Large deviation principle for a class of SPDE with locally monotone coefficients

Small time asymptotics for SPDEs with locally monotone coefficients

“…Recently, this framework has been substantially extended by the second named author and Röckner in [40,41,42,43] for more general class of SPDE with coefficients satisfying the generalized coercivity and local monotonicity conditions. In recent years, various properties for SPDEs with monotone or locally monotone coefficients has been intensively investigated in the literature, such as small noise LDP [39,45,49,59], random attractors [26,27,28,29], Harnack inequality and applications [38], Wong-Zakai approximation and support theorem [46], ultra-exponential convergence [58], and existence of optimal controls [16].…”

“…Then, applying the weak convergence approach developed by Budhiraja and Dupuis [10], it easy to get that ν ε satisfies the LDP with rate function I(•) given by (2) (see, e.g. [39,45]). Therefore, our task now is to show that the two families of probability measures µ ε and ν ε are exponentially equivalent, that is, for any δ > 0,…”

Large deviations for stochastic 3D Leray-<inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> model with fractional dissipation

“…Another difficulty here lies in dealing with the nonlinear term in (2) when θ 2 < 1 2 , because in this case the dissipation term is not strong enough to control the nonlinear term which is completely different with the well-studied case within the variational framework (see e.g. [19,36,39,41,46]). We overcome this difficulty by using the regularity of solutions of the deterministic equation to control the nonlinear term and considering the solution starting from a smaller state space H δ for δ = 2(1 − θ 2 ) when θ 2 < 1 2 .…”

“…[28]). The LDP has been intensively investigated in the literature for different type of stochastic partial differential equations (SPDEs), for example, include stochastic reaction-diffusion equations [14,49], stochastic 2D Navier-Stokes equations [9,50], the Boussinesq equations [24], the quasi-geostrophic equations [40], the primitive equations [23], 2D hydrodynamical type models [19], and equations with general monotone and locally monotone drift [36,41,46,55]. For more references on LDP for SPDEs we may refer [11,13,15,34,42,47,53,58] and more references therein.…”

Large deviation principle for stochastic Burgers type equation with reflection

“…It is well-known that the weak convergence approach (see, e.g., [1,3,4]) is a powerful tool to prove the large and moderate deviations for the SPDEs, see, e.g., [9,11,19,27] and so on. We refer to [2] for the excellent reviews during the past decade in this field.…”