Gaussian probability distribution for a delta correlated random variable.

Question

I read in a paper that I have a vector $\bf{x}$ such that $$ ⟨x_i(t)x_j(t′)⟩=δ_{ij}δ(t−t′) $$ and the probability distribution for $\bf{x}$(t) is a Gaussian distribution and it can be written as: $$ \exp\left[−\int_0^T dt\sum_{i,j}A_{i,j} x_i(t)x_j(t)\right] $$ where $A_{i,j}$ is the variance matrix. Can someone please help how this Gaussian distribution is written; $\bf{especially ~why ~there ~is~ an~ integral~ in~ the~ exponent. }$