Homogenization of random parabolic operator with large potential

Campillo, Fabien; Kleptsyna, Marina; Piatnitski, Andrey

doi:10.1016/s0304-4149(00)00095-8

Cited by 28 publications

(28 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Case p = 1 is technically more demanding since then the perturbation techniques used for p < 1 are no longer valid. However, a similar result has been proven for p = 1 and spatially periodic drift driven in time by a finite dimensional diffusion in a recent paper by Campillo et al [5]. It has been shown there that the solutions of (1.1) approximate in the weak sense a solution of a deterministic, constant coefficient, heat equation.…”

Section: Introductionsupporting

confidence: 71%

“…It has been shown there that the solutions of (1.1) approximate in the weak sense a solution of a deterministic, constant coefficient, heat equation. The proof presented in [5] is a version of a martingale argument that appears often in the homogenization theory. It relies on the construction of a random field, called a corrector, having a property that for a certain perturbation ε (t), t ≥ 0, which is a linear functional of the corrector and vanishes, as ε ↓ 0, one can write…”

Section: Introductionmentioning

confidence: 99%

“…Here ·, · is the standard L 2 (R d ) scalar product and g is an appropriate test function. The fact that the randomness of the drift is generated by a finite dimensional diffusion is a crucial ingredient of the construction of the corrector and the aforementioned decomposition of (1.4) presented in [5]. Our paper deals with a genuine infinite dimensional velocity field and therefore we cannot directly apply the method of ibid.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Diffusion approximation for the convection-diffusion equation with random drift

Komorowski¹

2001

Probab Theory Relat Fields

View full text Add to dashboard Cite

We consider the asymptotic behavior of the solutions of scaled convectiondiffusion equations ∂ t u ε (t, x) = κ x u ε (t, x) + 1/εV(t/ε 2 , x/ε) · ∇ x u ε (t, x) with the initial condition u ε (0, x) = u 0 (x) as the parameter ε ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈ R d is a d-dimensional, stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u ε (t, ·), t ≥ 0 in an appropriate functional space converge weakly, as ε ↓ 0, to a δ-type measure concentrated on a solution of a certain constant coefficient heat equation.

show abstract

Section: Introductionsupporting

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Diffusion approximation for the convection-diffusion equation with random drift

Komorowski¹

2001

Probab Theory Relat Fields

View full text Add to dashboard Cite

show abstract

“…The case when V is time-independent was considered in [1,8]. The articles [4,5] considered a situation where V is a stationary process as a function of time, but periodic in space.…”

Section: Introductionmentioning

confidence: 99%

Random homogenisation of a highly oscillatory singular potential

Hairer

Pardoux

Piatnitski

2013

Stoch PDE: Anal Comp

Self Cite

View full text Add to dashboard Cite

In this article, we consider the problem of homogenising the linear heat equation perturbed by a rapidly oscillating random potential. We consider the situation where the space-time scaling of the potential's oscillations is not given by the diffusion scaling that leaves the heat equation invariant. Instead, we treat the case where spatial oscillations are much faster than temporal oscillations. Under suitable scaling of the amplitude of the potential, we prove convergence to a deterministic heat equation with constant potential, thus completing the results previously obtained in Pardoux and Piatnitski (Ann Probab, 40(3):1316-1356, 2012.

show abstract

“…In this case, the homogenization procedure for elliptic and parabolic problems is essentially the same. The averaging problems for parabolic equations with rapidly oscillating coefficients both in space and time variables, have been considered in [3,6,10] or [13] for example. It was shown in [3,10] and [13], that for divergence form operators, "usual" homogenization results hold.…”

Section: Introductionmentioning

confidence: 99%

Homogenization in perforated domains with rapidly pulsing perforations

Cioranescu

Piatnitski

2003

ESAIM: COCV

View full text Add to dashboard Cite

Abstract. The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period ε of the holes goes to zero. Since standard conservation laws do not hold in this model, a first difficulty is to get a priori estimates of the solutions. We obtain them in a weighted space where the weight is the principal eigenfunction of an "adjoint" periodic time-dependent eigenvalue problem. This problem is not a classical one, and its investigation is an important part of this work. Then, by using the multiple scale method, we construct the leading terms of a formal expansion (with respect to ε) of the solution and give the limit "homogenized" problem. An interesting peculiarity of the model is that, depending on the geometry of the holes, a large convection term may appear in the limit equation.Mathematics Subject Classification. 35B27, 74Q10, 76M50.

show abstract

Homogenization of random parabolic operator with large potential

Cited by 28 publications

References 10 publications

Diffusion approximation for the convection-diffusion equation with random drift

Diffusion approximation for the convection-diffusion equation with random drift

Random homogenisation of a highly oscillatory singular potential

Homogenization in perforated domains with rapidly pulsing perforations

Contact Info

Product

Resources

About