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## ON HOMOGENIZATION OF A DIFFUSION PERTURBED BY A PERIODIC REFLECTION INVARIANT VECTOR FIELD (2006)

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316 |
Homogenization of differential operators and integral functionals,”
- Zhikov, Kozlov, et al.
- 1994
(Show Context)
Citation Context ...2]. Lawler [14] and Anshelevich et al [1] proved homogenization for a discrete version. See the books of Bolthausen-Sznitman [2] for an account of the theory in a discrete setting and of Zhikov et al =-=[21]-=- for the continuous case.sHOMOGENIZATION OF A DIFFUSION PERTURBED BY A VECTOR FIELD 3 In this paper we shall be concerned with a discrete version of the homogenization problem described by (1.1), (1.2... |

196 |
Boundary value problems with rapidly oscillating random coefficients.
- Papanicolaou, Varadhan
- 1979
(Show Context)
Citation Context ...these cases is therefore considerably simpler than for the problem (1.1), (1.2). The first proofs of homogenization for divergence form equations were obtained by Kozlov [12] and PapanicolaouVaradhan =-=[15]-=- in the continuous case. Künneman [13] proved a corresponding result for the discrete case. For non-divergence form equations with zero drift the first proofs in the continuous case were given by Papa... |

103 |
The limiting behavior of a one-dimensional random walk in a random medium. Theory Prob.
- Sinaı
- 1982
(Show Context)
Citation Context ...re appears to have only been studied in the case where Ω is an infinite space for which the variables b(τx ·), x ∈ R, are uncorrelated on a scale larger than O(1). The problem was introduced by Sinai =-=[17]-=- in a discrete setting. He proved that in dimension d = 1 a scaling limit of the random walk corresponding to a finite difference approximation to (1.1) exists with probability 1 in Ω. The limiting pr... |

52 |
The limit distribution of Sinai’s random walk in random environment, Phys
- Kesten
- 1986
(Show Context)
Citation Context ...caling limit of the random walk corresponding to a finite difference approximation to (1.1) exists with probability 1 in Ω. The limiting process is strongly subdiffusive. In a subsequent paper Kesten =-=[11]-=- obtained an explicit formula for the distribution of the scaling limit. For dimension d ≥ 3 Fisher [10] and Derrida-Lück [8] predicted that a homogenized limit exists as in (1.2) with 0 < q(b) < ∞. T... |

51 | Slowdown estimates and central limit theorem for random walks in random environment. - Sznitman - 2000 |

44 |
Random walks in asymmetric random environments
- Bricmont, Kupiainen
- 1991
(Show Context)
Citation Context ...ing limit. For dimension d ≥ 3 Fisher [10] and Derrida-Lück [8] predicted that a homogenized limit exists as in (1.2) with 0 < q(b) < ∞. This was proved for sufficiently small b by Bricmont-Kupiainen =-=[3]-=- and Sznitman-Zeitouni [20] using a very difficult induction argument. A formal perturbation expansion for q(b) was obtained in [4, 5] where it was shown that each term of the expansion is finite if d... |

39 |
Weak convergence of random walk in random environments
- Lawler
- 1982
(Show Context)
Citation Context ...g result for the discrete case. For non-divergence form equations with zero drift the first proofs in the continuous case were given by Papanicolaou-Varadhan [16] and Zhikov-Sirazhudinov [22]. Lawler =-=[14]-=- and Anshelevich et al [1] proved homogenization for a discrete version. See the books of Bolthausen-Sznitman [2] for an account of the theory in a discrete setting and of Zhikov et al [21] for the co... |

24 | An Invariance principle for Isotropic Diffusions in Random Environment.
- Sznitman, Zeitouni
- 2006
(Show Context)
Citation Context ... ≥ 3 Fisher [10] and Derrida-Lück [8] predicted that a homogenized limit exists as in (1.2) with 0 < q(b) < ∞. This was proved for sufficiently small b by Bricmont-Kupiainen [3] and Sznitman-Zeitouni =-=[20]-=- using a very difficult induction argument. A formal perturbation expansion for q(b) was obtained in [4, 5] where it was shown that each term of the expansion is finite if d ≥ 3. One does not expect t... |

