## Homogenization of periodic linear degenerate PDEs (2007)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Hairer07homogenizationof,

author = {Martin Hairer and Etienne Pardoux},

title = {Homogenization of periodic linear degenerate PDEs},

year = {2007}

}

### OpenURL

### Abstract

It is well-known under the name of ‘periodic homogenization ’ that, under a centering condition of the drift, a periodic diffusion process on R d converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set. As a consequence, it is no longer the case that the effective diffusivity matrix is necessarily non-degenerate. It turns out that, provided that some very weak regularity conditions are met, the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion. In particular, this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank. We also discuss the application of our results to the homogenization of a class of elliptic and parabolic PDEs. 1

### Citations

270 |
Stochastic flows and stochastic differential equation
- Kunita
- 1990
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- Jikov, Kozlov, et al.
- 1994
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The Malliavin calculus and related topics, Probability and its Applications (New
- Nualart
- 1995
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70 | Stochastic calculus of variations and hypoelliptic operators - MALLIAVIN - 1976 |

64 | Applications of the Malliavin calculus - KUSUOKA, STROOCK - 1987 |

24 |
Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients, J Funct Anal 167
- Pardoux
- 1999
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- Hairer, Mattingly
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On the Poisson equation and diffusion
- Pardoux, Veretennikov
- 2001
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Homogenization of degenerate elliptic-parabolic equations
- PARONETTO
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Homogenization for degenerate operators with periodical coefficients with respect to the heisenberg
- Biroli, Mosco, et al.
- 1996
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- ARCANGELIS, CASSANO
- 1992
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- BELLIEUD, BOUCHITTÉ
- 1998
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