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?
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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.
