Exponential convergence to the stationary measure and hyperbolicity of the minimisers for random Lagrangian Systems

Boritchev, Alexandre

  • Proceedings of Equadiff 14, Publisher: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing(Bratislava), page 117-126

Abstract

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We consider a class of 1d Lagrangian systems with random forcing in the spaceperiodic setting: φ t + φ x 2 / 2 = F ω , x ∈ S 1 = ℝ / ℤ . These systems have been studied since the 1990s by Khanin, Sinai and their collaborators [7, 9, 11, 12, 15]. Here we give an overview of their results and then we expose our recent proof of the exponential convergence to the stationary measure [6]. This is the first such result in a classical setting, i.e. in the dual-Lipschitz metric with respect to the Lebesgue space L p for finite p , partially answering the conjecture formulated in [11]. In the multidimensional setting, a more technically involved proof has been recently given by Iturriaga, Khanin and Zhang [13].

How to cite

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Boritchev, Alexandre. "Exponential convergence to the stationary measure and hyperbolicity of the minimisers for random Lagrangian Systems." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 117-126. <http://eudml.org/doc/294952>.

@inProceedings{Boritchev2017,
abstract = {We consider a class of 1d Lagrangian systems with random forcing in the spaceperiodic setting: \begin\{equation\} \nonumber \phi \_t+\phi \_x^2/2=F^\{\omega \},\ x \in S^1=\mathbb \{R\}/ \mathbb \{Z\}. \end\{equation\} These systems have been studied since the 1990s by Khanin, Sinai and their collaborators [7, 9, 11, 12, 15]. Here we give an overview of their results and then we expose our recent proof of the exponential convergence to the stationary measure [6]. This is the first such result in a classical setting, i.e. in the dual-Lipschitz metric with respect to the Lebesgue space $L_p$ for finite $p$, partially answering the conjecture formulated in [11]. In the multidimensional setting, a more technically involved proof has been recently given by Iturriaga, Khanin and Zhang [13].},
author = {Boritchev, Alexandre},
booktitle = {Proceedings of Equadiff 14},
keywords = {Lagrangian dynamics, Random dynamical systems, Invariant measure, Hyperbolicity},
location = {Bratislava},
pages = {117-126},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {Exponential convergence to the stationary measure and hyperbolicity of the minimisers for random Lagrangian Systems},
url = {http://eudml.org/doc/294952},
year = {2017},
}

TY - CLSWK
AU - Boritchev, Alexandre
TI - Exponential convergence to the stationary measure and hyperbolicity of the minimisers for random Lagrangian Systems
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 117
EP - 126
AB - We consider a class of 1d Lagrangian systems with random forcing in the spaceperiodic setting: \begin{equation} \nonumber \phi _t+\phi _x^2/2=F^{\omega },\ x \in S^1=\mathbb {R}/ \mathbb {Z}. \end{equation} These systems have been studied since the 1990s by Khanin, Sinai and their collaborators [7, 9, 11, 12, 15]. Here we give an overview of their results and then we expose our recent proof of the exponential convergence to the stationary measure [6]. This is the first such result in a classical setting, i.e. in the dual-Lipschitz metric with respect to the Lebesgue space $L_p$ for finite $p$, partially answering the conjecture formulated in [11]. In the multidimensional setting, a more technically involved proof has been recently given by Iturriaga, Khanin and Zhang [13].
KW - Lagrangian dynamics, Random dynamical systems, Invariant measure, Hyperbolicity
UR - http://eudml.org/doc/294952
ER -

References

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  2. Boritchev, A., Estimates for solutions of a low-viscosity kick-forced generalised Burgers equation, , Proceedings of the Royal Society of Edinburgh A, 143(2) (2013), pp. 253–268. MR3039811
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  6. Boritchev, A., Exponential convergence to the stationary measure for a class of 1D Lagrangian systems with random forcing, , accepted to Stochastic and Partial Differential Equations: Analysis and Computations. MR3768996
  7. Boritchev, A., Khanin, K., On the hyperbolicity of minimizers for 1D random Lagrangian systems, , Nonlinearity, 26(1) (2013), pp. 65–80. MR3001762
  8. Doering, C., Gibbon, J. D., Applied analysis of the Navier-Stokes equations, , Cambridge Texts in Applied Mathematics, Cambridge University Press, 1995. MR1325465
  9. E, Weinan, Khanin, K., Mazel, A., HASH(0x24dff00), Sinai, Ya., Invariant measures for Burgers equation with stochastic forcing, , Annals of Mathematics, 151 (2000), pp. 877–960. MR1779561
  10. Fathi, A., Weak KAM Theorem in Lagrangian Dynamics, , preliminary version, 2005. 
  11. Gomes, D., Iturriaga, R., Khanin, K., HASH(0x24e24c0), Padilla, P., Viscosity limit of stationary distributions for the random forced Burgers equation, , Moscow Mathematical Journal, 5 (2005), pp. 613–631. MR2241814
  12. Iturriaga, R., Khanin, K., Burgers turbulence and random Lagrangian systems, , Communications in Mathematical Physics, 232:3 (2003), pp. 377–428. MR1952472
  13. Iturriaga, R., Khanin, K., HASH(0x24e6f80), Zhang, K., Exponential convergence of solutions for random Hamilton-Jacobi equation, , Preprint, arxiv: 1703.10218, 2017. 
  14. Iturriaga, R., Sanchez-Morgado, H., Hyperbolicity and exponential convergence of the Lax-Oleinik semigroup, , Journal of Differential Equations, 246(5) (2009), pp. 1744–1753. MR2494686
  15. Khanin, K., Zhang, K., Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations, , Communications in Mathematical Physics, 355 (2017), pp. 803. MR3681391
  16. Sinai, Y., Two results concerning asymptotic behavior of solutions of the Burgers equation with force, , Journal of Statistical Physics, 64, 1991, pp. 1–12. MR1117645

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