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Transactions of the American Mathematical Society

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Uniform large deviation principles for Banach space valued stochastic evolution equations


Authors: Michael Salins, Amarjit Budhiraja and Paul Dupuis
Journal: Trans. Amer. Math. Soc. 372 (2019), 8363-8421
MSC (2010): Primary 60F10, 60H15, 35R60
DOI: https://doi.org/10.1090/tran/7872
Published electronically: July 16, 2019
MathSciNet review: 4029700
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Abstract: We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDEs) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform LDP over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite-dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak-$ \star $ compactness of closed bounded sets in the double dual space. We prove that a modified version of our SDE satisfies a uniform Laplace principle over weak-$ \star $ compact sets and consequently a uniform over bounded sets LDP. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite-dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and two-dimensional stochastic Navier-Stokes equations with multiplicative noise.


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Additional Information

Michael Salins
Affiliation: Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215
Email: msalins@bu.edu

Amarjit Budhiraja
Affiliation: Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, North Carolina 27599
Email: budhiraj@email.unc.edu

Paul Dupuis
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: Paul_Dupuis@brown.edu

DOI: https://doi.org/10.1090/tran/7872
Keywords: Uniform large deviations, variational representations, uniform Laplace principle, stochastic partial differential equations, small noise asymptotics, exit time asymptotics, stochastic reaction-diffusion equations, stochastic Navier--Stokes equations.
Received by editor(s): March 2, 2018
Received by editor(s) in revised form: January 18, 2019, and March 11, 2019
Published electronically: July 16, 2019
Additional Notes: The second author’s research was supported in part by the National Science Foundation (DMS-1305120, DMS-1814894) and the Army Research Office (W911NF-14-1-0331).
The third author’s research was supported in part by the National Science Foundation (DMS-1317199).
Article copyright: © Copyright 2019 American Mathematical Society