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We explore probabilistic approaches to the deterministic energy equality for the forced Surface Quasi-Geostrophic (SQG) equation on a torus. First, we prove the zero-noise dynamical large deviations for a corresponding stochastic SQG equation, where the lower bound matches the upper bound on a certain closure of the weak-strong uniqueness class for the deterministic forced SQG equation. Furthermore, we show that the energy equality for the deterministic SQG equation holds on arbitrary time-reversible subsets of the domain, where we match the upper bound and the lower bound. Conversely, the violation of the deterministic energy equality breaks the lower bound of large deviations. These results extend the existing techniques in Gess, Heydecker, and the second author \cite{arXiv:2311.02223} to generalized Sobolev spaces with negative indices. Finally, we provide an analysis of the restricted quasi-potential and prove a conditional equivalence compared to the rate function of large deviations for the Gaussian distribution. This suggests a potential connection between non-Gaussian large deviations in equilibrium for the stochastic SQG equation and the open problem regarding the uniqueness of the deterministic SQG equation., Comment: 42 pages

Higher order fluctuation expansions for stochastic heat equations (SHE) with nonlinear, non-conservative and conservative noise are obtained. These Edgeworth-type expansions describe the asymptotic behavior of solutions in suitable joint scaling regimes of small noise intensity and diverging singularity. The results include both the case of the SHE with regular and irregular diffusion coefficients. In particular, this includes the correlated Dawson-Watanabe and Dean-Kawasaki SPDEs, as well as SPDEs corresponding to the Fleming-Viot and symmetric simple exclusion processes., Comment: 51 pages

The dynamical large deviations principle for the three-dimensional incompressible Landau-Lifschitz-Navier-Stokes equations is shown, in the joint scaling regime of vanishing noise intensity and correlation length. This proves the consistency of the large deviations in lattice gas models \cite{QY}, with Landau-Lifschitz fluctuating hydrodynamics \cite{LL87}. Secondly, in the course of the proof, we unveil a novel relation between the validity of the deterministic energy equality for the deterministic forced Navier-Stokes equations and matching large deviations upper and lower bounds. In particular, we conclude that time-reversible uniqueness to the forced Navier-Stokes equations implies the validity of the energy equality, thus generalising the classical Lions-Ladyzhenskaya result. Thirdly, we prove that no non-trivial large deviations result can be true for local-in-time strong solutions., Comment: Updated and shortened version

Inspired by [Fehrman, Gess; Invent. Math., 2023], we provide a fine analysis of the McKean-Vlasov PDE with singular interactions and drift terms of square root form. As the corresponding skeleton equation of Dean-Kawasaki equation with singular interactions (a stochastic, conservative PDE), it determines the rate function of small noise large deviations. By imposing Ladyzhenskaya-Prodi-Serrin type conditions on the interaction kernel, we establish the large deviations in the framework of stochastic renormalized kinetic solution, when the intensity and the correlation of the noise are simultaneously sent to $0$ under a suitable scaling. This result contributes to demonstrating the consistency between the macroscopic fluctuation theory associated with singular interacting mean-field systems and fluctuating hydrodynamics related to the Dean-Kawasaki equation. As an application, we also obtain large deviations for the other stochastic conservative PDE called fluctuating Ising-Kac-Kawasaki dynamics. It is of great importance in exploring fluctuations of Kawasaki dynamical Ising-Kac model, since they formally exhibits the same key features in terms of Gaussian fluctuations, large deviations, and scaling limits near criticality., Comment: In the new version, we have changed the research perspective and added an application to Kawasaki dynamical Ising-Kac model

Inspired by [Fehrman, Gess; Invent. Math., 2023] and [Fehrman, Gess; Arch. Ration. Mech. Anal., 2024], we consider the Dean-Kawasaki equation with singular interactions and correlated noise which can be viewed as fluctuating mean-field limits. By imposing the Ladyzhenskaya-Prodi-Serrin condition on the interaction kernel, the existence of probabilistic weak renormalized kinetic solutions is established. Further, under an additional integrability assumption on the divergence of the interaction kernel, a kinetic formulation approach is applied to derive pathwise uniqueness, leading to the strong well-posedness of the equation. As an application, we obtain the well-posedness of a conservative stochastic partial differential equations known as fluctuating Ising-Kac-Kawasaki dynamics, which paves a step on the conjecture concerning nonlinear fluctuations of Kawasaki dynamics proposed by [Giacomin, Lebowitz, Presutti; Math. Surveys Monogr., 1999]., Comment: A new section has been added, and the main result can be applied to establish the well-posedness of a conservative SPDE known as fluctuating Ising-Kac-Kawasaki dynamics

We establish a central limit theorem and prove a moderate deviation principle for stochastic scalar conservation laws. Due to the lack of viscous term, this is done in the framework of kinetic solution. The weak convergence method and doubling variables method play a key role., Comment: Published at https://doi.org/10.1016/j.jmaa.2022.126445 in the Journal of Mathematical Analysis and Applications