Averaging of Backward Stochastic Differential Equations and Homogenization of Partial Differential Equations with Periodic Coefficients

Abstract

We study the limit of the solutions of systems of semi-linear partial differential equations (PDEs) of second order of parabolic type, with rapidly oscillating periodic coefficients, a singular drift, and singular coefficients of the zero and second order terms. Our basic tool is the approach given by Pardoux [14 Pardoux , E. 1999 . Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach . J. Funct. Anal. 167 : 498 – 520 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]]. In particular, we use the weak convergence of an associated backward stochastic differential equation (BSDE).     

ON THE NORM OF THE FOURIER-JACOBI PROJECTION

We extend the results of Pollard [7] Pollard, H. 1949. The mean convergence of orthogonal series. III. Duke Math. J., 16: 189–191. [Crossref], [Web of Science ®] , [Google Scholar] and give asymptotic estimates for the norm of the Fourier-Jacobi projection operator in the appropriate weighted L_p space.     

REPRESENTATION OF THE SPECTRAL SHIFT FUNCTION AND SPECTRAL ASYMPTOTICS FOR TRAPPING PERTURBATIONS

We obtain in the semi-classical setup of “black-box” long-range perturbations a representation for the derivative of spectral shift function ξ(λ) related to two self-adjoint operators L _j (h), j = 1,2. We show that the derivative ξ′(λ) is estimated by the norms of the cut-off resolvents of the operators L _j (h). Finally, we establish a Weyl type formula for the spectral shift function ξ(λ) generalizing the results of Robert [19] Robert, D. 1994. Relative time-delay for perturbations of elliptic operators and semiclassical asymptotics. J. Funct. Anal., 126: 36–82. [Crossref], [Web of Science ®] , [Google Scholar] and Christiansen [5] Christiansen, T. 1998. Spectral asymptotics for general compactly supported perturbations of the Laplacian on Rⁿ. Comm. P.D.E., 23: 933–947. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar].     

THE NICHOLS ZOELLER THEOREM FOR HOPF ALGEBRAS IN THE CATEGORY OF YETTER DRINFELD MODULES

In 1989 Nichols and Zoeller [NZ] Nichols, W. D. and Zoeller, M. B. 1989. A Hopf algebra freeness theorem. Amer. J. Math., 111: 381–385. [Crossref], [Web of Science ®] , [Google Scholar] showed that finite dimensional k-Hopf algebras are free over Hopf subalgebras. An analog result for Yetter Drinfeld Hopf algebras was not known. In this paper the existence of such a basis will be proved. Moreover the existence of a basis in a certain categorial sense cannot be expected.     

Evaluating a model of forest succession using fuzzy analysis

In this paper we discuss how to select the optimal policy from a set of possible policies for a model of forest succession, which can be characterized by a set of trees and the corresponding average life-span with each possible tree transition. The transition probabilities are estimated by counting the numbers of sapling trees of each species under a canopy tree. [1] Horn, Henry S. 1975. Forest Succession. Sci. Amer., : 90–98.  [Google Scholar]. In our setting the transition matrix is defined by using the linguistic terms and as a consequence, the expected longevity of each tree is fuzzy. We use the Dempster–Shafer theory [8] Shafer, G. 1976. A Mathematical Theory of Evidence Princeton University Press.  [Google Scholar] ('76) together with techniques of Norton [7] Norton, J. 1988. Limit Theorems for Dempster's Rule of Combination. Theory and Decision, 25(3): 287–313. [Crossref], [Web of Science ®] , [Google Scholar] ('88) and Smetz [9] Smetz, P. 1990. Belief Functions versus Probability Functions. Uncertainty in Artificial Intelligence, 5: 1–8.  [Google Scholar] ('76) to approximate the transition probabilities.     

ON FREE PRODUCTS OF ABSOLUTE GALOIS GROUPS. II

In [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar], the author gave a positive solution to the problem in the survey of Jarden [2] Jarden, M. 1996. “Infinite Galois Theory”. In Handbook of Algebra I Amsterdam: Elsevier Sci.. [Crossref] , [Google Scholar] on the closedness of the class of profinite groups that are isomorphic to absolute Galois groups of fields with respect to finite free products. In [3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar], O. V. Mel'nikov solved this problem for separable profinite groups ([3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar] was done earlier than [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar]). In the same case, a more exact result on the absolute Galois groups of fields of fixed characteristic was obtained there. The proof proposed in 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.   is simpler than that in [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar] and, in addition, provides the results of Mel'nikov.

On February, 2000, the author (knowing nothing about 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.  ) found one more proof of these results. In the author opinion, this proof is the simplest and the construction used in the proof, as well as its properties (cf. Propositio n 1) can have other applications.     

