On the rate of convergence of an unstable solution of a stochastic differential equation

We study the rate of convergence of the process(tT)/T to the processw(t)/ asT , where(t) is a solution of the stochastic differential equationd(t)=a((t))dt+((t))dw(t) Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1424–1427, October, 1994.     

A characterization of Gaussian measures on locally compact Abelian groups

Let and be independent random variables having equal variance. In order that + and – be independent, it is necessary and sufficient that and have normal distributions. This result of Bernshtein [1] is carried over in [7] to the case when and take values in a locally compact Abelian group. In the present note, a characterization of Gaussian measures on locally compact Abelian groups is given in which in place of + and –, functions of and are considered which satisfy the associativity equation.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 759–762, November, 1977.     

Hitting probability of a Gaussian vector with values in a Hilbert space in a sphere of small radius

Let be a Gaussian random vector with values in a Hilbert space H. We denote by(a, z) the ball in H with center ata and of radius z. Some asymptotic formulas for I (a, z)=P (a, z), z 0, are presented in the paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 75–93, 1979.     

Oscillating processes with independent increments and nondegenerate Wiener component

For an oscillating process _z(t) (_z(0)=2,t0), which is defined with the help of two homogeneous processes ₁(t) and ₂(t) with independent increments and nondegemerate Wiener components, under certain restrictions we establish a relation of the form and find the characteristic function of the ergodic distribution of the process considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1415–1421, October, 1990.     

An explicit solution of the linear filtering problem for certain classes of nonrational spectral densities

We present an explicit solution of the problem of optimal linear filtering: the recovery of the useful signal(s) at the instantt+, (>0,<0, or=0) from known values of the received signal(s)=(s)+(s) in the past, i.e., at the instantt–s, s0. In doing so we assume the random processes(s) and /gr(s) are stationary and jointly stationary, while the stationary process of noise (s) with zero mean is assumed to be mutually correlated and jointly stationary with the process(s) under the assumption that there exists a common spectral densityf() for these processes.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 83–91, 1986.     

Special directions on contact metric three-manifolds

Blair [5] has introduced special directions on a contact metric 3-manifolds with negative sectional curvature for plane sections containing the characteristic vector field and, when is Anosov, compared such directions with the Anosov directions. In this paper we introduce the notion of Anosov-like special directions on a contact metric 3-manifold. Such directions exist, on contact metric manifolds with negative -Ricci curvature, if and only if the torsion is -parallel, namely (1.1) is satisfied. If a contact metric 3-manifold M admits Anosov-like special directions, and is -parallel, where is the Berger-Ebin operator, then is Anosov and the universal covering of M is the Lie group (2,R). We note that the notion of Anosov-like special directions is related to that of conformally Anosow flow introduced in [9] and [14] (see [6]).Supported by funds of the M.U.R.S.T. and of the University of Lecce. 1991.     

Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure with the Radon–Nikodym derivative d/d possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the -finite measures charging no set that is both -polar and -negligible (U being the potential component of ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azéma, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.     

The problem of controlling a Wiener process when the number of switchings is bounded

We are considering the problem of controlling a one-dimensional Wiener process (t) (0)=0,E=0,D= ²t.Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 53–55.     

Smooth Measures and Regular Strongly Supermedian Kernels Generating Sub-Markovian Resolvents

In the context of a transient Borel right Markov process with a fixed excessive measure , we characterize the regular strongly supermedian kernels, producing smooth measures by the Revuz correspondence. In the case of the measures charging no -semipolar sets, this is the analytical counterpart of a probabilistic result of Revuz, Fukushima, and Getoor and Fitzsimmons, concerning the positive continuous additive functionals. We also consider the case of the measures charging no set that is both -polar and -negligible (U being the potential part of ), answering to a problem of Revuz.     

