Uniform large deviation principles for Banach space valued stochastic evolution equations
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- by Michael Salins, Amarjit Budhiraja and Paul Dupuis PDF
- Trans. Amer. Math. Soc. 372 (2019), 8363-8421 Request permission
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.References
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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
- 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). - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 8363-8421
- MSC (2010): Primary 60F10, 60H15, 35R60
- DOI: https://doi.org/10.1090/tran/7872
- MathSciNet review: 4029700