Fibering method in some probabilistic problems

The paper is devoted to a systematic study of the distributions of functionals of stochastic processes by the fibering method and to a survey of results obtained in this direction in recent years. Principal attention is given to distinguishing conditions ensuring: a) absolute continuity; b) the existence of a bounded density; c) applicability of the local limit theorem for the distributions of functionals. Smooth, convex functionals and functionals of integral type are considered in detail.Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 22, pp. 61–158, 1984.     

The gauge theorem for a class of additive functionals of zero energy

Summary In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form ₀ ^t q(B _s )ds, whereq is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.Research supported in part by NSA grant MDA-92-H-30324     

Lorentz-Covariant Ultradistributions, Hyperfunctions, and Analytic Functionals

We generalize the theory of Lorentz-covariant distributions to broader classes of functionals including ultradistributions, hyperfunctions, and analytic functionals with a tempered growth. We prove that Lorentz-covariant functionals with essential singularities can be decomposed into polynomial covariants and establish the possibility of the invariant decomposition of their carrier cones. We describe the properties of odd highly singular generalized functions. These results are used to investigate the vacuum expectation values of nonlocal quantum fields with an arbitrary high-energy behavior and to extend the spin–statistics theorem to nonlocal field theory.     

A Limit Theorem for Integral Functionals of an Extremum of Independent Random Processes

We prove a theorem on the convergence of integral functionals of an extremum of independent stochastic processes to a degenerate law of distributions.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 214–221, February, 2005.     

Boundary Functionals of a Semicontinuous Process with Independent Increments on an Interval

We investigate boundary functionals of a semicontinuous process with independent increments on an interval with two reflecting boundaries. We determine the transition and ergodic distributions of the process, as well as the distributions of boundary functionals of the process, namely, the time of first hitting the upper (lower) boundary, the number of hittings of the boundaries, the number of intersections of the interval, and the total sojourn time of the process on the boundaries and inside the interval. We also present a limit theorem for the ergodic distribution of the process and asymptotic formulas for the mean values of the distributions considered.     

On supremal functionals defined in <Emphasis Type="Italic">BV</Emphasis>

This paper is devoted to the study of the lower semicontinuity properties of suitable supremal functionals defined on the space of the functions of bounded variation. The main tool in the proofs is a lower semicontinuity theorem for supremal functionals defined on measures that is inspired to the classical Reshetnyak theorem in which integral functionals are involved.      

Characterizing joint distributions of random sets by multivariate capacities

By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes.     

Distribution of functionals of bridges for diffusions with jumps

The paper deals with a method of calculation of distributions for functionals of bridges of a process which is a generalization of a diffusion with jumps. The approach to calculation of distributions for integral functionals of bridges is the same as for the diffusion itself. This approach is based on calculation of the Laplace transform of distributions of the integral functionals. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 34–47.     

THE EXTENSION THEOREM OF-LINEAR RANDOM FUNCTIONALS AND ITS APPLICATIONS

The extension theorem for linear random functionals in linear apaces is proved. And the representation theorem for bounded linear random functionals and the continuous linear random functionals are studied     

B -值白噪声广义泛函的解析刻画

Banach空间值白噪声广义泛函是一类重要的向量值白噪声广义泛函. 该文建立了Banach空间值白噪声广义泛函的一个解析刻画定理, 并给出了此结果的若干应用.     

Bounds for a class of linear functionals with applications to Hermite interpolation

A general estimation theorem is given for a class of linear functionals on Sobolev spaces. The functionals considered are those which annihilate certain classes of polynomials. An interpolation scheme of Hermite type is defined inN-dimensions and the accuracy in approximation is bounded by means of the above mentioned theorem. In one and two dimensions our schemes reduce to the usual ones, however our estimates in two dimensions are new in that they involve only the pure partial derivatives.This research was supported in part by the National Science Foundation under grant number N.S.F.-G.P.-9467.     

关于一个新泛函的锥拉伸与锥压缩不动点定理及应用

引进了一个新的泛函,它包含了已有的很多类型的泛函.关于此泛函建立了一个泛函形式的锥拉伸与锥压缩不动点定理,并应用此定理研究了一类二阶两点边值问题,得到了该问题至少一个正解的存在性.     

Decomposition theorems

The countable-decomposition theorem for linear functionals has become a useful tool in the theory of representing measures (see [4–7]). The original proof of this theorem was based on a rather involved study of extreme points in the state space of a convex cone. Recently M. Neumann [9] gave an independent proof using a refined form of Simons convergence lemma and Choquet's theorem. In this paper a (relatively) short proof of an extension (to a more abstract situation) of the countable-decomposition theorem is given. Furthermore a decomposition criterion is obtained which even works in the case when not all states are decomposable. All the work is based on a complete characterization of those states which are partially decomposable with respect to a given sequence of sublinear functionals.     

Integral-type functionals of first hitting times for continuous-time Markov chains

We investigate integral-type functionals of the first hitting times for continuous-time Markov chains. Recursive formulas and drift conditions for calculating or bounding integral-type functionals are obtained. The connection between the subexponential integral-type functionals and the subexponential ergodicity is established. Moreover, these results are applied to the birth-death processes. Polynomial integral-type functionals and polynomial ergodicity are studied, and a sufficient criterion for a central limit theorem is also presented.     

Critical points for nondifferentiable functions in presence of splitting

A classical critical point theorem in presence of splitting established by Brézis-Nirenberg is extended to functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. The obtained result is then exploited to prove a multiplicity theorem for a family of elliptic variational-hemivariational eigenvalue problems.     

Localization of limit sets of solutions of nonautonomous delay equations by nonmonotone functionals

We present new generalizations of the Barbashin-Krasovskii theorem which also apply to delay equations with unbounded right-hand side. These generalizations are based on information on the localization of the limit sets of solutions, which is obtained with the use of two classes of not necessarily monotone Lyapunov functionals. The classes of functionals to be used contain sign-definite Lyapunov-Krasovskii functionals as well as Lyapunov-Razumikhin functions.     

A Moment Characterization of B-Valued Generalized Functionals of White Noise

Kernel theorems are established for Bananch space-valued multilinear mappings, A moment characterization theorem for Banach space-valued generalized functionals of white noise is proved by using the above kernel theorems. A necessary and sufficient condition in terms of moments is given for sequences of Banach space-valued generalized functionals of white noise to converge strongly. The integration is also discussed of functions valued in the space of Banach space-valued generalized functionals.     

Distributions of Some Functionals of the Brownian Local Time

Some functionals of the Brownian local time are considered. For these functionals, we propose methods of computation of distributions of the functionals. Bibliography: 5 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 298, 2003, pp. 22–35.     

Stability in first approximation of stochastic systems with after effect : PMM vol. 40, n 6, 1976, pp. 1116–1121

The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved.The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained.Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.