A modified Kardar-Parisi-Zhang model
Résumé
A one dimensional stochastic differential equation of the form dX = AXdt + 1/2 (A) partial derivative[((A) X)(2)] dt + partial derivative dW(t), X( 0) = x is considered, where A = 1/2 partial derivative(2). The equation is equipped with periodic boundary conditions. When = 0 this equation arises in the Kardar-Parisi-Zhang model. For not equal 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 > 1/8.