Sequential estimation of the parameters of diffusion processes

For the parameter of a diffusion process(t), satisfying the stochastic differential equation d(t)=f (t,)dt+dw(l), we propose an effective sequential estimation plan with an unbiased and normally distributed estimate. The proposed sequential plan is discussed in detail for the example of a process (t) having a linear stochastic differential.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 627–638, November, 1972.In conclusion the author wishes to express his deep gratitude to A. N. Shiryaev for formulating the problem and for useful observations     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

Two classes of conformally flat contact metric 3-manifolds

We give a classification of 3—dimensional conformally flat contact metric manifolds satisfying: =0(=L g) orR(Y, Z)=k[(Z)Y–(Y)Z]+[(Z)hY]–(Y)hZ] wherek and are functions. It is proved that they are flat (the non-Sasakian case) or of constant curvature 1 (the Sasakian case).     

Equations de convolution et formes quadratiques

Résumé On étudie, sans hypothèse de convexité, les équations f=g, f=g et f=g.

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Summary We study, without any convexity hypothesis, equations f=g, f=g and f=g where and respectively denote infimal convolution and deconvolution. We give an explicit formulation of these results in the quadratic hilbertian frame, and we interpret them in terms of parallel addition and subtraction of non necessarily semi-definite positive operators.

     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

Über sphärische Blätterungen und die Vollständigkeit ihrer Blätter

Let M be a Riemannian manifold, a differentiable form on M with values in a bundle Hom (TM, ), and let G be an open subset of M such that Ker forms a vector bundle of constant fibre dimension k>0 over G. We prove: If satisfies some analytical conditions, then is completely integrable, the integral manifolds of are spherically bent in M, and in some interesting cases they are complete.     

Contact metric manifolds

In this paper we study contact metric manifoldsM ²ⁿ⁺¹(, , ,g) with characteristic vector field belonging to thek-nullity distribution. Moreover we prove that there exist i) nonK-contact, contact metric manifolds of dimension greater than 3 with Ricci operator commuting with and ii) 3-dimensional contact metric manifolds with non-zero constant -sectional curvature.     

Estimate of a random field based on the trajectory of a point moving in it

We consider the stochastic differential equationd _t =( _t )dt+ _t ( _t )dw _t in Euclidean space, where (x) is a Gaussian random field andw _t is a standard Wiener process. Let f _t ={ _s ,st}. Equations are obtained for the conditional meansm _t (x)=f _t } andB _t (x, y)=M{(x)(y)|f _t }.Translated fromTeariya Sluchaínykh Protsessov, Vol. 14, pp. 7–9, 1986.     

Limit distribution function of inhomogeneities in regions with random boundary. I

A study is made of the interaction of systems of charged particles with a membrane consisting of inhomogeneities randomly distributed in accordance with the same law in the neighborhoods of corresponding sites of a planar crystal lattice. A system of equations for the self-consistent potentialU ₁(x, ⁰,..., ^N ,...) and density of surface changes (x, ⁰,..., ^N ,...) is derived and solved.Institute of Nuclear Physics, Uzbek Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 98–112, July, 1992.     

On a class of 3-dimensional contact metric manifolds

Let M³ be a 3-dimensional contact metric manifold with contact structure (, , , g), such that and =R(.,)) commute. Such a manifold is called 3--manifold. We prove that every 3--manifold with -parallel Weyl tensor is either flat or a Sasakian manifold with constant curvature 1.     

On pseudo-differential operators and smoothness of special Lie-group representations

Two algebras of global pseudo-differential operators over ⁿ are investigated, with corresponding classes of symbols A⁰=CB (all (x, )-derivatives bounded over ²ⁿ), and A₁ (all finite applications of _xj, _j, and _pq=_pq–_px_p on the symbol are in A₀). The class A₁ consists of classical symbols, i.e., ^x a= 0((1+||)–^||) for x Kc ;ⁿ, K, compact. It is shown that a bounded operator A of 210C=L²(Rⁿ) is a pseudo-differential operator with symbol aA_j if and only if the map AG^–1AG, G g_j is infinitely differentiable, from a certain Lie-group g_j c GL(210C) to (210C) with operator norm. g₀ is the Weyl (or Heisenberg) group. Extensions to operators of arbitrary order are discussed. Applications to follow in a subsequent paper.Dedicated to Hans Lewy and Charles B. Morrey, Jr.