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Homogenization of random parabolic operator with large potential

Author

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  • Campillo, Fabien
  • Kleptsyna, Marina
  • Piatnitski, Andrey

Abstract

We study the averaging problem for a divergence form random parabolic operators with a large potential and with coefficients rapidly oscillating both in space and time variables. We assume that the medium possesses the periodic microscopic structure while the dynamics of the system is random and, moreover, diffusive. A parameter [alpha] will represent the ratio between space and time microscopic length scales. A parameter [beta] will represent the effect of the potential term. The relation between [alpha] and [beta] is of great importance. In a trivial case the presence of the potential term will be "neglectable". If not, the problem will have a meaning if a balance between these two parameters is achieved, then the averaging results hold while the structure of the limit problem depends crucially on [alpha] (with three limit cases: one classical and two given under martingale problems form). These results show that the presence of stochastic dynamics might change essentially the limit behavior of solutions.

Suggested Citation

  • Campillo, Fabien & Kleptsyna, Marina & Piatnitski, Andrey, 2001. "Homogenization of random parabolic operator with large potential," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 57-85, May.
    • Handle: RePEc:eee:spapps:v:93:y:2001:i:1:p:57-85
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    References listed on IDEAS

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    1. Veretennikov, A. Yu., 1997. "On polynomial mixing bounds for stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 115-127, October.
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    1. Kleptsyna, M. & Piatnitski, A. & Popier, A., 2015. "Homogenization of random parabolic operators. Diffusion approximation," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1926-1944.
    2. Amirat, Youcef & Bodart, Olivier & Chechkin, Gregory A. & Piatnitski, Andrey L., 2011. "Boundary homogenization in domains with randomly oscillating boundary," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 1-23, January.

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