Exponential mixing for stochastic PDEs: the non-additive case

By C. Odasso. 2005

56 citations recorded by the source corpus.

We establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two representative examples: 2D Navier–Stokes (NS) equations and Complex Ginzburg–Landau (CGL) equation with a locally Lipschitz noise. Due to the possible degeneracy of the noise, Doob theorem cannot be applied. Hence, a coupling method is used in the spirit of Kuksin and Shirikyan (J. Math. Pures Appl. 1:567–602, 2002) and Mattingly (Commun. Math. Phys. 230:421–462, 2002).Previous results require assumptions on the covariance of the noise which might seem restrictive and artificial. For instance, for NS and CGL, the covariance operator is supposed to be diagonal in the eigenbasis of the Laplacian and not depending on the high modes of the solutions. The method developed in the present paper gets rid of such assumptions and only requires that the range of the covariance operator contains the low modes.

Subfields: Control and Systems Engineering, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/e3acfb1b4785e845327abfe51b88360f730d77f6

How researchers see this paper

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In Control and Systems Engineering

1. Arnaud Debussche
2. Jerzy Zabczyk
3. Armen Shirikyan
4. Lihu Xu
5. Enrico Priola
6. Xicheng Zhang
7. Jonathan C. Mattingly
8. Feng-Yu Wang

In Finance

1. Arnaud Debussche
2. Jerzy Zabczyk
3. Enrico Priola
4. Armen Shirikyan
5. Lihu Xu
6. Franco Flandoli
7. Martin Hairer
8. Charles-Edouard Br'ehier

In Modeling and Simulation

1. Arnaud Debussche
2. Jerzy Zabczyk
3. David Applebaum
4. Markus Riedle
5. Fabian Hornung

Citations in the corpus

• Ergodicity Results for the Stochastic Navier–Stokes Equations: An Introduction (2013, background): “Then, these arguments were generalized in several directions and the theory was extended to many other equations (see for instance [23], [35], [37], [43], [41], [44], [45] , [46], [56], [57], [63]).”
• Exponential ergodicity and regularity for equations with Lévy noise (2011, background): “We refer the reader to [13,19,35] for discrete-time random perturbations, to [21,11,14,36,24] for a white noise, to [22] for a compound Poisson process, and to the book [15] for further references on this subject.”