Class Of Additive Functions Research Articles

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.

Estimate of the concentration function for a class of additive functions

We show how methods for estimating the concentration function of the sum of independent random variables can be used to obtain estimates of the concentration function for the values of additive functions under suitable conditions. Previously obtained estimates of the concentration function are consequences of the estimate obtained in the present paper for functions from of the class under consideration.

On summatory functions of additive functions and regular variation

An overview of results and problems concerning the asymptotic behaviour for summatory functions of a certain class of additive functions is given. The class of functions in question involves regular variation. Some new Abelian and Tauberian results for additive functions of the form F(n)=Sigma ah(p), p^alpha||n, are obtained. .

First Exit Time from a Bounded Interval for a Certain Class of Additive Functionals of Brownian Motion

Let (Bt)t≥0 be standard Brownian motion starting at y, Xt = x + ∫t0V(Bs) ds for x ∈ (a, b), with V(y) = yγ if y≥0, V(y)=−K(−y)γ if y≤0, where γ>0 and K is a given positive constant. Set τab=inf{t>0: Xt∉(a, b)} and σ0=inf{t>0: Bt=0}. In this paper we give several informations about the random variable τab. We namely evaluate the moments of the random variables \(B_{\tau _{ab} } and B_{\tau _{ab} \wedge \sigma _0 } \), and also show how to calculate the expectations \({\mathbb{E}}\left( {\tau _{ab}^m B_{\tau _{ab} }^n } \right) and {\mathbb{E}}\left( {\left( {\tau _{ab} \wedge \sigma _0 } \right)^m B_{\tau _{ab} \wedge \sigma _0 }^n } \right)\). Then, we explicitly determine the probability laws of the random variables \(B_{{\tau }_{ab} } and B_{{\tau }_{ab} \wedge \sigma _0 }\) as well as the probability \({\mathbb{P}}\left\{ {X_{\tau _{ab} } = a\left( {or b} \right)} \right\}\) by means of special functions.

Reciprocal sums of a class of additive functions in short intervals

USING the method of probabilistic number-theory, De ~oninck"] , and De Koninck and ~ a l a q b o s ' ~ ] studied the reciprocal sum of the additive function f ( n ) satisfying f ( n )>,to >O ( n 2 2 ) and f ( p -1 and obtained an asymptotic formula, where t o is an absolute positive constant. Let B ( x ) denote the number of n <x not satisfying the inequality loglogn R ( x ) < f ( n ) < loglogn + R ( x ) , ( 1 where R ( x ) is a function tending to infinity. Then in refs. [ I , 21, it is proved that if R ( x ) = o ( loglogx ) and B ( x ) = o ( xlloglogx ) , and

Focker-Planck equation on a manifold. Effective diffusion and spectrum

The main objectives of the present paper are: (i) to construct a proper generalization of Ornstein-Uhlenbeck process for the case of a smooth Riemannian manifold (also under the action of an external potential field); (ii) to prove a Central Limit Theorem for a certain class of additive functionals, making use of the estimations how fast the process converges to equilibrium; (iii) to investigate spectral properties of the corresponding generator; and (iv) to discuss some geometrical aspects of averaging for this process.

Comportement Asymptotique de

AbstractLet {x} denote the fractional part of x. We find an asymptotic formula of , where k is any positive integer and a is any real number ≥ 1, and so for the sum ∑n&lt;xf(n), where f(n) belongs to a class of additive functions.

An elementary method for estimating deviations from the normal distribution in a certain class of additive functions

An elementary method for estimating deviations from the normal distribution in a certain class of additive functions

A Characterization of Finitely Monotonic Additive Functions

Letf(m) be a real-valued, number theoretic function . We say thatf(m) is additive if f(mn) = f(m) +f(n) whenever (m, n) = 1 . If f(m) satisfies the additional restriction that f(p) = f(p2) _ f(p3) = . . ., then we say that f(m) is strongly additive . We denote the class of additive functions by .4. A function fEF .2/ is called finitely monotonic if there exists an infinite sequence xk -+ oo and a positive constant A, so that for each xk there are integers

On the existence of one class of additive functionals of diffusion processes

On the existence of one class of additive functionals of diffusion processes

A Class of Additive Functions

A Class of Additive Functions

A Class of Additive Functions

A Class of Additive Functions

Additive Functionals on Lp Spaces

In (1) a representation theorem was proved for a class of additive functionals defined on the continuous real-valued functions with domain S = [0, 1]. The theorem was extended to the case where S is an arbitrary compact metric space in (3). Our present purpose is to consider the corresponding class of additive functionals defined on Lp spaces, p &gt; 0. In (4) Martin and Mizel have considered functionals defined on the class of bounded measurable functions which, however, satisfy a certain “stochastic” condition which we do not require.