Abstract
We discuss coupled KPZ (Kardar-Parisi-Zhang) equations. The motivation comes from the study of nonlinear fluctuating hydrodynamics, cf. [11, 12]. We first give a quick overview of results of Funaki and Hoshino [6], in particular, two approximating equations, trilinear condition (T) for coupling constants \(\varGamma \), invariant measures and global-in-time existence of solutions. Then, we study at heuristic level the role of the trilinear condition (T) in view of invariant measures and renormalizations for 4th order terms. Ertaş and Kardar [2] gave an example which does not satisfy (T) but has an invariant measure. We finally discuss the cross-diffusion case.
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References
Da Prato, G., Debussche, A.: Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196, 180–210 (2002)
Ertaş, D., Kardar, M.: Dynamic roughening of directed lines. Phys. Rev. Lett. 69, 929–932 (1992)
Ferrari, P.L., Sasamoto, T., Spohn, H.: Coupled Kardar-Parisi-Zhang equations in one dimension. J. Stat. Phys. 153, 377–399 (2013)
Funaki, T.: Infinitesimal invariance for the coupled KPZ equations. In: Memoriam Marc Yor – Séminaire de Probabilités XLVII. Lecture Notes in Mathematics, vol. 2137, pp. 37–47. Springer (2015)
Funaki, T.: Lectures on Random Interfaces. SpringerBriefs in Probability and Mathematical Statistics. Springer, Singapore (2016)
Funaki, T., Hoshino, M.: A coupled KPZ equation, its two types of approximations and existence of global solutions. J. Funct. Anal. 273, 1165–1204 (2017)
Funaki, T., Quastel, J.: KPZ equation, its renormalization and invariant measures. Stoch. PDE Anal. Comp. 3, 159–220 (2015)
Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi. 3, 1–75 (2015)
Hairer, M., Mattingly, J.: The strong Feller property for singular stochastic PDEs. Ann. Inst. Henri Poincaré Probab. Stat. 54, 1314–1340 (2018)
Kupiainen, A., Marcozzi, M.: Renormalization of generalized KPZ equation. J. Stat. Phys. 166, 876–902 (2017)
Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191–1227 (2014)
Spohn, H., Stoltz, G.: Nonlinear fluctuating hydrodynamics in one dimension: the case of two conserved fields. J. Stat. Phys. 160, 861–884 (2015)
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Funaki, T. (2019). Invariant Measures in Coupled KPZ Equations. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds) Stochastic Dynamics Out of Equilibrium. IHPStochDyn 2017. Springer Proceedings in Mathematics & Statistics, vol 282. Springer, Cham. https://doi.org/10.1007/978-3-030-15096-9_20
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DOI: https://doi.org/10.1007/978-3-030-15096-9_20
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