I read in a paper that I have a vector $\bf{x}$ such that
$$ \langle x_i (t) x_j (t')\rangle=\delta_{ij}\delta(t-t') $$ and the probability distribution for $\bf{x(t)}$ is a Gaussian distribution and it can be written as: $$ \exp\left[-\int_0^T\,\mathrm dt \sum_{i,j} A^{i,j}x_i(t)x_j(t)\right] $$ where $A^{i,j}$ is the variance matrix. Can someone please help how this Gaussian distribution is written; especially why there is an integral in the exponent.