TY - JOUR

T1 - Space-time invariant measures, entropy, and dimension for stochastic Ginzburg-Landau equations

AU - Rougemont, Jacques

PY - 2002

Y1 - 2002

N2 - We consider a randomly forced Ginzburg-Landau equation on an unbounded domain. The forcing is smooth and homogeneous in space and white noise in time. We prove existence and smoothness of solutions, existence of an invariant measure for the corresponding Markov process and we define the spatial densities of topological entropy, of measure-theoretic entropy, and of upper box-counting dimension. We prove inequalities relating these different quantities. The proof of existence of an invariant measure uses the compact embedding of some space of uniformly smooth functions into the space of locally square-integrable functions and a priori bounds on the semi-flow in these spaces. The bounds on the entropy follow from spatially localised estimates on the rate of divergence of nearby orbits and on the smoothing effect of the evolution. © Springer-Verlag 2002.

AB - We consider a randomly forced Ginzburg-Landau equation on an unbounded domain. The forcing is smooth and homogeneous in space and white noise in time. We prove existence and smoothness of solutions, existence of an invariant measure for the corresponding Markov process and we define the spatial densities of topological entropy, of measure-theoretic entropy, and of upper box-counting dimension. We prove inequalities relating these different quantities. The proof of existence of an invariant measure uses the compact embedding of some space of uniformly smooth functions into the space of locally square-integrable functions and a priori bounds on the semi-flow in these spaces. The bounds on the entropy follow from spatially localised estimates on the rate of divergence of nearby orbits and on the smoothing effect of the evolution. © Springer-Verlag 2002.

UR - http://www.scopus.com/inward/record.url?scp=0036101263&partnerID=8YFLogxK

U2 - 10.1007/s002200100586

DO - 10.1007/s002200100586

M3 - Article

SN - 1432-0916

VL - 225

SP - 423

EP - 448

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -