Flow Decomposition and Large Deviations

Abstract

We study large deviations properties related to the behavior, as = goes to 0, of diffusion processes generated by = 2 L 1 +L 2 , where L 1 and L 2 are two second-order differential operators, extending recent results of Doss and Stroock and Rabeherimanana. The main tool is the decomposition theorem for flows of stochastic differential equations proved by Bismut and Kunita. We give another application of flow decomposition in a nonlinear filtering problem.

Large Deviations for Stochastic Flows of Diffeomorphisms

“…existence, uniqueness) of the uncontrolled versions of the flows. Unlike in [18,3] and [5] (which consider only finite dimensional flows), the proof of the LDP does not require any exponential probability estimates or discretization/approximation of the original model.…”

“…The results of the current paper are the first step towards the study of the analogous problem for infinite dimensional SDEs. The paper [5] used the LDP for stochastic diffeomorphic flows to study large deviation properties, as ε → 0, of finite dimensional diffusions generated by εL 1 + L 2 , where L 1 , L 2 are two second order differential operators. The analogous problem for infinite dimensional diffusions is currently open; a key ingredient is again the LDP for infinite dimensional flows obtained in the current paper.…”

“…For ε ∈ (0, ∞), when f i is replaced by εf i in (1.1), we write the corresponding flow as φ ε . Large deviations for φ ε inŴ k , as ε → 0, have been studied for the case k = 0 in [18,3] and for general k in [5].…”

“…One such example is given in Section 5. Thus, following Kunita's [15] notation for stochastic integration with respect to semi-martingales with a spatial parameter, the study of general Brownian flows of C k -diffeomorphisms leads to SDEs of the form [18,3,5], which consider stochastic flows driven by only finitely many real Brownian motions. The main technical difficulty in establishing the LDP inŴ k × W k is the proof of a result analogous to Proposition 4.10, which establishes tightness of certain controlled processes, when k − 1 is replaced by k.…”

Sur les grandes déviations en théorie de filtrage non linéaire

“…D'ailleurs depuis le fameux résultat de M. Schilder [13], plusieurs auteurs ont travaillé dans ce sens ; on rappelle notamment les résultats de H. Doss [6], Baldi et al [2], G. Ben Arous et M. Ledoux [4], et G. Ben Arous et F. Castelle [3].…”