# Assumptions on inputs Gaussian Process Regression

Let $$F:\mathcal{X} \to \mathcal{Y}$$ a function one seeks to approximate. You have $$N$$ observations of this function and you want to predict the value at some other points. In the Gaussian Process Regression framework, does one assume that the inputs $$X\in\mathcal{X}$$ follow a Gaussian random distribution?

The Gaussian process is a generalization normal distribution, where the $$\boldsymbol{\mu}, \boldsymbol{\Sigma}$$ are functions of the inputs $$\mathbf{x}$$. This makes the resulting Gaussian process is a distribution over functions since the realization of each such function of $$\mathbf{x}$$ is a random variable distributed according to a Gaussian distribution.
$$\left[ {\begin{array}{c} f(x_1) \\ f(x_2) \\ \vdots \\ f(x_N) \\ \end{array}} \right] \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma}) = \mathcal{GP}\left(m(\mathbf{x}),\, k(\mathbf{x}, \mathbf{x}')\right)$$
This formulation says nothing about the distribution of the $$\mathbf{x}$$'s.