On Infinite Systems of Linear Autonomous and Nonautonomous Stochastic Equations

The solvability of autonomous and nonautonomous stochastic linear differential equations in is studied. The existence of strong continuous (L^p-continuous) solutions of autonomous linear stochastic differential equations in with continuous (L^p-continuous) right-hand sides is proved. Uniqueness conditions are obtained. We give examples showing that both deterministic and stochastic linear nonautonomous differential equations with the same operator in may fail to have a solution. We also establish existence and uniqueness conditions for nonautonomous equations.     

A White Noise Approach to Stochastic Neumann Boundary-Value Problems

We illustrate the use of white noise analysis in the solution of stochastic partial differential equations by explicitly solving the stochastic Neumann boundary-value problem LU(x)–c(x)U(x)=0, xDR ^d ,(x)U(x)=–W(x), xD, where L is a uniformly elliptic linear partial differential operator and W(x), xR ^d , is d-parameter white noise.     

Large Deviations of Solutions of Hyperbolic SPDE's in the Hölder Norm

In this paper, we prove the large deviations principle for solutions of a hyperbolic stochastic partial differential equation, in the Hölder topology of index for all 0 < . This result generalizes those in [5] and [10] to the Hölder norm, and the result in [3] for solutions of a class fo stochastic differential equations involving a two-parameter Wiener process. These solutions are obtained by small perturbations of the noise.     

Convergence of the Marcus–Lushnikov Process

The Smoluchowski coagulation equation describes the concentration c(t,x) of particles of mass x [ 0,] at the instant t 0, in an infinite system of coalescing particles. It is well-known that in some cases, gelation occurs: a particle with infinite mass appears. But this infinite particle is inert, in the sense that it does not interact with finite particles. We consider the so-called Marcus–Lushnikov process, which is a stochastic finite system of coalescing particles. This process is expected to converge, as the number of particles tends to infinity, to a solution of the Smoluchowski coagulation equation. We show that it actually converges, for t [0,], to a modified Smoluchowski equation, which takes into account a possible interaction between finite and infinite particles.     

Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

We consider the Hamiltonian H (K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator H₀(K) lying below z0 with respect to the total quasimomentum K0 and the spectral parameter z–0.     

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

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     

Koszul–Sullivan Models and the Cohomology of Certain Solvmanifolds

In this paper we develop a technique of working with graded differential algebra models of solvmanifolds, overcoming the main difficulty arising from the non-nilpotency of the corresponding Mostow fibrations. A graded differential model for solvmanifolds of the form G/ with G=RN is presented (N is a nilpotent Lie group, G is a semi-direct product). As an application, we prove the Benson–Gordon conjecture in dimension four.     

A pursuit-evasion differential game with noisy measurements of the evader's bearing from the pursuer

A stochastic pursuit-evasion differential game involving two players, E and P, moving in the plane is considered. It is assumed that player E (the evader) has complete observation of the position and velocity of player P, whereas player P (the pursuer) can measure the distanced (P, E) between P and E but receives noise-corrupted measurements of the bearing of E from P. Three cases are dealt with: (a) using the noise-corrupted measurements of , player P applies the proportional navigation guidance law; (b) P has complete observation ofd (P, E) and (this case is treated for the sake of completeness); (c) using the noise-corrupted measurements of , P applies an erroneous line-of sight guidance law. For each of the cases, sufficient conditions on optimal strategies are derived. In each of the cases, these conditions require the solution of a nonlinear partial differential equation on a in ². Finally, optimal strategies are computed by solving the corresponding equations numerically.     

Lévy–Khintchine Formula and Dual Convolution Semigroups Associated with Laguerre and Bessel Functions

In this work we consider a system of partial differential operators D ₁,D ₂ on K=[0,+[×R, whose eigenfunctions are the functions (x,t), (x,t)K, =((R0)×N)(0×[0,+[), which are related to the Laguerre functions for ((R 0)×N)(0,0) and which are the Bessel functions for (0×[0,+[). We provide K and with a convolution structure. We prove a Lévy–Khintchine formula on K, which permits us to characterize dual convolution semigroups on .     

On certain imbedding and extension theorems for a weighted class of functions

n- , S n–1 m (0m<n), () S . () , ().     

