Large and moderate deviations for intersection local times

We study the large and moderate deviations for intersection local times generated by, respectively, independent Brownian local times and independent local times of symmetric random walks. Our result in the Brownian case generalizes the large deviation principle achieved in Mansmann (1991) for the L ₂-norm of Brownian local times, and coincides with the large deviation obtained by Csörgö, Shi and Yor (1991) for self intersection local times of Brownian bridges. Our approach relies on a Feynman-Kac type large deviation for Brownian occupation time, certain localization techniques from Donsker-Varadhan (1975) and Mansmann (1991), and some general methods developed along the line of probability in Banach space. Our treatment in the case of random walks also involves rescaling, spectral representation and invariance principle. The law of the iterated logarithm for intersection local times is given as an application of our deviation results.Supported in part by NSF Grant DMS-0102238Supported in part by NSF Grant DMS-0204513 Mathematics Subject Classification (2000):Primary: 60J55; Secondary: 60B12, 60F05, 60F10, 60F15, 60F25, 60G17, 60J65     

Sharp bounds on the density,Green function and jumping function of subordinate killed BM

Subordination of a killed Brownian motion in a domain D^d via an /2-stable subordinator gives rise to a process Z_t whose infinitesimal generator is –(–|_D)^/2, the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Z_t when D is either a bounded C^1,1 domain or an exterior C^1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.Mathematics Subject Classification (2000):Primary 60J45, Secondary 60J75, 31C25     

-measures for branching exit Markov systems and their applications to differential equations

Semilinear equations Lu=(u) where L is an elliptic differential operator and is a positive function can be investigated by using (L,)-superdiffusions. In a special case u=u² a powerful probabilistic tool – the Brownian snake – introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation. Some of these results (but not all of them) were extended by Dynkin and Kuznetsov to general equations by applying superprocesses. An important role in the theory of the Brownian snake and its applications is played by measures _x on the space of continuous paths. Our goal is to introduce analogous measures related to superprocesses (and to general branching exit Markov systems). They are defined on the space of measures and we call them -measures. Using -measures allows to combine some advantages of Brownian snakes and of superprocesses as tools for a study of semilinear PDEs.Partially supported by National Science Foundation Grant DMS-0204237 and DMS-9971009Mathematics Subject Classification (2000): Primary 31C15, Secondary 35J65, 60J60     

Solutions of Δ<Emphasis Type="Italic">u</Emphasis>=4<Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">2</Emphasis></Superscript> with Neumann’s conditions using the Brownian snake

We consider a Brownian snake (W_s,s0) with underlying process a reflected Brownian motion in a bounded domain D. We construct a continuous additive functional (L_s,s0) of the Brownian snake which counts the time spent by the end points _s of the Brownian snake paths on D. The random measure Z=_sdLs is supported by D. Then we represent the solution v of u=4u² in D with weak Neumann boundary condition 0 by using exponential moment of (Z,) under the excursion measure of the Brownian snake. We then derive an integral equation for v. For small it is then possible to describe negative solution of u=4u² in D with weak Neumann boundary condition . In contrast to the exit measure of the Brownian snake out of D, the measure Z is more regular. In particular we show it is absolutely continuous with respect to the surface measure on D for dimension 2 and 3.Mathematics Subject Classification (2000):60J55, 60J80, 60H30, 60G57, 35C15, 35J65     

Essential spectral radius for Markov semigroups (I): discrete time case

Using two new measures of non-compactness (P) and _w (P) for a positive kernel P on a Polish space E, we obtain a new formula of Nussbaum-Gelfand type for the essential spectral radius r _ess (P) on b. Using that formula we show that different known sufficient conditions for geometric ergodicity such as Doeblins condition, drift condition by means of Lyapunov function, geometric recurrence etc lead to variational formulas of the essential spectral radius. All those can be easily transported on the weighted space b _u . Some related results on L ² () are also obtained, especially in the symmetric case. Moreover we prove that for a strongly Feller and topologically transitive Markov kernel, the large deviation principle of Donsker-Varadhan for occupation measures of the associated Markov process holds if and only if the essential spectral radius is zero; this result allows us to show that the sufficient condition of Donsker-Varadhan for the large deviation principle is in fact necessary. The knowledge of r _ess (P) allows us to estimate eigenvalues of P in L ² in the symmetric case, and to estimate the geometric convergence rate by means of that in the metric of Wasserstein. Applications to different concrete models are provided for illustrating those general results. Mathematics Subject Classification (2000):60J05, 60F10, 47A10, 47D07     

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over ℤ6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ₆-codes. Then extremal self-dual ₆-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.     

Comparison theorem and estimates for transition probability densities of diffusion processes

We establish several comparison theorems for the transition probability density p _b (x,t,y) of Brownian motion with drift b, and deduce explicit, sharp lower and upper bounds for p _b (x,t,y) in terms of the norms of the vector field b. The main results are obtained through carefully estimating the mixed moments of Bessel processes. All constants are explicit in our lower and upper bounds, which is different from most of the previous estimates, and is important in many applications for example in statistical inferences for diffusion processes.Research partially supported by N.S.F. Grants DMS-0203823, and by Doctoral Program Fundation of the Ministry of Education of China, Grant No. 20020269015. Mathematics Subject Classification (2000):Primary: 60H10, 60H30; Secondary: 35K05     

On the Completeness and Decidability of the Horn‐Like Fragment of the First‐Order Linear Temporal Logic

