Convergence In L2 Research Articles

Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions

We study the Taylor expansion for the solution of a differential equation driven by a multi-dimensional Hölder path with exponent $H> 1/2$. We derive a convergence criterion that enables us to write the solution as an infinite sum of iterated integrals on a non empty interval. We apply our deterministic results to stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H > 1/2$. We also study the convergence in L2 of the stochastic Taylor expansion by using L2 estimates of iterated integrals and Borel-Cantelli type arguments.

Remarks on asymptotic behavior of weighted quadratic variation of subfractional Brownian motion

The present note is devoted to prove, by means of Malliavin calculus, the convergence in L2 of some properly renormalized weighted quadratic variation of sub-fractional Brownian motion SH with parameter H<14.

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron–Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds and demonstrate L1 convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.

A remark on a condition raised by Tikhonov: An example in L 1 convergence of fourier series

By constructing an example, the present note will show that the condition \( \overline {GM} \) raised by Tikhonov [2] to study L∞ convergence case cannot keep the classical theorem working in L1 convergence case.

Well‐posed problem of a pollutant model of the Kazhikhov–Smagulov type

AbstractIn this paper, we consider a well‐posed problem of a pollutant model of the Kazhikhov–Smagulov type, which is derived by Bresch et al. (J. Math. Fluid Mech. 2007; 9:377–397). For proper smooth data, existence and uniqueness are stated on a time interval, which become independent of the diffusion coefficient λ when λ goes to zero. A blow‐up criterion involving the norm of the gradient of the velocity in L1(0, T; L∞) is also proved. Besides, we show that if the density‐dependent Euler system has a smooth solution on a given time interval [0, T0] , then the pollutant model of the Kazhikhov–Smagulov type with the same data and small diffusion coefficient has a smooth solution on [0, T0] . The diffusion solution tends to the Euler solution when the diffusion coefficient λ goes to zero. The rate of the convergence in L2 is of order λ . Copyright © 2008 John Wiley &amp; Sons, Ltd.

L1-convergence of smoothing densities in non-parametric state space models

This paper addresses the problem of reconstructing partially observed stochastic processes. The L1 convergence of the filtering and smoothing densities in state space models is studied, when the transition and emission densities are estimated using non parametric kernel estimates. An application to real data is proposed, in which a wave time series is forecasted given a wind time series.

Error estimates for the numerical approximation of Neumann control problems

We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193-219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L2(Γ). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45-61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided.

Strong Traces for Solutions to Scalar Conservation Laws with General Flux

In this paper we consider bounded weak solutions u of scalar conservation laws, not necessarily of class BV, defined in a subset \(\Omega\subset{\mathbb{R}}^+\times{\mathbb{R}}\) . We define a strong notion of trace at the boundary of \(\Omega\) reached by L1 convergence for a large class of functionals of u, G(u). The functionals G depend on the flux function of the conservation law and on the boundary of \(\Omega\). The result holds for a general flux function and a general subset.

On L1 convergence of Fourier series of complex valued functions

In the present paper we give a brief review of L1 -convergence of trigonometric series. Previous known results in this direction are improved and generalized by establishing a new condition.

On the quasi-nodal map for the Sturm–Liouville problem

We show that the space of Sturm–Liouville operators characterized by H = (q, α, β) ∈ L1 (0, 1) × [0, π)2 such that is homeomorphic to the partition set of the space of all admissible sequences which form sequences that converge to q, α, and β individually. This space, Γ, of quasi-nodal sequences is a superset of, and is more natural than, the space of asymptotically nodal sequences defined in Law and Tsay (On the well-posedness of the inverse nodal problem. Inv. Probl.17 (2001), 1493–1512). The definition of Γ relies on the L1 convergence of the reconstruction formula for q by the exactly nodal sequence.

On Hessian Measures for Non-Commuting Vector Fields

Previous results on Hessian measures by Trudinger and Wang are extended to the subelliptic case. Specifically we prove the weak continuity of the 2-Hessian operator, with respect to local L1 convergence, for a system of m vector fields of step 2 and derive gradient estimates for the corresponding k-convex functions, k=1,2....m, for arbitrary step.

Finite element approximation for degenerate parabolic equations. an application of nonlinear semigroup theory

Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L1 contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in L1 and L∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L1 convergence without any convergence rate. The validity of theoretical results is confirmed by numerical examples.

