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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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