We prove the completeness and decidability of the Hornlike sequents, specifically, the socalled D₂sequents (of the firstorder linear temporal logic) considered in the author's paper [Lith. Math. J., 41(3), 266–281 (2001)]. In this paper, with the help of the infinitary calculus G_L, grounded by the author in his earlier papers, for D₂sequents we construct a D₂Sat calculus of the socalled saturated type consisting of decidable deductive procedures replacing the omegarule for the always operator. In the present paper, in order to prove the completeness and decidability of the calculus D₂Sat, we construct the socalled invariant decidable calculus D₂IN. We prove the equivalence of the calculi D₂IN, D₂Sat, and G _L ^** for the socalled saturated D₂sequents. From this equivalence, by reducing an arbitrary D₂sequent to a saturated D₂sequent, and also from the completeness of the G _L ^** calculus and decidability of the invariant calculus D₂IN, we deduce the completeness and decidability of the calculus D₂Sat in the class of D₂sequents.     

A note on induced cycles in Kneser graphs

Letg(n,r) be the maximal order of an induced cycle in the Knesser graph Kn([n] r), whose vertices are ther-sets of [n]={1, ...,n} and whose adjacency relation is disjointness. Thusg(n, r) is the largestm for which there is a sequenceA ₁,A ₂,...,A _m [n] ofr-sets withA _i A _j= if and only if |i-j|=1 orm–1. We prove that there is an absolute constantc>0 for which

Expected Sums of General Parking Functions

A (u₁, u₂, . . . )-parking function of length n is a sequence (x₁, x₂, . . . , x_n) whose order statistics (the sequence (x₍₁₎, x₍₂₎, . . . , x_(n)) obtained by rearranging the original sequence in non-decreasing order) satisfy x_(i) u_(i). The Gonarov polynomials g _n (x; a₀, a ₁, . . . , a _n-1) are polynomials biorthogonal to the linear functionals (a _i) Dⁱ, where (a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Gonarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u _i=a+ (i-1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u_i+1 - u_i. Finally, we give analogues of our results for real-valued parking functions.AMS Subject Classification: 05A15, 05A19, 05A20, 05E35.     

The geometric invariants $${\Omega^{n}}$$ of a product of groups

We compute the geometric invariants of a product G × H of groups in terms of and . This gives a sufficient condition in terms of and for a normal subgroup of G × H with abelian quotient to be of type F _n . We give an example involving the direct product of the Baumslag–Solitar group BS_1,2 with itself.      

The Brownian loop soup

We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to chronologically add Brownian loops to simple curves in the plane.Cornell University; Research supported in part by the National Science FoundationUniversité Paris-Sud and IUFMathematics Subject Classification (2000):60J65, 81T40     

Variational Formula for Dirichlet Forms and Estimates of Principal Eigenvalues for Symmetric α-stable Processes

In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure charging no set of zero capacity, we give a new proof of an L²()-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed -stable processes and obtain conditions for some measures being gaugeable. Mathematics Subject Classifications (2000) Primary 31C25; Secondary 34L15, 60G52.     

Uniqueness for isotropic diffusions with a linear drift

This paper proves that-valued solutions to the SDE are unique in distribution, when D^d is convex and open, D, c>0, is positive and locally Lipschitz on D and zero on D, and {xD:g(x)r} is convex for r sufficiently small. The proof (for =0) is based on the transformation X_te^ctX_t, which removes the drift, and a random time change. Although the set-up is rather specialized the result gives uniqueness for some SDEs that cannot be treated by any of the conventional techniques.Mathematics Subject Classification (2000):60J60, 60H10     

Induced product representations of extended Cuntz algebras

Let _H be the extended Cuntz algebra over the Hilbert space H. Since its zero grade part _H⁰ is the C^*-inductive limit of B(H^r), we look for some family of representations on an inductive limit of H^r as r. When such construction is shaped according to the structure of _H⁰, von Neumanns notion of a reference sequence of unit vectors for Hilbert infinite tensor products emerges; after a further Rieffel induction step, a class IPR[H] of representations of _H arises. For any two such representations, we describe explicitly their associated intertwiners. Any two representations in IPR[H] are either disjoint or unitarily equivalent. Actions of the group by translation on sequences of unit vectors are involved, as well as the ideals of .     

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and -convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (P _h) of optimal control problems and a control problem (P), we determine some general conditions, involvingG-convergence and -convergence, under which the sequence of the optimal pairs of the problems (P _h) converges to the optimal pair of problem (P).The authors wish to thank Professor E. De Giorgi for many stimulating discussions.     

M/G/∞ with alternating renewal breakdowns

We consider an M/G/ queue where the service station is subject to occasional interruptions which form an alternating renewal process ofup anddown periods. We show that under some natural conditions the random measure process associated with the residual service times of the customers is regenerative in the strict sense, and study its steady state characteristics. In particular we show that the steady state distribution of this random measure is a convolution of two distributions of (independent) random measures, one of which is associated with a standard M/G/ queue.     

Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

Some new examples of Markov processes which enjoy the time-inversion property

In this paper we give a sufficient condition on the semi group densities of an homogeneous Markov process taking values in ⁿ which ensures that it enjoys the time-inversion property. Our condition covers all previously known examples of Markov processes satisfying this property. As new examples we present a class of Markov processes with jumps, the Dunkl processes and their radial parts.Mathematics Subject Classification (2000): 60J25, 60J60, 60J65, 60J99