Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ) such thatu ^–1 is operator monotone. We then look for another functionv such thatv(u ^–1) is operator monotone, namely,u(A)u(B) impliesv(A)v(B) for self-adjoint operatorsA andB.     

∑II∑ threshold formulas

A II formula has the form, where eachL is either a variable or a negated variable. In this paper we study the computation of threshold functions by II formulas. By combining the proof of the Fredman-Komlós bound [5, 10] and a counting argument, we show that fork andn large andkn/2, every II formula computing the threshold functionT _k ⁿ has size at least exp . Fork andn large andkn ^2/3, we show that there exist II formulas for computingT _k ⁿ with size at most exp .     

Quadrature error expansions

Summary We continue the work of Part I, treating in detail the theory of numerical quadrature over a square [0, 1]² using anm ² copy,Q ^(m), of a one-point quadrature rule. As before, we determine the nature of an asymptotic expansion for the quadrature error functionalQ ^(m) F—IF in inverse powers ofm and related functions, valid for specified classes of the integrand functionF. The extreme case treated here is one in which the integrand function has a full-corner algebraic singularity. This has the formx y r, (x, y). Here , , and need not be integer, andr is (x ²+y ²)^/2 or some other similar homogeneous function. The error expansion forms the theoretic basis for the use of extrapolation, for this kind of integrand.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38     

The Gaussian measure of shifted balls

Summary Let be a centered Gaussian measure on a Hilbert spaceH and let be the centered ball of radiusR>0. ForaH and , we give the exact asymptotics of (B _R(t)+t·a) ast. Also, upper and lower bounds are given when is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates.Supported in part by NSF grant number DMS-9024961     

The addition formula for theta functions

Summary In this paper we solve the functional equationx(u + v)(u – v) = f ₁(u)g₁(v) + f₂(u)g₂(v) under the assumption thatx, , f ₁, f₂, g₁, g₂ are complex-valued functions onR ⁿ ,n N arbitrary, and 0 and 0 are continuous. Our main result shows that, apart from degeneracy and some obvious modifications, theta functions of one complex variable are the only continuous solutions of this functional equation.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Bernstein functions, complete hyperexpansivity and subnormality—I

Special classes of functions on the classical semigroupN of non-negative integers, as defined using the classical backward and forward difference operators, get associated in a natural way with special classes of bounded linear operators on Hilbert spaces. In particular, the class of completely monotone functions, which is a subclass of the class of positive definite functions ofN, gets associated with subnormal operators, and the class of completely alternating functions, which is a subclass of the class of negative definite functions onN, with completely hyper-expansive operators. The interplay between the theories of completely monotone and completely alternating functions has previously been exploited to unravel some interesting connections between subnormals and completely hyperexpansive operators. For example, it is known that a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {1/_n}(n0). The present paper discovers some new connections between the two classes of operators by building upon some well-known results in the literature that relate positive and negative definite functions on cartesian products of arbitrary sets using Bernstein functions. In particular, it is observed that the weight sequence of a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {_n+1/_n}(n0). It is also established that the weight sequence of any completely hyperexpansive weighted shift is a Hausdorff moment sequence. Further, the connection of Bernstein functions with Stieltjes functions and generalizations thereof is exploited to link certain classes of subnormal weighted shifts to completely hyperexpansive ones.     

Sur les équations fonctionnelles auxq-différences

Summary In this paper, we study the convergence of formal power series solutions of functional equations of the formP _k(x)(^[k](x))=(x), where ^[k] (x) denotes thek-th iterate of the function.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differential equations: if(0) = 0, and(0) =q is a nonzero complex number with absolute value less than one then, if(x)=a(n)x ⁿ is a divergent solution, there exists a positive real numbers such that the power seriesa(n)q ^sn(n+1)2 x ⁿ has a finite and nonzero radius of convergence.

     

Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

Increase of Dimension by Homogenization

The paper deals with the homogenization of a Neumann's problem in a thin periodic weakly connected domain of R ³. The domain _n is composed of a large number n of disjoint periodic connected components linked by a periodic lattice _n of very thin bridges. According to the distribution and to the size of the linking bridges, the limit problem as n tends to infinity is either a 4d Neumann's problem or a 4d nonlocal problem. The additional term corresponding to the increase of dimension is due to the connection effect of the bridges.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceH ^p (), 1p<, over a strictly pseudoconvex domain in ⁿ with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesT with suitable symbols onH ^p () have property (). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.     

