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)


A one dimensional stochastic differential equation of the form \[dX=A X dt+\tfrac12 (-A)^{-\alpha}\partial_\xi[((-A)^{-\alpha}X)^2]dt+\partial_\xi dW(t),\qquad X(0)=x\] is considered, where $A=\tfrac12 \partial^2_\xi$. The equation is equipped with periodic boundary conditions. When $\alpha=0$ this equation arises in the Kardar-Parisi-Zhang model. For $\alpha\ne 0$, 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 $\alpha>\frac18$.

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Pages: 442-453

Publication Date: November 28, 2007

DOI: 10.1214/ECP.v12-1333


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