Nonparametric Inference for a Class of Stochastic Partial Differential Equations II

Consider the stochastic partial differential equationdu (t,x) = (t)u (t, x)dt + dW _Q(t,x), 0 t T where = ₂/x ₂, and is a class of positive valued functions. We obtain an estimator for the linear multiplier (t) and establish the consistency, rate of convergence and asymptotic normality of this estimator as 0.     

Structure of Equations Solvable by the Inverse Scattering Transform for the Schrödinger Operator

We propose a new approach for deriving nonlinear evolution equations solvable by the inverse scattering transform. The starting point of this approach is consideration of the evolution equations for the scattering data generated by solutions of an arbitrary nonlinear evolution equation that rapidly decrease as x±. Using this approach, we find all nonlinear evolution equations whose integration reduces to investigation of the scattering-data evolution equations that are differential equations (in either ordinary or partial derivatives). In this case, the evolution equations for the scattering data themselves are linear and moreover solvable in the finite form.     

Decrease orders of theL p -moduli of continuity (0<p≦∞)

k — , 0<p, _k(p) — k- L _p ¹ ; >0, - , (0, ), (t)0 (t0), (t) ,t ^–(t) . _k(p), , 0<p1 _{k(p) (k– 1)p+1}, 1<p — _k , .     

Global bifurcation of fixed points and the Poincaré translation operator on manifolds

Let M be a differentiable manifold and [0, )×MM be a C¹ map satisfying the condition (0, p)=p for all pM. Among other results, we prove that when the degree (also called Hopf index or Euler characteristic) of the tangent vector field wMTM, given by w(p)=(/)(0, p), is well defined and nonzero, then the set (of nontrivial pairs) admits a connected subset whose closure is not compact and meets the slice {0}×M of [0, )×M. This extends known results regarding the existence of harmonic solutions of periodic ordinary differential equations on manifolds.     

A sufficient condition for near-optimal stochastic controls and its application to manufacturing systems

This paper presents an asymptotic analysis of control models governed by stochastic ordinary differential equations. A sufficient condition of near-optimal controls is given based on Ekeland's principle. It is shown that, under some concavity assumptions, the-maximum condition in terms of the Hamiltonian implies the -optimality. By applying this result to a general manufacturing system, the common practice of hierarchical controls employed in order to reduce the overall complexity of the system is justified on a rigorous basis. A near-optimal control for the operational level is constructed from a near-optimal control at the corporate level, and an asymptotic error bound is obtained. A stochastic extension of the classical HMMS model is treated as a specific example. The approach of this paper is different from those in the literature, and it allows us to handle some previously unsolved problems with nonlinear state equations as well as nonseparable cost functions.This work was partly supported by NSERC Grant A4619, URIF, and the Manufacturing Research Corporation of Ontario.     

Exact Anomalous Dimensions of Composite Operators in the Obukhov–Kraichnan Model

We consider two stochastic equations that describe the turbulent transfer of a passive scalar field (x)(t,x) and generalize the known Obukhov–Kraichnan model to the case of a possible compressibility and large-scale anisotropy. The pair correlation function of the field (x) is characterized by an infinite collection of anomalous indices, which have previously been found exactly using the zero-mode method. In the quantum field formulation, these indices are identified with the critical dimensions of an infinite family of tensor composite operators that are quadratic in the field (x), which allows obtaining exact values for the latter (the values not restricted to the -expansion) and then using them to find the corresponding renormalization constants. The identification of the correlation function indices with the composite-operator dimensions itself is supported by a direct calculation of the critical dimensions in the one-loop approximation.     

Ornstein–Uhlenbeck Path Integral and Its Application

Introducing a path integral for the Ornstein–Uhlenbeck process distorted by a potential V(x), we find out the T limit of the probability distributions of X[]:=1/T ₀ ^T V((t))dt for Ornstein–Uhlenbeck process (t), with appropriate values of the exponent that depend on V. The results are compared with those for the Wiener process.     

Periodic behavior of the stochastic Brusselator in the mean-field limit

Summary We prove a propagation of chaos result for the mean-field limit of a model for a trimolecular chemical reaction called Brusselator. Then we show that the pair of nonlinear (i.e. law-dependent) stochastic differential equations describing the evolution of the concentration of the molecules at a given site in the mean field limit has a solution with a periodic law (in t). Finally we identify the limit and establish a central limit theorem for the periodic law in the case where the noise tends to zero.Part of this work was performed while on leave at the Department of Mathematics and Statistics, Carleton University, Ottawa, Canada and supported by NSERC operating grants of M. Csörgö and D. Dawson