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Let $M$ be an arbitrary finite-dimensional smooth manifold. For simplicity, let's assume that $M$ has no boundary. Does there always exist a gaussian random field with constant variance on $M$? If not, does there exist a theorem which states sufficient (or necessary and sufficient) conditions on $M$ under which a gaussian random field will always exist?

My definition of a gaussian random field is a random function $f:M\to \mathbb{R}^{n}$ such that for arbitrary points $t_1, \dots, t_k$ in $M$, the images $f(t_1),\dots, f(t_k)$ form a multivariate guassian random vector. Adler, Taylor, and Worsley in Applications of Random Fields and Geoemtry assume that $f(t)$ has constant variance for all $t$.

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    $\begingroup$ What do you take a Gaussian Random Field to be on a manifold? I imagine I could answer your question if I knew the definition. Or are you talking about Riemann manifolds, like in the Adler paper? A plain manifold has too little structure to make much sense of a thing like a Gaussian Random Field, I imagine. $\endgroup$ – Ryan Budney Oct 21 '15 at 6:43
  • $\begingroup$ That's kind of the heart of my question. What structure do you need to ensure there always is one? As you say, an arbitrary smooth manifold has too little structure. Is a Riemannian metric enough? I added the definition of a random field to the question. $\endgroup$ – Alex Lapanowski Oct 21 '15 at 16:20
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    $\begingroup$ What continuity properties should $f$ satisfy? So far, you definition only refers to the underlying set of $M$. You better refer to the smooth structure of $M$ somewhere. $\endgroup$ – André Henriques Oct 21 '15 at 16:36
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You can construct examples a dime a dozen. $\newcommand{\bR}{\mathbb{R}}$

Here is a first simple way. Fix $N$ smooth functions $f_1,f_2,\dotsc, f_N:M\to\bR$ and $N$ independent Gaussian random variables $X_1,\dotsc, X_N$. Then

$$f(x)=\sum_{k=1}^N X_k f_k(x) $$

is a Gaussian random field and the sample functions are a.s. smooth.

You can allow infinitely many functions in the above example, i.e., $N=\infty$, but then you need to make some assumptions on the functions and on the variances of the variables $X_k$ to guarantee the convergence of the resulting series.

We know from Kolmogorov that the convergence is a $0-1$ phenomenon. The three-series theorem tells us what these conditions should be. (In the Gaussian case we can say a bit more.)

Here is an example of this kind when $M$ is compact. Fix a Riemann metric $g$ on $M$, denote by $\Delta$ the resulting Laplacian. Its eigenvalues (multiplicities included) are

$$ 0=\lambda_0<\lambda_1\leq \lambda_2\leq \cdots. $$

Fix an orthonormal eigenbasis $(\psi_k)_{k\geq 0}$ of $L^2(M, dV_g)$

$$\Delta \psi_k=\lambda_k,\;\; \Vert\psi_k\Vert_{L^2}=1,\;\;\forall k\geq 0. $$

Next choose independent Gaussian random variables $(X_k)_{k\geq 0}$. We denote by $v_k$ the variance of $X_k$. If $v_k$ goes to $0$ sufficiently fast, then the random series

$$f(x)=\sum_{k\geq 0} X_k \psi_k(x) $$

defines a Gaussian random field on $M$. The regularity of the sample functions of this random field depends on the decay rate of $v_k$. The faster $v_k$ decays as $k\to\infty$, the more regular is the random function. For example, if

$$\lim_{k\to \infty}k^\alpha v_k=0,\;\;\forall \alpha>0, $$

then the random function $f(x)$ is a.s. smooth.

Ultimately, the most general construction of a Gaussian random function on a manifold is via Gaussian measures on the space of distributions (i.e. generalized functions) on $M$. I refer to this paper for more details and additional references. I would start with reference [6] in this paper.

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