a Markov family of solutions for the 3D Navier-Stokes equations perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [3]. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Imprecise Markov chains with absorption

We consider convergence of Markov chains with uncertain parameters, known as imprecise Markov chains, which contain an absorbing state. We prove that under conditioning on non-absorption the imprecise conditional probabilities converge independently of the initial imprecise probability distribution if some regularity conditions are assumed. This is a generalisation of a known result from the classical theory of Markov chains by Darroch and Seneta [6].     

Spatial complexity in multi-layer cellular neural networks

This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input.     

Smooth singular flows in dimension 2 with the minimal self-joining property

It is proved that some velocity changes in flows on the torus determined by quasi-periodic Hamiltonians on : where α₁/α₂ is an irrational number with bounded partial quotients, lead to singular flows on with an ergodic component having a minimal set of self-joinings. Authors’ address: K. Frączek and M. Lemańczyk, Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland Research partially supported by KBN grant 1 P03A and by Marie Curie “Transfer of Knowledge” program, project MTKD-CT-2005-030042 (TODEQ) 03826.     

Curvature of Poisson pencils in dimension three

A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if the associated Veronese web is trivial. We suggest a simpler obstruction to flatness, which we call the curvature form of a Poisson pencil. This form can be defined in two ways: either via the Blaschke curvature form of the associated web, or via the Ricci tensor of a connection compatible with the pencil.We show that the curvature form of a Poisson pencil can be given by a simple explicit formula. This allows us to study flatness of linear pencils on three-dimensional Lie algebras, in particular those related to the argument translation method. Many of them appear to be non-flat.     

Nonoscillation for second order sublinear dynamic equations on time scales

Consider the Emden-Fowler sublinear dynamic equation

(0.1)     

Type III factors with unique Cartan decomposition

We prove that for any free ergodic nonsingular nonamenable action Γ?(X,μ)$\Gamma ?(X,\mu )$ of all Γ   in a large class of groups including all hyperbolic groups, the associated group measure space von Neumann algebra L^∞(X)?Γ${\mathrm{L}}^{\infty }(X)?\Gamma$ has L^∞(X)${\mathrm{L}}^{\infty }(X)$ as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in Popa and Vaes (in press) [38]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products _M1?BM2$M_1{?}_B{M}_2$ over a subalgebra B of type I.     

Convergence of the iterated Aluthge transform sequence for diagonalizable matrices

Given an r×r complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined by

Δ(T)=|T|^1/2U|T|^1/2.     

Self-duality and codings for expansive automorphisms

Lind and Schmidt have shown that for certain ergodic Zk-actions on a compact abelian group Γ, the homoclinic group H is isomorphic to the Pontryagin dual of Γ. Einsiedler and Schmidt extended these results and showed that Γ is a quotient of a locally compact ring R modulo H. In this paper, we present a dynamical interpretation of R if k=1: it is a product of the stable group and the unstable group of Γ, under a suitable topology. As applications, we give a topological interpretation of the Pisot-Vijayaraghavan theorem and we link the results to tessellation theory.     

Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the -valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the -valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.     

On the genericity of chaos

Let M be a compact manifold with dimM?2. We prove that some iteration of the generic homeomorphism on M is semiconjugated to the shift map and has infinite topological entropy (Theorem 1.1).     

Distributional limit theorems in infinite ergodic theory

We present a unified approach to the Darling-Kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure. Our method is based on control of the transfer operator up to the first entrance to a suitable reference set rather than on the full asymptotics of the operator. We illustrate our abstract results by showing that they easily apply to a significant class of infinite measure preserving interval maps. We also show that some of the tools introduced here are useful in the setup of pointwise dual ergodic transformations.     

Variational inequalities and the pricing of American options

This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called Brennan-Schwartz algorithm for the valuation of American put options.Research supported in part by a contract from Banque INDOSUEZ.     

Periodic points classify a family of Markov shifts

Ledrappier introduced the following type of space of doubly indexed sequences over a finite abelian group G,

     

Topological entropy for set valued maps

Any continuous map T on a compact metric space X induces in a natural way a continuous map on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map is zero or infinity. Moreover, the topological entropy of is zero, where C(X) denotes the space of all non-empty compact and connected subsets of X. For general continuous maps on compact metric spaces these results are not valid.     

Invariant measure for a class of parabolic degenerate equations

In this paper we consider a stochastic flow in Rⁿ which leaves a closed convex set K invariant. By using a recent characterization of the invariance, involving the distance function, we study the corresponding transition semigroup P_t and its infinitesimal generator N. Due to the invariance property, N is a degenerate elliptic operator. We study existence of an invariant measure ν of P_t and the realization of N in L² (H, ν).     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).