Distributions of additive functionals of the Brownian motion stopped at various random moments

Distributions of additive functionals of the Brownian motion stopped at random moments are investigated. The moments are constructed by the maximum and minimum operations from well-known random times, such as the moment inverse to some additive functionals and first exit time. Bibliography: 5 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 55–76.This research was supported in part by the Russian Foundation for Basic Research, grants 02-01-00265 and 00-15-96019.Translated by V. N. Sudakov.     

Distributions of functionals of the Brownian motion stopped at the moment inverse to the sojourn time

The paper deals with methods of computing the distributions of functionals of the Brownian motion stopped at some random moments. The moments are obtained by means of the minimum and maximum operations from the moments inverse to some additive functionals. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 39–56.     

Distributions of functionals of diffusions with jumps

The paper deals with a generalization of diffusion with jumps. One of the main points is that values of jumps depend on positions of the diffusion before the jump. The next generalization concerns moments of jumps. These moments occur in accordance with the compound Poisson process or with jumping moments constructed by inverse integral functionals of the diffusion. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 15–36.     

Additive functionals with application to sojourn times in infinite-server and processor sharing systems

We deal with additive functionals of stationary processes. It is shown that under some assumptions a stationary model of the time-changed process exists. Further, bounds for the expectation of functions of additive functionals are derived. As an application we analyze virtual sojourn times in an infinite-server system where the service speed is governed by a stationary process. It turns out that the sojourn time of some kind of virtual requests equals in distribution an additive functional of a stationary time-changed process, which provides bounds for the expectation of functions of virtual sojourn times, in particular bounds for fractional moments and the distribution function. Interpreting the GI(n)/GI(n)/∞ system or equivalently the GI(n)/GI system under state-dependent processor sharing as an infinite-server system where the service speed is governed by the number n of requests in the system provides results for sojourn times of virtual requests. In the case of M(n)/GI(n)/∞, the sojourn times of arriving and added requests equal in distribution sojourn times of virtual requests in modified systems, which yields several results for the sojourn times of arriving and added requests. In case of positive integer moments, the bounds generalize earlier results for M/GI(n)/∞. In particular, the mean sojourn times of arriving and added requests in M(n)/GI(n)/∞ are proportional to the required service time, generalizing Cohen’s famous result for M/GI(n)/∞.     

Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations

The estimate of the probability of the large deviation or the statistical random field is the key to ensure the convergence of moments of the associated estimator, and it also plays an essential role to prove mathematical validity of the asymptotic expansion of the estimator. For non-linear stochastic processes, it involves technical difficulties to show a standard exponential type estimate; besides, it is not necessary for these purposes. In this paper, we propose a polynomial-type large deviation inequality which is easily verified by the L ^p -boundedness of certain functionals; usually they are simple additive functionals. We treat a statistical random field with multi-grades and discuss M and Bayesian type estimators. As an application, we show the behavior of those estimators, including convergence of moments, for the statistical random field in the quasi-likelihood analysis of the stochastic differential equation that is possibly multi-dimensional and non-linear. The results are new even for stochastic differential equations, while they obviously apply to other various statistical models.     

On distribution of functionals of Brownian motion with linear drift

The paper deals with methods of calculation of the distributions of functionals of Brownian motion with linear drift. Various stopping time moments are considered. The cases where the moments take values equal to infinity are of special interest. Bibliography:5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 205–217. Translated by A. Sudakov.     

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     

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the L^p functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the L^p norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p ₀ = 2+4π ² /β ² ω ₂ , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p ₀ and p > p ₀ . We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.     

Scaling Limits of Additive Functionals of Interacting Particle Systems

Using the renormalization method introduced by the authors, we prove what we call the local Boltzmann‐Gibbs principle for conservative, stationary interacting particle systems in dimension d = 1. As applications of this result, we obtain various scaling limits of additive functionals of particle systems, like the occupation time of a given site or extensive additive fields of the dynamics. As a by‐product of these results, we also construct a novel process, related to the stationary solution of the stochastic Burgers equation. © 2013 Wiley Periodicals, Inc.     

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.     

Representation of martingale additive functionals on Banach spaces

Assume thatB is a separable real Banach space andX(t) is a diffusion process onB. In this paper, we will establish the representation theorem of martingale additive functionals ofX(t).     

Some categorical properties of the functors O τ and O R of weakly additive functionals

In the paper, the spaces of weakly additive τ-smooth and Radon functionals are investigated. It is proved that the functors of weakly additive τ-smooth and Radon functionals weakly preserve the density of Tychonoff spaces, and the functor of weakly additive τ-smooth functionals forms a monad in the category of Tychonoff spaces and their continuous mappings. Examples and remarks are given showing that these functors fail to satisfy certain Shchepin normality conditions. Problems having positive solutions for normal functors are presented.     

On the limit laws of the second order for additive functionals of Harris recurrent Markov chains

Let {X _n } _{n ≥0} be a Harris recurrent Markov chain with state space E and invariant measure π. The law of the iterated logarithm and the law of weak convergence are given for the additive functionals of the form

Über Zusammenhänge zwischen Übertragungsfunktion und zeitlichem Verhalten eines linearen Netzwerkes

Summary Proceeding from the wellknown definitions of Elmore [3], characterizing delay and rise time of a circuit by the centre of gravity and the inertial moment of the pulse response, we have studied the influence of the higher moments. The moments are integral functions of the pulse response and thus characterize the time domain behaviour of the network. The assumption that the higher moments (or combinations of such moments) are additive for cascaded networks yields simple relations with the attenuation and phase expanded in a power series of the angular frequency around the origin. A simple procedure is given for the calculation of the expansion coefficients from the transfer functionF(p).Several examples of filters are used in a discussion of the quantities defined. In the case of the cascade connection ofn identical sections the conditions are given for which the pulse response approximates a Gaussian forn.     

Transfer theorems and asymptotic distributional results for m‐ary search trees

We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m‐ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so‐called shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that: (i) for small toll sequences (t_n) [roughly, t_n = O(n^1/2)] we have asymptotic normality if m ≤ 26 and typically periodic behavior if m ≥ 27; (ii) for moderate toll sequences [roughly, t_n = ω(n^1/2) but t_n = o(n)] we have convergence to nonnormal distributions if m ≤ m₀ (where m₀ ≥ 26) and typically periodic behavior if m ≥ m₀ + 1; and (iii) for large toll sequences [roughly, t_n = ω(n)] we have convergence to nonnormal distributions for all values of m. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Local times and random time changes

Summary The purpose of this paper is to prove an integral representation theorem for continuous additive functionals (of a Hunt process satisfying hypothesis (F)) as integrals of local times (when they exist) with respect to certain measures. The effect of random time changes on the local times and on the integral representation is investigated.Research sponsored by the National Science Foundation, GP 5217.     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

Some Properties of the Functor O β

It is proved that the space O_β(X) of weakly additive order-preserving normed functionals with compact supports is a convex subset of the space C_p(C_b(X)). Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 313, 2004, pp. 131–134.     

On a representation of additive functionals of zero quadratic variation

For a Markov process X associated to a Dirichlet form, we use continuous additive functionals obtained by Fukushima decompositions in order to represent the class of additive functionals of zero quadratic variation. We do not assume that X is symmetric.     

A note on discontinuous time changes of Markov processes

The “time change” of a Markov process via the inverse of a discontinuous additive functional A_t can be accomplished in two steps. First, perform a time change via the inverse of the strictly increasing discontinuous additive functional obtained by replacing the continuous part of A_t by t. The second step is an ordinary time change via the inverse of a continuous additive functional. Decomposing the time change in this way is useful in studying the time changed process.