UNIFORM FILTERS AND SEMI-COMPATIBILITY

In [5] García Román, M., Márquez Hernández, M. and Verschoren, A. 1997. Structure Sheaves and Noncommutative Topologies. J. of Algebra, 194: 224–244. [Crossref], [Web of Science ®] , [Google Scholar] we used functors which are compositions of localization functors to construct sheaves over an arbitrary ring R. These functors share some properties with localization, and questions like when is the composition of localizations a localization functor? arise naturally. In this note we answer this question and some related ones using the key concept of semi-compatibility.

     

ON RESULTANT CRITERIA AND FORMULAS FOR THE INVERSION OF A POLYNOMIAL MAP

Concerning the inversion of a polynomial map F: K ² ? K ² over an arbitrary field K, it is natural to consider the following questions: (1) Can we find a necessary and sufficient criterion in terms of resultants for F to be invertible with polynomial ((2) resp. rational) inverse such that, this criterion gives an explicit formula to compute the inverse of F in this case? MacKay and Wang [5] McKay, J. and Wang, S. S. 1986. An Inversion Formula for Two Polynomials in Two Variables. J. of Pure and Appl. Algebra., 40: 245–257. [Crossref], [Web of Science ®] , [Google Scholar] gave a partial answer to question (1), by giving an explicit expression of the inverse of F, when F is invertible without constant terms. On the other hand, Adjamagbo and van den Essen [3] Adjamagbo, K. and van den Essen, A. 1990. A Resultant Criterion and Formula for the Inversion of a Polynomial Map in Two Variables. J. of Pure and Appl. Algebra., 64: 1–6. North-Holland [Google Scholar] have fully answered question (2) and have furnished a necessary and sufficient criterion which relies on the existence of some constants λ₁, λ₂ in K ^*. We improve this result by giving an explicit relation between λ₁, λ₂ and constants of the Theorem of MacKay and Wang [5] McKay, J. and Wang, S. S. 1986. An Inversion Formula for Two Polynomials in Two Variables. J. of Pure and Appl. Algebra., 40: 245–257. [Crossref], [Web of Science ®] , [Google Scholar].

Concerning question (2), Adjamagbo and Boury [2] Adjamagbo, K. and Boury, P. 1992. A Resultant Criterion and Formula for the Inversion of a Rational Map in Two Variables. J. of Pure and Appl. Algebra., 79: 1–13. North-Holland [Google Scholar] give a criterion for rational maps which relies on the existence of two polynomials λ₁, λ₂. We also improve this result, by expliciting the relations between these λ₁, λ₂ and the coefficients of F. This improvement enables us, first to give an explicit proof of the corresponding Theorem of Abhyankhar [1] Abhyankar, S. S. 1990. Algebraic Geometry for Scientists and Engineers. Math. Surveys and Monographs., 5: 267–273.  [Google Scholar], and secondly, to give a counter example where these λ₁, λ₂ are not in K ^*, contrary to claim of Yu [6] Yu, J.-T. 1993. Computing Minimal Polynomials and the Inverse via GCP. Comm. Algebra, 21(No.7): 2279–2294.  [Google Scholar].     

Asymptotic results for finite superpositions of Ornstein–Uhlenbeck processes

A model of intermittency based on superposition of Lévy driven Ornstein–Uhlenbeck processes is studied in [6 Grahovac, D., Leonenko, N., Sikorskii, A., and Te?niak, I. 2016. Intermittency of superpositions of Ornstein–Uhlenbeck type processes. J. Stat. Phys. 165:390–408.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, as shown in Theorem 5.1 in that paper, finite superpositions obey a (sample path) central limit theorem under suitable hypotheses. In this paper we prove large (and moderate) deviation results associated with this central limit theorem.     

On a discrete-time model with replicator dynamics in renewable resource exploitation

We consider a discrete-time version of the model proposed by Lamantia and Radi [15 F. Lamantia, and D. Radi, Exploitation of renewable resources with differentiated technologies: an evolutionary analysis, Math. Comput. Simulation 108 (2015), pp. 155–174. doi:10.1016/j.matcom.2013.09.013.[Crossref], [Web of Science ®] , [Google Scholar]] to describe a fishery where a population regulated by a logistic growth function is exploited by a pool of agents that can choose, at each time period, between two different harvesting strategies according to a profit-driven evolutionary selection rule. The resulting discrete dynamical system, represented by a two-dimensional nonlinear map, is characterized by the presence of invariant lines on which the dynamics are governed by one-dimensional restrictions that represent pure, i.e. adopted by all players, strategies. However, interesting dynamics related to interior attractors, where players playing both strategies coexist, are evidenced by analytical as well as numerical methods that reveal local and global bifurcations.     

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ? R ² ? J ? Rⁿ (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R ² such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = G_k = (2) × ? × (2).

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