23 |
The diffusion limit for reversible jump processes on Z d with ergodic random bond conductivities
- Künnemann
(Show Context)
Citation Context ...simpler than for the problem (1.1), (1.2). The first proofs of homogenization for divergence form equations were obtained by Kozlov [12] and PapanicolaouVaradhan [15] in the continuous case. Künneman =-=[13]-=- proved a corresponding result for the discrete case. For non-divergence form equations with zero drift the first proofs in the continuous case were given by Papanicolaou-Varadhan [16] and Zhikov-Sira... |

20 | Slowdown and neutral pockets for a random walk in random environment - Sznitman |

16 |
Diffusions with random coefficients. In Statistics and probability: essays in honor of
- Papanicolaou, Varadhan
- 1982
(Show Context)
Citation Context ... case. Künneman [13] proved a corresponding result for the discrete case. For non-divergence form equations with zero drift the first proofs in the continuous case were given by Papanicolaou-Varadhan =-=[16]-=- and Zhikov-Sirazhudinov [22]. Lawler [14] and Anshelevich et al [1] proved homogenization for a discrete version. See the books of Bolthausen-Sznitman [2] for an account of the theory in a discrete s... |

16 | Ten lectures on random media. - Bolthausen, Sznitman - 2002 |

10 |
lectures on random media
- Bolthausen, Sznitman
- 2002
(Show Context)
Citation Context ...us case were given by Papanicolaou-Varadhan [16] and Zhikov-Sirazhudinov [22]. Lawler [14] and Anshelevich et al [1] proved homogenization for a discrete version. See the books of Bolthausen-Sznitman =-=[2]-=- for an account of the theory in a discrete setting and of Zhikov et al [21] for the continuous case.sHOMOGENIZATION OF A DIFFUSION PERTURBED BY A VECTOR FIELD 3 In this paper we shall be concerned wi... |

9 |
Convection-Enhanced Diffusion for Random Flows,”
- Fannjiang, Papanicolaou
- 1997
(Show Context)
Citation Context ...ctor fields. The homogenized limit of diffusion perturbed by a divergence free vector field necessarily yields an effective diffusion constant which is larger than the constant for the pure diffusion =-=[9]-=-. The homogenization problem considered here appears to have only been studied in the case where Ω is an infinite space for which the variables b(τx ·), x ∈ R, are uncorrelated on a scale larger than ... |

7 |
Symmetric random walks in random environments
- Sinai
- 1982
(Show Context)
Citation Context ...ase. For non-divergence form equations with zero drift the first proofs in the continuous case were given by Papanicolaou-Varadhan [16] and Zhikov-Sirazhudinov [22]. Lawler [14] and Anshelevich et al =-=[1]-=- proved homogenization for a discrete version. See the books of Bolthausen-Sznitman [2] for an account of the theory in a discrete setting and of Zhikov et al [21] for the continuous case.sHOMOGENIZAT... |

6 |
Random walks in random environment
- Fisher
- 1984
(Show Context)
Citation Context ...probability 1 in Ω. The limiting process is strongly subdiffusive. In a subsequent paper Kesten [11] obtained an explicit formula for the distribution of the scaling limit. For dimension d ≥ 3 Fisher =-=[10]-=- and Derrida-Lück [8] predicted that a homogenized limit exists as in (1.2) with 0 < q(b) < ∞. This was proved for sufficiently small b by Bricmont-Kupiainen [3] and Sznitman-Zeitouni [20] using a ver... |

