Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

By Wei Liu. 2008

142 citations recorded by the source corpus.

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2cff1ec365a87faab041cda51fed32620b36951f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Wei Liu
2. Michael Röckner
3. Tusheng Zhang
4. Xicheng Zhang
5. Wei David Liu
6. Shijie Shang
7. Benjamin Gess
8. Igor D. Chueshov

In Finance

1. Wei Liu
2. Tusheng Zhang
3. Michael Röckner
4. Xicheng Zhang
5. A. Budhiraja
6. Paul Dupuis
7. Shijie Shang
8. Jiang Chen

In Mathematical Physics

1. Tusheng Zhang
2. Wei Liu
3. Xicheng Zhang
4. Michael Röckner
5. Annie Millet
6. A. Budhiraja
7. Paul Dupuis
8. Wei David Liu

Citations in the corpus

• Freidlin-Wentzell's Large Deviations for Stochastic Evolution Equations (2008, background): “The authors also thanks Dr. Wei Liu for sending us his preprint [ 19 ].”
• Stochastic 3D tamed Navier-Stokes equations: Existence, uniqueness and small time large deviation principles (2012, result): “For example, for SPDE with monotone coefficients under very general conditions this LDP has been proved in [21], strongly generalizing a corresponding former result by P.”
• Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations (2010, methodology): “Our proof is inspired by Liu’s method [ 14 ] and also works for 2D stochastic Navier-Stokes equations.”
• Well-posedness of stochastic partial differential equations with fully local monotone coefficients (2022, background): “see [48, 27, 5] for regularity of solutions, [18, 74, 47, 19, 88, 52, 61, 87, 37, 38] for large deviation principle, [31, 32, 33, 78, 65, 4, 90] for numerical approximation, [84, 46, 85, 86] for the Harnack inequality, gradient estimates, invariant measure and ergodicity, [28, 29] for random attract”
• Large Deviation Principles of Obstacle Problems for Quasilinear Stochastic PDEs (2017, methodology): “ and Zhang /Quasilinear SPDEs 3 powerful tool which has been applied by many people to prove large deviation principles for various dynamical systems driven by Gaussian noises, see e.g. [BD-1], [DM], [L], [MSS], [RZZ],[BDM1]. In order to apply the weak convergence approach to the obstacle problems, ”
• Large Deviations for Stochastic Partial Differential Equations Driven by a Poisson Random Measure (2012, methodology): “In recent years an approach based on certain variational representation formulas for moments of nonnegative functionals of Brownian motions [7] has been increasingly used for the study of the small noise large deviation problem for Brownian motion driven infinite dimensional systems [2, 7, 8, 10, 13”
• Strong solutions for SPDE with locally monotone coefficients driven by L\'{e}vy noise (2011, unknown): “[32, 33]).”
• Random attractors for a class of stochastic partial differential equations driven by general additive noise (2010, background): “see (H1) − (H4) in [24, 25, 34, 35, 36, 42] and (H2) in [19, 34, 36].”
• Variational representations for continuous time processes (2011, methodology): “For example, Brownian motion representations have been used by [1, 6, 7, 8, 10, 11, 18, 19, 21, 22, 24, 25, 26, 29, 30] in the large deviation analysis of solutions to SPDEs in the small noise limit, and recently in [5] for interacting particle limits.”
• Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations (2008, methodology): “g, [31] for 2D Navier-Stokes equations, [13] for 2D Bénard convection, [4] for stochastic reactiondiffusion system, [24] for stochastic p-Laplacian equation, [26] for the GOY shell model of turbulence).”
• Harnack inequality and applications for stochastic evolution equations with monotone drifts (2008, background): “The following SPDE has been studied in [19, 20], in which the main part of drift in the equation is a high order generalization of the Laplace operator.”
• Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations (2008, methodology): “In Section 4, using the weak convergence method, we prove the Freidlin-Wentzell large deviation principle for the small perturbations of stochastic Volterra equations under a compactness assumption and some uniform non-explosion conditions for the controlled equations.”
• SPDE in Hilbert space with locally monotone coefficients (2010, methodology): “see [8, 19, 28, 32] for the small noise large deviation principle, [13, 14] for discretization approximation schemes to the solutions of SPDE, [34, 17, 18] for the dimension-free Harnack inequality and resulting ergodicity, compactness and contractivity properties of the associated transition semigr”