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.
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