6 |
Averaging of random structures, Dokl
- Kozlov
- 1978
(Show Context)
Citation Context ...he proof of homogenization in these cases is therefore considerably simpler than for the problem (1.1), (1.2). The first proofs of homogenization for divergence form equations were obtained by Kozlov =-=[12]-=- and PapanicolaouVaradhan [15] in the continuous case. Künneman [13] proved a corresponding result for the discrete case. For non-divergence form equations with zero drift the first proofs in the cont... |

5 |
Diffusion on a random lattice: Weak-disorder expansion in arbitrary dimension, Phys
- Derrida, Luck
- 1983
(Show Context)
Citation Context ...e limiting process is strongly subdiffusive. In a subsequent paper Kesten [11] obtained an explicit formula for the distribution of the scaling limit. For dimension d ≥ 3 Fisher [10] and Derrida-Lück =-=[8]-=- predicted that a homogenized limit exists as in (1.2) with 0 < q(b) < ∞. This was proved for sufficiently small b by Bricmont-Kupiainen [3] and Sznitman-Zeitouni [20] using a very difficult induction... |

3 | Homogenization of random walk in asymmetric random environment
- Conlon
(Show Context)
Citation Context ... < ∞. This was proved for sufficiently small b by Bricmont-Kupiainen [3] and Sznitman-Zeitouni [20] using a very difficult induction argument. A formal perturbation expansion for q(b) was obtained in =-=[4, 5]-=- where it was shown that each term of the expansion is finite if d ≥ 3. One does not expect the series to converge however. For d = 1, 2 there are individual terms in the perturbation expansion which ... |

3 |
On homogenisation of elliptic equations with random coefficients
- Conlon, Naddaf
(Show Context)
Citation Context ...ce Laplacian on functions with domain Ωd−1. The normalization of ψ is chosen so that q(0) = 1/2d. The general formula (1.7) is proven in §4. 2. Proof of Theorem 1.1 We follow the method introduced in =-=[7]-=- to obtain homogenized limits. Thus in (1.5) we put uε(x, ω) = vε(x, τ x/ε ω) whence (1.5) becomes (2.1) vε(x, ω) − 2d� i=1 1 2d [vε(x + εei, τei ω) + vε(x − εei, τ−e i ω)] − b(ω) [vε(x + εe1, τe1 ω) ... |

2 |
G compactness of a class of second-order nondivergence elliptic operators
- Zhikov, Sirazhudinov
- 1981
(Show Context)
Citation Context ... corresponding result for the discrete case. For non-divergence form equations with zero drift the first proofs in the continuous case were given by Papanicolaou-Varadhan [16] and Zhikov-Sirazhudinov =-=[22]-=-. Lawler [14] and Anshelevich et al [1] proved homogenization for a discrete version. See the books of Bolthausen-Sznitman [2] for an account of the theory in a discrete setting and of Zhikov et al [2... |

1 | Perturbation theory for random walk in asymmetric environment
- Conlon
(Show Context)
Citation Context ... < ∞. This was proved for sufficiently small b by Bricmont-Kupiainen [3] and Sznitman-Zeitouni [20] using a very difficult induction argument. A formal perturbation expansion for q(b) was obtained in =-=[4, 5]-=- where it was shown that each term of the expansion is finite if d ≥ 3. One does not expect the series to converge however. For d = 1, 2 there are individual terms in the perturbation expansion which ... |

1 |
On homogenization of non-divergence form partial difference equations, Electron
- Conlon, Pulizzotto
(Show Context)
Citation Context ...atisfies the homogenous equation (2.48) with the boundary conditions (2.53). The result follows then from (2.61). � Proof of Theorem 1.1. The proof proceeds identically to the proof of Theorem 1.1 of =-=[6]-=-, on using lemmas 2.1-2.4. � Finally we wish to show that Theorem 1.2 holds to leading order in perturbation theory.sHOMOGENIZATION OF A DIFFUSION PERTURBED BY A VECTOR FIELD 15 Theorem 2.1. There exi... |

1 | Quantum mechanics. A modern and concise introductory course - Bes |