Divergence,convergence and moments of some integral functionals of diffusions

Summary In this paper we study some integral functionals of diffusions. We obtain criteria for their divergence and their convergence and we investigate the existence of their moments.     

Weak convergence of infinite-dimensional diffusions 1

Continuous dependence - in the sense of weak convergence of laws - of martingale solutions to stochastic partial differential equations on coefficients is studied, the results obtained being applicable to equations with rapidly oscillating coefficients. In the proofs, Gatarek's and Goldys’ recent approach to martingale solutions is substantially used     

Rates of convergence of diffusions with drifted Brownian potentials

We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.

     

On the convergence of policy iteration for controlled diffusions

The convergence of an approximation scheme known as policy iteration has been demonstrated for controlled diffusions by Fleming, Puterman, and Bismut. In this paper, we show that this approximation scheme is equivalent to the Newton-Kantorovich iteration for solving the optimality equation and exploit this equivalence to obtain a new proof of convergence. Estimates of the rate of convergence of this procedure are also obtained.This research was partially supported by NRC Grant No. A-3609.     

Boundedness and convergence of martingales in rearrangement-invariant function spaces

Let (W, F, P)(\Omega, \cal F, P) be a complete nonatomic probability space. We shall give a characterization of rearrangement-invariant spaces X over W\Omega with the property that every martingale f = (f_n)_{n \geqq 0}f = (f_n)_{n \geqq 0} bounded in X converges with respect to the norm topology of X. Using the results, we shall consider the summability of martingales by Toeplitz matrices.     

Pointwise convergence to the initial data for nonlocal dyadic diffusions

We solve the initial value problem for the diffusion induced by dyadic fractional derivative s in ?⁺. First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator. As a consequence we obtain the pointwise convergence to the initial data.     

Weak convergence of Markov chain sampling methods and annealing algorithms to diffusions

Simulated annealing algorithms have traditionally been developed and analyzed along two distinct lines: Metropolis-type Markov chain algorithms and Langevin-type Markov diffusion algorithms. Here, we analyze the dynamics of continuous state Markov chains which arise from a particular implementation of the Metropolis and heat-bath Markov chain sampling methods. It is shown that certain continuous-time interpolations of the Metropolis and heat-bath chains converge weakly to Langevin diffusions running at different time scales. This exposes a close and potentially useful relationship between the Markov chain and diffusion versions of simulated annealing.The research reported here has been supported by the Whirlpool Foundation, by the Air Force Office of Scientific Research under Contract 89-0276, and by the Army Research Office under Contract DAAL-03-86-K-0171 (Center for Intelligent Control Systems).     

Stochastic partial differential equations for some measure-valued diffusions

Summary We consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R ¹, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.     

Boundedness and convergence for the non-Liénard type differential equation

In this article, the author studies the boundedness and convergence for the {(x) = a(y) - f(x),(y) = b(y)β(x) - g(x) e(t),where a(y), b(y), f(x),g(x),β(x) are real continuous functions in y ∈ R or x ∈ R,β(x) ≥ 0 for all x and e(t) is a real continuous function on R = {t: t ≥ 0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.     

Boundedness of some multipliers on stratified Lie groups

Let ￡ be the sub-Laplacian on a stratified Lie group G, and let m be a function defined on [0,+∞). We give the boundedness of the multiplier operators m(￡) on Herz-type Hardy spaces on G.     

Boundedness and convergence to zero of solutions of a forced second-order nonlinear differential equation

Sufficient conditions for continuability, boundedness, and convergence to zero of solutions of (a(t)x′)′ + h(t, x, x′) + q(t) f(x) g(x′) = e(t, x, x′) are given.     

Measure-valued immigration diffusions and generalized ornstein-uhlenbeck diffusions

1.IntroductionLetAbeaboundedlinearoperatoronS(R')whichadmitsanon-positivedefiniteself-adjoillteXtensiononL'(R').AssumethatAgeneratesastronglycontinuoussellilgroupicofboundedlinearoperatorsonS(R').LetBbeaboundedlinearoperatoron).aamtheresultsof[l],wek...     

Boundedness theorems for some fourth order differential equations

Summary In this paper a new approach involving the use of two signum functions together with a suitably chosen Lyapunov function is employed to investigate the boundedness property of solutions of two special cases of(1.3). This approach makes for considerable reduction in the conditions imposed on f, g in an earlier paper[1]. Entrata in Redazione il 25 ottobre 1970.     

Boundedness of the extremal solution for some p-Laplacian problems

We investigate the regularity of extremal solutions to some p-Laplacian Dirichlet problems with strong nonlinearities. Under adequate assumptions we prove the smoothness of the extremal solutions for some classes of nonlinearities. Our results suggest that the extremal solution’s boundedness for some range of dimensions depends on the nonlinearity f.     

一类分数次积分算子在Herz型Hardy空间上的有界性

通过放宽尺寸条件,得到了一类具有分数次积分性质的次线性算子从Herz型Hardy空间到(弱)Herz型Hardy空间有界性的判定条件,以及端点处的弱型估计.