On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

Brownian Martingales and Some New Results on Interpolation of Hardy Spaces

Let H₁( ) be the usual Hardy space on . We show that the couple (H₁( ), L( ) is a Calderón couple. This result immediately follows from the following stronger one: Given any fH₁( ) +L( ) there exist two linear operators U and V satisfying the properties: (i) Uf=Nf (Nf being the non-tangential maximal function of f) and U is contractive from H₁( ) to L₁( ) and also from L( ) to L( ); (ii) V(Nf)=f, V is similtaneously bounded from L₁( ) to H₁( ) and from L( ) to L( ) and the norms of V on these spaces are controlled by a universal constant. We also have similar results on the couple (L_p( ), BMO ( )) for every 1相似文献   

An open mapping theorem for measures

LetX andY be Hausdorff spaces and denote byM (X) andM (Y) the corresponding spaces of finite and non-negative Borel measures, endowed with the weak topology. A Borel map :XY induces the map :M (X)M (Y). We give necessary and sufficient conditions for to be open. In case of being a surjection between Suslin spaces, is open if and only if is.     

Extrapolation of transformations of random processes perturbed by white noise

We consider the problem of linear mean square optimal estimation of transformation of a stationary random process (t) in observations of process (t) + n(t) for t < – 0, where (t) is white noise uncorrelated with (t). We find least favorable spectral densities f₀() D and minimax (robust) spectral characteristics of an optimal estimator of transformation A for various classesD of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 216–223, February, 1991.     

Stochastic Partial Differential Equations on Manifolds~II. Nonlinear Filtering

The conditional law of an unobservable component x(t) of a diffusion (x(t),y(t)) given the observations {y(s):s[0,t]} is investigated when x(t) lives on a submanifold of . The existence of the conditional density with respect to a given measure on is shown under fairly general conditions, and the analytical properties of this density are characterized in terms of the Sobolev spaces used in the first part of this series.     

Rates of Decay and h-Processes for One Dimensional Diffusions Conditioned on Non-Absorption

Let (X _t ) be a one dimensional diffusion corresponding to the operator , starting from x>0 and T ₀ be the hitting time of 0. Consider the family of positive solutions of the equation with (0, ), where . We show that the distribution of the h-process induced by any such is , for a suitable sequence of stopping times (S _M : M0) related to which converges to with M. We also give analytical conditions for , where is the smallest point of increase of the spectral measure associated to .     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

The order of approximation to functions of the Zα. Class by means of positive linear operators

Let C_n (, ) be the upper bound for deviations of periodic functions which form the Zygmund class Z,0 0<<2 from a class of positive linear operators. A study is made of the conditions under which there exists a limit nC_n(, )=C(, ). An explicit expression is given for the functions C(,).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 201–210, August, 1968.     

An existence theorem for generalized quasi-variational inequalities

In this paper, we deal with the following generalized quasi-variational inequality problem: given a closed convex subsetX ⁿ , a multifunction :X 2 ⁿ and a multifunction :X 2 ^X , find a point ( ) X × ⁿ such that We prove an existence theorem in which, in particular, the multifunction is not supposed to be upper semicontinuous.     

Perturbation of Dirichlet forms by measures

Perturbations of a Dirichlet form by measures are studied. The perturbed form –_–+₊ is defined for _– in a suitable Kato class and ₊ absolutely continuous with respect to capacity. L _p-properties of the corresponding semigroups are derived by approximating _– by functions. For treating ₊, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has L _p -L _q -smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ₁ the same is shown to be true for the perturbed semigroup, for a large class of measures.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

On the L∞ Norm of the First Eigenfunction of the Dirichlet Laplacian

We obtain the asymptotic behaviour for the L norm of the first eigenfunction of the Dirichlet Laplace operator on a conic sector over a geodesic disc in as . We are led to conjecture that for an open, bounded and convex set D with inradius and diameter d, where and      

The compact support property for solutions to the heat equation with noise

Summary We consider all solutions of a martingale problem associated with the stochastic pde and show thatu(t,·) has compact support for allt0 ifu(0,·) does and if <1. This extends a result of T. Shiga who derived this compact support property for 1/2 and complements a result of C. Mueller who proved this property fails if 1.The author's research was supported by an NSF grant and an NSERC operating grantThe author's research was supported by an NSERC operating grant     

On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Conformally Invariant Regularization and Skeleton Expansions in Gauge Theory

We consider a conformally invariant regularization of an Abelian gauge theory in an Euclidean space of even dimension D 4 and regularized skeleton expansions for vertices and higher Green's functions. We set the respective regularized fields and with the scaling dimensions and into correspondence to the gauge field A and Euclidean current j . We postulate special rules for the limiting transition 0. These rules are different for the transversal and longitudinal components of the field and the current . We show that in the limit 0, there appear conformally invariant fields A and j each of which is transformed by a direct sum of two irreducible representations of the conformal group. Removing the regularization, we obtain a well-defined skeleton theory constructed from conformal two- and three-point correlation functions. We consider skeleton equations on the transversal component of the vertex operator and of the spinor propagator in conformal quantum electrodynamics. For simplicity, we restrict the consideration to an Abelian gauge field A , but generalization to a non-Abelian theory is straightforward.     

On regularities of the distribution of special sequences

Ifp2 is an integer, then every nonnegative integerk is represented by an expression of the form with integersa _i (k), 0a _i (k)p–1,i=0.1,...,s. The radical-inverse function to the basep, _p (k), is defined by . The sequence is uniformly distributed modulo 1 (it may be called a one-dimensional Halton sequence). In the casep=2 it is the van der Corput sequence. The set of all numbers (0, 1] such that the local discrepancy is bounded inn is determined.     

Time-Localization of Random Distributions on Wiener Space

A nuclear space of distributions on Wiener space was constructed by Gorostiza and Nualart [10] as a framework for studying weak convergence of trajectorial fluctuations of particle systems. A basic problem in recovering the usual time-evolution results from the trajectorial ones consists in associating in a unique way an -valued process to a random distribution on by localizing it at each time t [0,1]. In this paper we solve this problem for a large class of random distributions which includes trajectorial fluctuation limits of some systems of diffusions.