# pr.probability – Divergence-free Gaussian vector field with given mean magnitude and correlation function

My general question is how one might construct an isotropic random vector field $$vec f: mathbb{R}^3 to mathbb{R}^3$$ which has a given mean magnitude $$mathbb{E}(||vec f(vec x)||)=mu$$ such that vector magnitudes and direction are correlated up to some length scales $$l$$ (beyond which the correlation goes to zero). Furthermore, we wish that $$nabla cdot vec f=0$$, although a construction which does not satisfy this condition is already very interesting.

More precisely, we are given a vector $$vec{mu}$$ in $$mathbb{R^3}$$ and matrix-valued correlation function $$C(vec x_1,vec x_2): mathbb{R^3} times mathbb{R^3} to M_3(mathbb{R})$$ which is isotropic, i.e $$C(vec x_1,vec x_2)=C(||vec x_1-vec x_2||)$$. One may define a Gaussian Process $$f(vec x) sim GP(vec{mu},C(vec x_1,vec x_2))$$ such that $$mathbb{E}(vec f(vec x))=vecmu$$ and $$Cov(vec f(vec x_1),vec f(vec x_1))=C(vec x_1,vec x_2)$$. I believe it is known how to generate such a random field $$f$$. It seems the equivalent problem for a scalar field $$f$$ is well-studied, where one can for example draw the field first in Fourier space by normalizing a white noise field with the appropriate power spectrum $$P(k)$$, and then transform back to real space. However it would be useful if someone could describe here a simple procedure for the case of $$mathbb{R}^3$$, which I think is known but I haven’t found a clear description anywhere.

Question: how can one generate such a random field $$f$$ if one imposes a zero mean vector $$vec{mu}=vec 0$$, and in addition mean magntitude $$mathbb{E}(||vec f(vec x)||)=mu$$? And now if we impose also $$nabla cdot vec f=0$$?

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