TITLE: Why is the gaussian free field a distribution but Brownian motion is a function?
QUESTION [9 upvotes]: As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?

REPLY [3 votes]: A Gaussian free field, in any dimension, is a Gaussian field with covariance given by the Green's function, $G(x,y)$ satisfying $\Delta G(x,\cdot)=\delta(\cdot-x)$. In dimension $1$, $G(x,\cdot)$ is a continuous function - in fact, just a piecewise linear one. In higher dimensions, $G(x,\cdot)$ has a singularity at $x$. Therefore, the value of a field at a point cannot be well defined - on one hand, it would have to be Gaussian, but on the other hand, it must have infinite variance.