Error estimates for mixed finite element approximations to a linear Stefan problem

We introduce a semidiscrete mixed finite element approximation for the single-phase linear Stefan problem and show the unique existence of the approximation. And the optimal rate of convergence in L2 and H1 norms are derived.

Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

The convergence in L2(T) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with limn→∞1n∑n−1k=0|ak|=0, {an}n⩾0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|ϕn|2dσ}n⩾0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szegő class which consists of measures with ∑∞n=0(1−|an|2)1/2<∞ and describe it in terms of Hessenberg matrices.

Boundary Layers Associated with Incompressible Navier–Stokes Equations: The Noncharacteristic Boundary Case

The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier–Stokes equations in the case where the boundaries are characteristic (no-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus noncharacteristic. The form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e−Uz/ε where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and ε is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J.35, No. 2, 209–230) where the convergence in L2 of the solutions of the Navier–Stokes equations to that of the Euler equations at vanishing viscosity was established. In the two dimensional case we are able to derive the physically relevant uniform in space (L∞ norm) estimates of the boundary layer. The uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. To the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier–Stokes equations for incompressible fluids.

Weak Convergence of a Sequence of Semimartingales to a Diffusion with Discontinuous Drift and Diffusion Coefficients

This paper presents a set of sufficient conditions for a sequence of semimartingales to converge weakly to a solution of a stochastic differential equation (SDE) with discontinuous drift and diffusion coefficients. This result is closely related to a well-known weak-convergence theorem due to Liptser and Shiryayev (see [27]) which proves the weak convergence to a solution of a SDE with continuous drift and diffusion coefficients in the Skorokhod–Lindvall J1-topology. The goal of this paper is to obtain a stronger result in order to solve outstanding problems in the area of large-scale queueing networks – in which the weak convergence of normalized queueing length is a solution of a SDE with discontinuous coefficients. To do this we need to make the stronger assumptions: (1) replacing the convergence in probability of the triplets of a sequence of semimartingales in the original Liptser and Shiryayev's theorem by stronger convergence in L2, (2) assuming the diffusion coefficient is coercive, and (3) assuming the discontinuity sets of the coefficients of the limit diffusion processs are of Lebesgue measure zero.

APPROXIMATION BY THE FINITE VOLUME METHOD OF AN ELLIPTIC-PARABOLIC EQUATION ARISING IN ENVIRONMENTAL STUDIES

We prove the convergence of a finite volume scheme for the Richards equation β(p)t- div (Λ(β(p))(∇p-ρg))=0 together with a Dirichlet boundary condition and an initial condition in a bounded domain Ω ×(0, T). We consider the hydraulic charge [Formula: see text] as the main unknown function so that no upwinding is necessary. The convergence proof is based on the strong convergence in L2 of the water saturation β(p), which one obtains by estimating differences of space and time translates and applying Kolmogorov's theorem. This implies the convergence in L2 of the approximate water mobility towards Λ(β(p)) as the time and mesh steps tend to 0, which in turn implies the convergence of the approximate pressure to a weak solution p of the continuous problem.

Strong Traces for Solutions of Multidimensional Scalar Conservation Laws

In this paper we consider multidimensional scalar conservation laws without BV estimates defined in a subset Ω⊂ℝ+×ℝ d . We show that, with a non-degeneracy hypothesis on the flux, we can define a strong notion of trace at the boundary of Ω reached by L 1 convergence.

Numerical approximation of an elliptic-parabolic equation arising in environment

We prove the convergence of a finite volume scheme for the Richards equation β(p)t-div(ź(β(p)) (źp-źg) =0, together with a Dirichlet boundary condition and an initial condition, in a bounded domain. We consider the hydraulic charge u=-z as the main unknown function so that no upwinding is necessary. The convergence proof is based on the strong convergence in L2 of the water saturation β(p), which one obtains by estimating differences of space and time translates and applying Kolmogorov's theorem.

A Korovkin Theorem for Abstract Lebesgue Spaces

Wulbert and Meir have each obtained a Korovkin theorem for weak convergence of operators on an L1 space. Here we prove a result which includes both of these theorems and which provides a general setting for weak Korovkin type L1 convergence of operators which are not assumed positive.