TITLE: What structure is needed to define a Gaussian distribution on a given space?
QUESTION [9 upvotes]: In most textbooks, the normal distribution is defined on $\mathbb{R}^n$ by specifying its probability density function. This works perfectly well, but it isn't really amenable to generalisation.
I'm wondering what the minimal structure is that one must have on a given space $S$ before one can define an analogue of the normal distribution. If $S$ has a Riemannian metric defined on it, then one can define a Brownian motion on $S$ using the Laplace-Beltrami operator. The family of normal distributions on $S$ could then be defined as the one-dimensional marginals of the Brownian motion.
Alternatively, the normal distribution could be characterised as the distribution that maximises entropy when its mean and variance are known. This characterisation only seems to require a notion of "mean" (which suggests that $S$ must be a metric space).
Is there a more general construction of the normal distribution?

REPLY [5 votes]: Let $X$ be a Banach space with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon measure on $X$.  The continuous linear functionals on $X$ are measurable functions, hence random variables.  Let's assume that for all $f \in X^*$, $$\mathbb E|f|^2 = \int_X |f(x)|^2 \ \mathrm{d}\mathbb P(x) < \infty.$$ This assumption implies a powerful structure theorem (cf. Theorem III.2.1 of [1]):
Theorem.  There exists an element $\mu \in X$ (the mean element) and a continuous, symmetric operator $K : X^* \to X$ (the covariance operator) such that for all $f, g \in X^*$, $$f(\mu) = \mathbb E f \qquad \mathrm{and} \qquad f(Kg) = \mathbb E(fg) - f(\mu)g(\mu).$$
Furthermore, the support of the measure $\mathbb P$ is contained in the affine subspace spanned by the mean and covariance operator: $$\operatorname{supp} \mathbb P \subseteq \mu + \overline{KX^*},$$
and this space is separable.
As Mark Meckes said, we say that the measure $\mathbb P$ is Gaussian if each $f \in X^*$ is a real-valued Gaussian random variable (hence with mean $f(\mu)$ and variance $f(Kf)$.  In this case, the support is the full affine subspace $\mu + \overline{KX^*}$.  Furthermore, the measure is completely characterized by its characteristic functional $\varphi : X^* \to \mathbb R$:
$$\varphi(f) = \int_X \mathrm e^{\mathrm i f(x)} \ \mathrm d\mathbb P(x) = \mathrm e^{\mathrm i f(\mu) - f(Kf)/2}.$$
For more on this topic, I suggest the reference:
[1] Probability Distributions on Banach Spaces by Vakhania, Tarieladze and Chobanyan.