Invariant measures for Burgers equation with stochastic forcing

By E. Weinan, Kostya Khanin, A. Mazel, Y. Sinai. 2000

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In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some non-degeneracy conditions on the forcing, that almost surely there is a unique main shock and a unique global minimizer for the stationary solutions. Furthermore the global minimizer is a hyperbolic trajectory of the underlying system of characteristics.

Subfields: Finance, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7296ad3923d7cfce012acc149b07c63daa6855f7

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Citations in the corpus

Viscosity Limit of Stationary Distributions for the Random Forced Burgers Equation (2005, methodology): “Existence and Uniqueness of Solutions in the Case of Positive Viscosity In this section we follow closely the arguments in [6] which are originally inspired by [1] and [4].”
Burgers Turbulence (2007, methodology): “The relations between the Burgers equation with a time-periodic forcing and Aubry–Mather theory were discussed for the first time in [70] and in [38].”
Stochastic non-isotropic degenerate parabolic–hyperbolic equations (2016, methodology): “udied SPDE of the type (1.10) additionally including a second order quasilinear term. The long-time behavior of solutions to (1.10) with additive noise has been analyzed by E, Khanin, Mazel and Sinai [14] and Debussche and Vovelle [13]. STOCHASTIC PARABOLIC-HYPERBOLIC EQUATIONS 3 Organization of the”
Random periodic processes, periodic measures and ergodicity (2014, background): “ONG FENG AND HUAIZHONG ZHAO and invariant measures has lied in the central place of the study of stochastic partial diﬀerential equations and ergodic theory in the last few decades ([5],[6],[13],[15],[17],[25],[29],[33], [34],[36],[38],[47],[48],[53], to name but a few). The ﬁrst part of the deﬁniti”
Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, methodology): “], Burgers [9], Ryan [50] and Sinai [52]. For stochastic scalar conservation laws driven by additive noise, also including boundary value problems, we refer to the works by E, Khanin, Mazel and Sinai [22], Kim [37], Nakazawa [45], Saussereau and Stoica [51], Vallet and Wittbold [53] and the referenc”
Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems (2013, background): “For the regularized problem, several properties, like well-posedness of the Cauchy problem and existence of invariant measures can be proved ([40] and [14]).”
Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients (2006, background): “In particular, for the existence of stationary solutions for SPDEs, results are only known in very few cases ([7], [14], [21], [29], [30]).”
Degenerate parabolic stochastic partial differential equations (2013, background): “Such a term can be used for instance to account for numerical and empirical uncertainties and therefore stochastic conservation laws has been recently of growing interest, see [6], [16], [20], [36], [44], [70], [76], [78].”
Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding (2012, background): “For results about SPDEs, see Sinai [28], [29], Mattingly [21], E, Khanin, Mazel and Sinai [11], Caraballo, Kloeden and Schmalfuss [3], Liu and Zhao [20], Zhang and Zhao [32], [33], Duan, Lu and Schmalfuss [9], [10], Mohammed, Zhang and Zhao [22], Lian and Lu [19], though there are still many problem”
On Nonlinear Stochastic Balance Laws (2011, methodology): “For other related results, we refer to Sinai [16] and E-Khanin-MazelSinai [6] for the existence, uniqueness, and weak convergence of invariant measures for the one-dimensional Burgers equation with stochastic forcing which is periodic in x, as well as the structure and regularity properties of the s”