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 $t$.



Does every smooth manifold carry a gaussian random field?
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*](http://www3.nd.edu/~lnicolae/GB2.pdf) for more details and additional references. I would start with reference [6] in this paper.


