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A function space large deviation principle for certain stochastic integrals. (English) Zbl 0682.60018
Let X(t) be a diffusion process on a compact manifold and \[ Y(t)=\int^{t}_{0}Y_ 0(X(s))ds+\sum^{d}_{k=1}\int^{t}_{0}Y_ k(X(s))\circ d\beta_ k(s), \] where \(\beta_ k(\cdot)\), \(1\leq k\leq d\), are independent Wiener processes. The authors prove large deviations principles for \[ \{N^{- 1}\sum^{N-1}_{k=0}(Y(k+t)-Y(k)),\quad 0\leq t\leq 1\}_{N=1,2,3,...}\quad and\quad for\quad \{T^{-1}Y(tT),\quad 0\leq t\leq 1\}_{T\geq 0} \] and identify the corresponding rate functions on C([0,1],\({\mathbb{R}})\). An application to the asymptotics of the Lyapunov exponent of a homogeneous system is given.
Reviewer: L.Arnold

60F10 Large deviations
60H05 Stochastic integrals
Full Text: DOI
[1] Arnold, L., Kliemann, W.: Large deviations of linear stochastic differential equations. In: Engelbert, H.J., Schmidt, W. (eds.) Stochastic Differential systems. Proceedings, Eisenach 1986. (Lect. Notes Control Inf. Sci., vol. 96, pp. 117-151) Berlin Heidelberg New York: Springer 1987 · Zbl 0632.60021
[2] Baxendale, P., Stroock, D.W.: Large deviations and stochastic flows of diffeomorphisms. Probab. Theor. Rel. Fields 80, 169-215 (1988) · Zbl 0638.60035 · doi:10.1007/BF00356102
[3] Carlen, E.: Conservative diffusions. Commun. Math. Phys. 94, 293-315 (1984) · Zbl 0558.60059 · doi:10.1007/BF01224827
[4] Dawson, D., Gärtner, J.: Large deviations from the Mc Kean Vlasov limit for weakly interacting diffusions. Stochastics 20, 247-308 (1987) · Zbl 0613.60021
[5] Donsker, M., Varadhan, S.R.S.: Asymptotic evaluation of certain Markov process expectations for large time, III. Commun. Pure Appl. Math. 29, 389-461 (1976) · Zbl 0348.60032 · doi:10.1002/cpa.3160290405
[6] Fernique, X.: Régularité des fonctions aléatoires gaussiennes. (Lect. Notes Math., vol. 480, pp. 1-96). Berlin Heidelberg New York: Springer 1975
[7] Föllmer, H.: An entropy approach to the time reversal of diffusion processes. (Lect. Notes Control Inf. Sci., vol. 69, pp. 156-163). Berlin Heidelberg New York: Springer 1985 · Zbl 0562.60083
[8] Föllmer, H.: Ecole d’été de calcul des Probabilités XV?XVII, (Lect. Notes Math., vol. 1362, pp. 101-204). Berlin Heidelberg New York: Springer 1988
[9] Stroock, D.W.: An Introduction to the Theory of Large Deviations. Berlin Heidelberg New York: Springer 1984 · Zbl 0552.60022
[10] Stroock, D.W.: On the rate at which a homogeneous diffusion approaches a limit, an application of large deviation theory to certain stochastic integrals. Ann. Probab. 14, 840-859 (1986) · Zbl 0604.60076 · doi:10.1214/aop/1176992441
[11] Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer 1979 · Zbl 0426.60069
[12] Watanabe, S.: Stochastic differential equations and Malliavin Calculus, Tata Institute of Fundamental Research. Berlin Heidelberg New York: Springer 1984
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