On the Poisson Equation and Diffusion Approximation. I

Pardoux, Étienne; Veretennikov, A. Yu.

doi:10.1214/aop/1015345596

Cited by 267 publications

(319 citation statements)

References 11 publications

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“…Such a solution Ψ must exist again by the assumptions on the coefficients (see, for instance, Pardoux & Veretennikov, 2001). So applying Itô formula to…”

Section: Proofmentioning

confidence: 99%

On Jumps Stochastic Evolution Equations With Application of Homogenization and Large Deviations

Manga¹,

Coulibaly²,

Diédhiou³

2019

JMR

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We consider a class of jumps and diffusion stochastic differential equations which are perturbed by to two parameters:  ε (viscosity parameter) and δ (homogenization parameter) both tending to zero. We analyse the problem taking into account the combinatorial effects of the two parameters  ε and δ . We prove a Large Deviations Principle estimate for jumps stochastic evolution equation in case that homogenization dominates.

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“…Such a solution Ψ must exist again by the assumptions on the coefficients (see, for instance, Pardoux & Veretennikov, 2001). So applying Itô formula to…”

Section: Proofmentioning

confidence: 99%

On Jumps Stochastic Evolution Equations With Application of Homogenization and Large Deviations

Manga¹,

Coulibaly²,

Diédhiou³

2019

JMR

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“…Upon obtaining appropriate a-priori estimates for this Poisson equation, one can reduce the averaging principle or normal deviations to the analysis of an Itô's formula. Since we are working in the case when fast motion Y ε,η (t) is an OU process, when applying the corrector method, we are mostly close to the set-up of [23] (see also [22], [24], [7]). Our analysis of the normal deviations will be following the corrector method and based on a-priori bounds provided in [23].…”

Section: 2mentioning

confidence: 99%

“…In the paper [11], the authors considered the case when diffusion matrix is a scalar multiple of the identity matrix, and the Poisson equation there corresponds to hypo-elliptic diffusions. Our analysis relies more on estimates obtained in [23] (also see [22], [24]).…”

Section: 2mentioning

confidence: 99%

A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations

Hu¹,

Li²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We consider in this work a system of two stochastic differential equations named the perturbed compositional gradient flow. By introducing a separation of fast and slow scales of the two equations, we show that the limit of the slow motion is given by an averaged ordinary differential equation. We then demonstrate that the deviation of the slow motion from the averaged equation, after proper rescaling, converges to a stochastic process with Gaussian inputs. This indicates that the slow motion can be approximated in the weak sense by a standard perturbed gradient flow or the continuous-time stochastic gradient descent algorithm that solves the optimization problem for a composition of two functions. As an application, the perturbed compositional gradient flow corresponds to the diffusion limit of the Stochastic Composite Gradient Descent (SCGD) algorithm for minimizing a composition of two expected-value functions in the optimization literatures. For the strongly convex case, such an analysis implies that the SCGD algorithm has the same convergence time asymptotic as the classical stochastic gradient descent algorithm. Thus it validates, at the level of continuous approximation, the effectiveness of using the SCGD algorithm in the strongly convex case.2010 Mathematics Subject Classification. 34C29, 60J60, 62L20, 90C30.

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“…The authors derived similar results by using the Mori-Zwanzig projection decomposition method for stochastic climate models (see [14]). With the auxiliary Poisson equation, Pardoux proved the limit theorem in a series of articles (refer to [28], [29] and [30]). In [22] and [24], Majda used multi scale methods to analyze the atmospheric ocean model.…”

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confidence: 99%

Homogenization for stochastic reaction-diffusion equations with singular perturbation term

Shi¹,

Gao²

2022

DCDS-B

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The main purpose of this paper is to study the homogenization problem of stochastic reaction-diffusion equations with singular perturbation term. The difficulty in studying such problems is how to get the uniform estimates of the equations under the influence of the singularity term. Firstly, we use the properties of the elliptic equation corresponding to the generator to eliminate the influence of singular terms and obtain the uniform estimates of the slow equation and thus, get the tightness. Finally, we prove that under appropriate assumptions, the slow equation converges to a homogenization equation in law.

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On the Poisson Equation and Diffusion Approximation. I

Cited by 267 publications

References 11 publications

On Jumps Stochastic Evolution Equations With Application of Homogenization and Large Deviations

On Jumps Stochastic Evolution Equations With Application of Homogenization and Large Deviations

A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations

Homogenization for stochastic reaction-diffusion equations with singular perturbation term

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