The equivalence between the large deviations principle (LDP for short) and the Laplace-Varadhan
principle in the setting of Polish spaces is the starting point for the designated weak convergence approach
to large deviations theory. Instead of the usual approximation procedures and the cumbersome verification
of exponential tightness, this approach allows the user to derive simpler sufficient conditions for
establishing LDPs that rely on the verification of tightness for the laws of the processes involved and the
verification of the convergence, through compactness arguments, in well-known functional spaces, of the
associated controlled equations to the problem of finding the rate function for the LDP. This approach was
developed in different settings by Dupuis, Ellis, Budhiraja and collaborators (we refer the reader to the
book [3] and [2]). It is our purpose to derive a variational formula for functionals of Fractional Brownian
Motions (fBMs for short) and to establish a sufficient condition for the verification of LDPs for families
of measurable maps of fBMs. As a first application we have in mind the derivation of a LDP for an
infinite-dimensional system given by the stochastic fractional Navier Stokes equation perturbed in small
noise limit by the fBM. As a second application we have in mind the generalization of the work [1] giving
a nonlinear Feynman-Kac formula for nonlocal partial differential equations (PDEs for short) associated
to forward-backward stochastic differential equations driven by the fBM and studying, via probabilistic
arguments and Malliavin calculus for the densities, the homogenization regime of those PDEs. This is
ongoing work with prof. Pedro Catuogno.
References
[1] F. Baudoin. L. Coutin. Operators associated with a stochastic differential equation driven by fractional
Brownian motions. Stochastic Processes and their Applications Vol. 117(5), pp 550-574. (2007)
[2] A. Budhiraja, P. Dupuis, V. Maroulas. Large deviations for infinite dimensional stochastic dynamical systems.
Ann. Probab. vol.36(4),1390-1420 (2008)
[3] P. Dupuis, R. S. Ellis. A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in
Probability and Statistics. Wiley and Sons, New York (1997)
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