Draft:The Random Wave Model

The Random Plane Wave is a Gaussian process on [0,∞) with covariance function:

$K(t,s)=J_{0}(|t-s|)$

where J₀ is the zeroth-order Bessel function of the first kind.

Properties

The Random Plane Wave possesses both translation invariance (stationarity) and rotation invariance (isotropy), properties that follow directly from the J₀ kernel depending only on the distance between points. The process has smooth sample paths.

Mathematical structure

The kernel arises as the Fourier transform of the uniform measure on the unit circle $S^{1}$. This establishes that it is a positive semi-definite kernel.

The eigenfunctions satisfy the integral equation:

$\int _{0}^{\infty }J_{0}(|x-y|)f(y)\,dy=\lambda f(x)$

The explicit solution to this equation remains an open problem.

Interpretation

The Random Plane Wave is the limit of a superposition of plane waves uniformly distributed in all directions with equal magnitude and uniformly distributed random phase. This interpretation arises from considering a discrete approximation to the uniform measure on $S^{1}$. Let $\mu _{n}$ be a discrete measure that places equal mass at the $n$th roots of unity on $S^{1}$. The corresponding Gaussian process can then be written as:

$X_{n}(t)={\frac {1}{\sqrt {n}}}\sum _{k=1}^{n}\xi _{k}\cos(t\cdot \lambda _{k})$

where $\lambda _{k}=(\cos(2\pi k/n),\sin(2\pi k/n))$ and $\xi _{k}$ are independent standard normal random variables.

As $n\to \infty$, this process converges to the Random Plane Wave. In this limit, the process can be understood as a Gaussian superposition of cosine waves with unit wavelength in all possible directions.

See also

• Gaussian process
• Bessel function
• Karhunen–Loève theorem
• Moore–Aronszajn theorem
• Stationary process