Kardar–Parisi–Zhang equation

Remarks

The equation is formal: a solution $h(\cdot, t)$ is only Hölder continuous of every order less than $\frac{1}{2}$ in $x$, so $(\partial_x h)^2$ is not defined. For $\lambda \ne 0$, a standard meaning is the Cole–Hopf solution $h = \frac{2\nu}{\lambda}\log Z$, where $Z > 0$ solves the stochastic heat equation $\partial_t Z = \nu\, \partial_x^2 Z + \frac{\lambda\sqrt{D}}{2\nu} Z \xi$ in the Itô sense; see Cole–Hopf transformation. Hairer's theory of regularity structures gives a solution theory that agrees with the Cole–Hopf solution, up to an infinite renormalization constant. Example: with $\lambda = 0$ the equation is linear, the additive stochastic heat equation, and its solution is a Gaussian process given by a stochastic convolution with the heat kernel. Non-example: the deterministic equation $h_t = \nu h_{xx} + \frac{\lambda}{2}(h_x)^2$ has no noise term, so it is not the KPZ equation; it is a viscous Hamilton–Jacobi equation.