Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 12 (2007) > Paper 43 open journal systems 


A modified Kardar--Parisi--Zhang model

Giuseppe Da Prato, Scuola Normale Superiore PISA Italy
Arnaud Debussche, IRMAR, ENS Cachan Bretagne, CNRS, UEB
Luciano Tubaro, Dipartimento di Matematica, Università di Trento


Abstract
A one dimensional stochastic differential equation of the form
dX=A X dt+(1/2) (-A)-α∂ξ[((-A)-αX)2]dt+∂ξ dW(t),    X(0)=x
is considered, where A=(1/2) ∂ξ2. The equation is equipped with periodic boundary conditions. When α=0 this equation arises in the Kardar--Parisi--Zhang model. For α≠ 0, this equation conserves two important properties of the Kardar--Parisi--Zhang model: it contains a quadratic nonlinear term and has an explicit invariant measure which is gaussian. However, it is not as singular and using renormalization and a fixed point result we prove existence and uniqueness of a strong solution provided α>1/8.


Full text: PDF

Pages: 442-453

Published on: November 28, 2007
Bibliography
  1. S. Albeverio, A. B. Cruzeiro. Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids. Comm. Math. Phys. 129 (1990), no. 3, 431--444. MR1051499
  2. S. Albeverio, M. Röckner, M. Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Related Fields 89 (1991), no. 3, 347--386. MR1113223
  3. S. Assing. A pregenerator for Burgers equation forced by conservative noise. Comm. Math. Phys. 225 (2002), no. 3, 611--632. MR1888875
  4. L. Bertini, Lorenzo, G. Giacomin. Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 (1997), no. 3, 571--607. MR1462228
  5. V.S. Borkar, R.T. Chari, S.K. Mitter. Stochastic quantization of field theory in finite and infinite volume. J. Funct. Anal. 81 (1988), no. 1, 184--206. MR0967896
  6. J.-Y. Chemin. Fluides parfaits incompressibles. (French) [Incompressible perfect fluids] Astèrisque No. 230 (1995), 177 pp. MR1340046
  7. J.-Y. Chemin. About Navier-Stokes system. Prèpublication du Laboratoire d'Analyse Numèrique de l'Universitè Paris 6, R96023 (1996). Math. Review number not available.
  8. G. Da Prato, J. Zabczyk. Ergodicity for infinite-dimensional systems. London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. xii+339 pp. ISBN: 0-521-57900-7 MR1417491
  9. G. Da Prato, L. Tubaro. Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization. Probab. Theory Related Fields 118 (2000), no. 1, 131--145. MR1785456
  10. G. Da Prato, L. Tubaro. Introduction to Stochastic Quantization. Pubblicazione del Dipartimento di Matematica dell'Università di Trento. (2007). Math. Review number not available.
  11. G. Da Prato, A. Debussche. Two-dimensional Navier-Stokes equations driven by a space-time white noise. J. Funct. Anal. 196 (2002), no. 1, 180--210. MR1941997
  12. G. Da Prato, A. Debussche. Strong solutions to the stochastic quantization equations. Ann. Probab. 31 (2003), no. 4, 1900--1916. MR2016604
  13. A. Debussche. The 2D-Navier-Stokes equations perturbed by a delta correlated noise. Probabilistic methods in fluids, 115--129, World Sci. Publ., River Edge, NJ, 2003. MR2083368
  14. D. Gatarek, Dariusz, B. Goldys. Existence, uniqueness and ergodicity for the stochastic quantization equation. Studia Math. 119 (1996), no. 2, 179--193. MR1391475
  15. M. Kardar, G. Parisi, J.C. Zhang. Dynamical scaling of growing interfaces. Phys. Rev. Lett.56 (1986), 889--892. Math. Review number not available.
  16. R. Mikulevicius, B.L. Rozovskii. Martingale problems for stochastic PDE's. Stochastic partial differential equations: six perspectives, 243--325, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999. MR1661767
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X