First order properties of random posets

Let =(n,p) be a binary relation on the set [n]={1, 2, ..., n} such that (i,i) for every i and (i,j) with probability p, independently for each pair i,j [n], where i<j. Define as the transitive closure of and denote poset ([n], ) by R(n, p). We show that for any constant p probability of each first order property of R(n, p) converges as n .     

Bilateral difference method for solving the boundary value problem for an ordinary differential equation

A method is proposed for calculating the bilateral approximations of the solution of the boundary value problem on [0, 1] for the equation y+p(x)y-q(x)y=f(x) and the derivative of the solution having the maximum deviation O(h² (h)+h³) on {kh} _k ^N =0, where(t) is the sum of the continuity moduli of the functions p, q,f, on the set of points {kh} _k ^N =0, h=1/N by means of O(N) operations. The data obtained for fairly smooth p, q,f allow interpolation to be used for calculating the bilateral approximations of the solution and its higher derivatives having the maximum deviation O(h³) on [0, 1].Translated from Matematicheskie Zametkii, Vol. 11, No. 4, pp. 421–430, April, 1972.     

Rudiments of an average case complexity theory for piecewise-linear path following algorithms

We examine the efficiency of PL path following algorithms in followingF _T ^-1 (0), whereF _T is the PL approximation, induced by the simplicial triangulationT, to a mapf: ^{n n-1}. In particular, we consider the problem of determining an upper bound on the expected number of pivots made per unit length off ^–1(0) that is approximated. We show that if the sizes of the simplices ofT are sufficiently small, where sufficiently small is an explicitly given quantity dependent on measurements of how nicef is, then the average directional density ofT, as introduced by Todd, really does give a good approximation to the expected number of pivots made, confirming what researchers have believed on intuitive grounds for a decade. Because what constitutes sufficiently small is a precisely given quantity, i.e., non-asymptotic, we are able to provide some rigorous justification for the claim that the expected number of pivots grows only polynomially inn, the number of variables.Several other issues are also examined.Research supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. This research was performed while the author was a member of the Mathematical Sciences Research Institute, Berkeley, California.     

Nilpotent commuting varieties of reductive Lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p0, and =LieG. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let =() denote the nilpotent variety of , and ^nil():={(x,y)×|[x,y]=0}, the nilpotent commuting variety of . Our main goal in this paper is to show that the variety ^nil() is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme _n Hilb ⁿ (²) is irreducible over any algebraically closed field. Mathematics Subject Classification (2000) 20G05     

Eigenvalue distribution of some fractal semi-elliptic differential operators: Combinatorial approach

We obtain the sharp order of growth of the eigenvalue distribution function for the operator in the anisotropic Sobolev space , generated by the quadratic form _Q u² d, whereQ² is the unit square and is a probability self-affine fractal measure onQ. The geometry of Supp should be in a certain way consistent with the parameterst ₁ ,t ₂ .     

Rates of convergence for the approximation of dual shift-invariant systems in ℓ2(ℤ)

A shift-invariant system is a collection of functions {g_m,n} of the form g_m,n(k)=g_m(k–an). Such systems play an important role in time-frequency analysis and digital signal processing. A principal problem is to find a dual system _m,n(k)=_m(k–an) such that each functionf can be written asf= f, _m,ngm,n. The mathematical theory usually addresses this problem in infinite dimensions (typically in L² () or ²()), whereas numerical methods have to operate with a finite-dimensional model. Exploiting the link between the frame operator and Laurent operators with matrix-valued symbol, we apply the finite section method to show that the dual functions obtained by solving a finite-dimensional problem converge to the dual functions of the original infinite-dimensional problem in ²(). For compactly supported g_{m, n} (FIR filter banks) we prove an exponential rate of convergence and derive explicit expressions for the involved constants. Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures. Again we provide explicit estimates for the speed of convergence. Some remarks on tight frames complete the paper.Part of this work was done while the author was a visitor at the Department of Statistics at the Stanford University.The author has been partially supported by Erwin-Schrödinger scholarship J01388-MAT of the Austrian Science foundation FWF.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.