Harnack inequality and applications for stochastic evolution equations with monotone drifts

By Wei Liu. 2008

85 citations recorded by the source corpus.

As a Generalization to Wang (Ann Probab 35:1333–1350, 2007) where the dimension-free Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic p-Laplace equation in Hilbert space.

Subfields: Applied Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/edef2bb337bbd8d9d43ebd4d333e4814612fc7ad

How researchers see this paper

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In Applied Mathematics

1. Wei Liu
2. Michael Röckner
3. Feng-Yu Wang
4. Tusheng Zhang
5. Wei David Liu
6. Shijie Shang
7. Benjamin Gess
8. Jian Wang

In Finance

1. Wei Liu
2. Michael Röckner
3. Tusheng Zhang
4. Shijie Shang
5. Feng-Yu Wang
6. Benjamin Gess
7. Wei David Liu
8. Xicheng Zhang

Citations in the corpus

• Harnack inequalities for stochastic equations driven by Levy noise (2012, background): “roduced in [22] and the log-Harnack inequality introduced in [17, 25] have been intensively investigated and applied for stochastic (partial) differential equations driven by Gaussian noises, see e.g. [3, 23, 13, 4, 7, 8, 12, 24, 34, 9, 14, 29, 30, 10, 15, 31, 32, 33] and references within. Due to te”
• 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”
• Strong solutions for SPDE with locally monotone coefficients driven by L\'{e}vy noise (2011, unknown): “(4.7) has a unique solution. Proof. (1) Define the following Gelfand triple V := W1,p 0 (Λ) ⊆ H:= L2(Λ) ⊆ W−1,q(Λ), then it is well known that Pd i=1 Di (|Diu| p−2D iu) satisfy (H1)-(H4) with α= p(cf. [32, 33]). In particular, there exists a constant δ>0 such that (4.10) Xd i=1 V ∗hDi |Diu|p−2D iu”
• 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 Deviations for Stochastic Evolution Equations with Small Multiplicative Noise (2008, methodology): “ assumption (A2) does not hold. Hence like the case of stochastic fast diffusion equations, we should apply Theorem 2.2 to derive the LDP for (5.10) on C([0,T];H). 27 The following SPDE was studied in [21, 22]. The main part of drift is a high order generalization of the Laplace operator. Example 5.6”
• 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].”