I'm looking for a Large Deviation Principle for the Empirical Path Law. Someone must have done this right? That is, for $i\in \mathbb{N}$, $t\in[0,T]$, let $X^i(t)\in \mathbb{R}^d$ be (the position of) iid Brownian Particles, with say $X^i(0)=x_0$. Define the empirical path law
$$ L^n:=\frac{1}{n}\sum_{i=1}^n \delta_{X^i(\cdot)} $$
where $\delta_X$ is the dirac mass at $X$ measure on an appropriate path space $C([0,T];\mathbb{R}^d$). Is there a result for a LDP for (the measure associated to) $L^n$, if so where is it?!