### Abstract

We prove an enhanced limit theorem for additive functionals of a multidimensional Volterra process $(y_t)_ {t\geq 0}$. As an application, we establish weak convergence of the solutions of rough differential equations (RDE) of the form $$ dx^\varepsilon_t=\frac 1 {\sqrt \varepsilon} f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}}),dt+g(x_t^\varepsilon),d\mathbf{B}_t,$$ and identify their limits as solutions of an RDE driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. The equation models a passive tracer in a random field.
In particular if $h$ is random field such that $h(x, \cdot)$ a semi-martingale with spatial parameter $x$, we show that the solutions of the equations $$ dx^\varepsilon_t=\frac 1 {\sqrt \varepsilon} f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}}),dt+h(x_t^\varepsilon, dt),$$ converge weakly to that of a Kunita type Itô SDE $dx_t=G(x_t,dt)$ where $G(x,t)$ is a semi-martingale with spatial parameters. Furthermore the $N$-point motions converge.