In the article, the authors consider a diffusive stochastic model with evolution given by the following stochastic differential equations \[ \begin{cases} dX_t^{\varepsilon}=\varepsilon dB_t-Y_t^\varepsilon dt,\\ dY_t^\varepsilon=\lambda(\nabla U(X_t^\varepsilon)-Y_t^\varepsilon)dt, \end{cases} \] where \({\varepsilon,\lambda>0}\), \({B_t, t\geq0}\) is a standard \(d\)-dimensional Brownian motion and \({U:\mathbb{R}^d\to\mathbb{R}}\) is a smooth, positive and coercive function.
The Markov process \({Z_t^\varepsilon=(X_t^\varepsilon, Y_t^\varepsilon)}\) has the unique invariant measure \({\nu_\varepsilon}\) for which the large deviation principle is obtained in the article. Also, for \({\nu_\varepsilon}\) the authors prove the exponential tightness property and express the associated rate function as a solution of a control problem.