Large deviations principles for stochastic scalar conservation laws

• Published in: Probability theory and related fields Vol. 147; no. 3-4; pp. 607 - 648
• Main Author: Mariani, Mauro
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.07.2010; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Cauchy problems; Conservation laws; Deviation; Diffusion; Economics; Entropy; Exact sciences and technology; Finance; General topics; Insurance; Investigations; Limit theorems; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Noise; Operations Research/Decision Theory; Partial differential equations; Principles; Probability; Probability and statistics; Probability Theory and Stochastic Processes; Quantitative Finance; Sciences and techniques of general use; Statistics for Business; Studies; Theoretical; Viscosity; Conservation laws; 60F10; Stochastic PDE; Large deviations; Entropy functional; 60K35; 60H15; Probability theory; Entropy; Stochastic system; Partial differential equation; Large deviation; Functional; Conservation law; Young measure; Particle system; Vanishing viscosity; Large Deviations
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-009-0218-6

Large deviations principles for a family of scalar 1 + 1 dimensional conservative stochastic PDEs (viscous conservation laws) are investigated, in the limit of jointly vanishing noise and viscosity. A first large deviations principle is obtained in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. A second order large deviations principle is therefore investigated, however, this can be only partially proved. The second order rate functional provides a generalization for non-convex fluxes of the functional introduced by Jensen and Varadhan in a stochastic particles system setting.