Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach

Benchérif-Madani, Abdellatif; Pardoux, Étienne

doi:10.1051/ps:2007026

Cited by 9 publications

(5 citation statements)

References 10 publications

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“…For every n ∈ N * , the sequence (M ε, n , N ε, n , A ε, n , L ε, n ) ε>0 is tight on the space (C ([0, t], R)) 4 endowed with the topology of uniform convergence.…”

Section: A Sequence Of Auxiliary Processes Tightness and Convergencementioning

confidence: 99%

Averaging for BSDEs with null recurrent fast component. Application to homogenization in a non periodic media

Bahlali

Elouaflin²,

Pardoux

2017

Stochastic Processes and their Applications

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International audienceWe establish an averaging principle for a family of solutions (Xε,Yε) := (X1,ε, X2,ε, Yε) of asystem of decoupled forward backward stochastic differential equations (SDE-BSDE for short) with a nullrecurrent fast component X1,ε. In contrast to the classical periodic case, we can not rely on an invariantprobability and the slow forward component X2,ε cannot be approximated by a diffusion process. Onthe other hand, we assume that the coefficients admit a limit in a Cesa`ro sense. In such a case, the limitcoefficients may have discontinuity. We show that the triplet (X1,ε, X2,ε, Yε) converges in law to thesolution (X1, X2,Y) of a system of SDE–BSDE, where X := (X1, X2) is a Markov diffusion which isthe unique (in law) weak solution of the averaged forward component and Y is the unique solution to theaveraged backward component. This is done with a backward component whose generator depends on thevariable z. As application, we establish an homogenization result for semilinear PDEs when the coefficientscan be neither periodic nor ergodic. We show that the averaged BDSE is related to the averaged PDE via aprobabilistic representation of the (unique) Sobolev W 1,2(R+ × Rd )–solution of the limit PDEs. Our d +1,locapproach combines PDE methods and probabilistic arguments which are based on stability property and weak convergence of BSDEs in the S-topology

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“…For every n ∈ N * , the sequence (M ε, n , N ε, n , A ε, n , L ε, n ) ε>0 is tight on the space (C ([0, t], R)) 4 endowed with the topology of uniform convergence.…”

Section: A Sequence Of Auxiliary Processes Tightness and Convergencementioning

confidence: 99%

Averaging for BSDEs with null recurrent fast component. Application to homogenization in a non periodic media

Bahlali

Elouaflin²,

Pardoux

2017

Stochastic Processes and their Applications

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“…There is a huge literature that we cannot cite in details, but the interested reader can for instance see [1,5,4,11,18,15,16] and the references therein. Another aspect concerning homogenization of SDEs (stochastic differential equations) has also been studied by several authors (see for instance [20,13,6,25]). These problems are related to our problem when the SDE reduces to an ODE.…”

Section: Brief Review Of the Literaturementioning

confidence: 99%

“…Another aspect concerning homogenization of SDEs (stochastic differential equations) has also been studied by several authors (see for instance [20,13,6,25]). These problems are related to our problem when the SDE reduces to an ODE.…”

Section: Brief Review Of the Literaturementioning

confidence: 99%

On the Rate of Convergence in Periodic Homogenization of Scalar First-Order Ordinary Differential Equations

Ibrahim¹,

Monneau²

2010

SIAM J. Math. Anal.

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In this paper, we study the rate of convergence in periodic homogenization of scalar ordinary differential equations. We provide a quantitative error estimate between the solutions of a first-order ordinary differential equation with rapidly oscillating coefficients and the limiting homogenized solution. As an application of our result, we obtain an error estimate for the solution of some particular linear transport equations.

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“…Later the S topology was used in problems related to homogenization of stochastic differential equations (e.g. [1], [2], [5], [25], [30], [35], [32]), diffusion approximation of solutions to the Poisson equation ( [31]), stability of solutions to semilinear equations with Dirichlet operator ( [19]), martingale transport on the Skorokhod space ( [10]), the Skorokhod problem ( [23], [24], [28], [34]), econometrics ( [7]), control theory ( [3], [22]), linear models with heavy-tails ( [4]), continuity of semilinear Neumann-Dirichlet problems ( [27]), generalized Doob-Meyer decomposition ( [15]), modeling stochastic reaction networks ( [16]) and even in some considerations of more general character ( [8], [21]).…”

Section: Introductionmentioning

confidence: 99%

New characterizations of the $S$ topology on the Skorokhod space

Jakubowski¹

2018

Electron. Commun. Probab.

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The S topology on the Skorokhod space was introduced by the author in 1997 and since then it has proved to be a useful tool in several areas of the theory of stochastic processes. The paper brings complementary information on the S topology. It is shown that the convergence of sequences in the S topology admits a closed form description, exhibiting the locally convex character of the S topology. Morover, it is proved that the S topology is, up to some technicalities, finer than any linear topology which is coarser than Skorokhod's J 1 topology. The paper contains also definitions of extensions of the S topology to the Skorokhod space of functions defined on [0, +∞) and with multidimensional values.

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Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach

Cited by 9 publications

References 10 publications

Averaging for BSDEs with null recurrent fast component. Application to homogenization in a non periodic media

Averaging for BSDEs with null recurrent fast component. Application to homogenization in a non periodic media

On the Rate of Convergence in Periodic Homogenization of Scalar First-Order Ordinary Differential Equations

New characterizations of the $S$ topology on the Skorokhod space

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