Having established existence of the derivative of a SRVF, stochastic integrals can be defined as a mean-square Riemann integration. Let be a RVF on a domain and for all , let be a deterministic continuous and bounded function such that . The function can also be complex such that . The stochastic volume integral is the mean-square Riemann integral

The integral exists if the limit of the Riemann sum exists

where is a volume element centered on and . The integral then exists if

Since then . When then

This definition leads to the following corollary (See Gilrod ref…).

A SRVF is mean-square Riemann integrable iff

=

<

For a Gaussian RVF this is equivalent to

The definition can be extended to include line integrals. Let be a GRVF on a domain and for all and . Let be a deterministic continuous and bounded function such that .

The integral exists if the limit of the Riemann sum exists

,

where is a line element centered on and is the length of the line. The stochastic line integral then exists if

. If then

For a GRVF, the linking of two curves or knots is described by the correlations

The n-point correlation for a link comprised of n curves or knots

is

=

which can be expressed in a path integral form as

We can now examine the spectral properties of Gaussian random vector field.

Given the stochastic integral of (-), then setting with and gives a stochastic Fourier integral.

If there exists such a random vector field then the random vector field

is harmonizable. The integral exists in the mss iff

and for a Gaussian RVF. Equation (-) defines the cross spectral density function, provided the integral exists.

Given the transform pair

there is a technical difficulty in that the integrals do not actually converge when integrated over all space. The problem is that the random function extends over all space and does not decay to zero as , as is required for the existence of a classical Fourier transform. The resolution of this difficulty is essentially to define the transform of over a finite volume or domain of 3-space so that and consider the limit as the volume grows large and approaches infinity.

A finite-range transform and its

complex conjugate can be defined as

with and where is a cubic volume defined by , , and defines the sides of the cube. Taking the product of the transform and its conjugate then averaging gives the spectral moment

We then consider the behavior of (36) in the limit as or . The correlation is a function of owing to its homogeneity, and there is a decorrelation for . The length scale over which has significant non-zero values is with being the correlation length. Let be a vector separation of two points. Replacing as an integration variable in (36) gives

where depends on on and is defined by , , . The cube has sides and is centred on . The integral is only significantly different from zero when $| r |=O(l)$ and this region lies well within , provided that is many correlation lengths from the boundary of .Then as for most , the integration over can be carried to infinity.

The approximation of the integral over by applies everywhere except when is of the order of a correlation length from the boundary of .

We are most interested in the case when (38) has different values of . The spectral correlation is where the integration variable has been changed from to . Again. provided that is many correlation lengths from the boundary of the integral over can be

extended to infinity so that

where

The function has a maximum peak at and becomes increasingly sharply peaked about as with the integral always satisfied. It follows that has all the properties of a delta function so that as . This then leads

to the important result

We have replaced with since the delta function is effectively zero when . Since is real the complex conjugate of (-) gives so that the spectral correlation (in the infinite limit) becomes

Spectral analysis of turbulence for example, involves Fourier analysis of a stochastic Navier-Stokes flow on so that The essential idea is to decompose the turbulent fluctuations about the mean flow into sinusoidal components and study the distribution of turbulent energy amongst the different wave vectors representing the different scales of turbulence.

The spectral function is given by the Fourier transform

of the stress tensor .

*

and depends on the wavenumber vector and as the turbulence evolves. The inverse transform is , which is an integral over all 3-dimensional wavenumber space. Setting and i=j in (-) gives the stress tensor of (-)

The ‘turbulent kinetic energy’ can therefore be expressed as an integral over . In turbulence theory for example, where is a random Navier-Stokes flow, the function can be interpreted as the distribution of turbulent energy over the various wavenumbers. The stresses and spectral functions have the symmetries and . Also leading to , which implies that is a Hermitian matrix. Each of the diagonal terms is therefore real as should be the case for a quantity interpreted as the distribution of energy with wavenumber. The magnitude of the wavenumber vector has dimensions of inverse length. One can interpret the length scale as the spatial scale represented by wavenumber . For example if is an integral scale of turbulence the wavevectors represent the scales of turbulence that contain most of the kinetic energy. The spectral function then has its maximum values for and decays to zero as . The stress tensor can then be represented as the inverse Fourier transform

where and the integration range is taken over all space since we take the spectral function to be such that the Fourier integrals now converge. Equation (44) becomes

Now so that

Finally, for homogeneity and isotropy it is required that

so that the spectral function must satisfy . The most general form is then

The 2-point correlations then depend on the ansatzes for the spectral function.

Let be the spectral function corresponding to the 2-point function

The ansatz for the spectral function and relationship between the wave vectors then determines the form of the correlation for the GRVFS such that

[1] If then and

This correlation then describes a white-in-space noise.

[2] If and then

which is of the same mathematical form as the propagator in a Chern-Simons theory(ref). The following lemma on the possible 2-point correlations is also stated.

Let be a Gaussian random field defined for all with . Let and . If the 2-point function is a ‘white-in-space noise’ of the form

then

[1] There exists a random Gaussian field for all with the 2-point correlation

[2] There exists a random Gaussian field for all with the 2-point correlation

Let then

which follows from the Poisson equation

Hence

or . Next, given there is a Greens function such that

so that

Let and and are GRVFs on . As in (-), the spectral function choice gives a 2-point correlation for white noise.

Let be a ball of radius and . If the delta-function is ‘smeared-out’ over into a very narrow and highly peaked rectangular or top-hat function of width . Then

where

~for

for

so that the correlation vanishes outside . This is equivalent to choosing

for the spectral function so that the Fourier transform is

and then making the rescaling .

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