Due to the nonlinearity in the equation and the presence of space-time white noise, solutions to the KPZ equation are known to not be smooth or regular, but rather ' fractal ' or ' rough.' Even without the nonlinear term, the equation reduces to the stochastic heat equation, whose solution is not differentiable in the space variable but satisfies a Hölder condition with exponent less than 1/2. Thus, the nonlinear term ( ∇ ∇ h ) 2 is ill-defined in a classical sense.