A modified Kardar-Parisi-Zhang model
A modified Kardar-Parisi-Zhang model
复制标题
改进的 Kardar-Parisi-Zhang 模型
DOI：
10.1214/ecp.v12-1333
复制
发表时间：
2007
影响因子：
0.5
通讯作者：
L. Tubaro
中科院分区：
文献类型：
--
作者：
G. Prato;A. Debussche;L. Tubaro
考虑 \[dX=A X dt+\tfrac12 (-A)^{-\alpha}\partial_\xi[((-A)^{-\alpha}X)^2]dt+\partial_\xi dW(t),\qquad X(0)=x\] 形式的一维随机微分方程，其中 $A=\tfrac12 \partial^2_\xi$。该方程配备周期性边界条件。当 $\alpha=0$ 时，该方程出现在 Kardar-Parisi-Zhang 模型中。对于 $\alpha\ne 0$，该方程保留了 Kardar-Parisi-Zhang 模型的两个重要属性：它包含二次非线性项并具有显式的高斯不变测度。然而，它并不那么奇异，并且使用重整化和定点结果，我们证明了提供 $\alpha>\frac18$ 的强解的存在性和唯一性。
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 $\alpha>\frac18$.