## Large deviation principle for invariant distributions of memory gradient diffusions.(English)Zbl 1286.60033

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.

### MSC:

 60F10 Large deviations 60J60 Diffusion processes 60G10 Stationary stochastic processes 60K35 Interacting random processes; statistical mechanics type models; percolation theory 35H10 Hypoelliptic equations 93D30 Lyapunov and storage functions
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