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Register a new userLarge deviations for two scaled diffusions
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R. Liptser
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We formulate large deviations principle (LDP) for diffusion pair $(X^epsilon,xi^epsilon)=(X_t^epsilon,xi_t^epsilon)$, where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for $(X^epsilon, u^epsilon)$ with $ u^epsilon(dt,dz)$ being an occupation type measure corresponding to $xi_t^epsilon$. In some sense we obtain a combination of Freidlin-Wentzells and Donsker-Varadhans results. Our approach relies the concept of the exponential tightness and Puhalskiis theorem.
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We formulate the large deviations for a class of two scale chemical kinetic processes motivated from biological applications. The result is successfully applied to treat a genetic switching model with positive feedbacks. The corresponding Hamiltonian is convex with respect to the momentum variable as a by-product of the large deviation theory. This property ensures its superiority in the rare event simulations compared with the result obtained by formal WKB asymptotics. The result is of general interest to understand the large deviations for multiscale problems.
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