Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients

Yong Liu; Bin Tang; Rangrang Zhang

Home/Archive/arXiv

arXiv · 2604.22461

Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients

Yong Liu·Bin Tang·Rangrang Zhang

View original source ↗Read source document ↗Export metadata ↓

Abstract

We establish the well-posedness of stationary solutions for a class of SPDEs with locally monotone coefficients, and prove the Freidlin--Wentzell large deviation principle (LDP) for these stationary solutions. The LDP for the associated invariant measures then follows via the contraction principle, avoiding the need to construct the quasi-potential and verify the Dembo--Zeitouni uniform LDP over bounded sets. By working directly with stationary solutions, we bypass these technical difficulties, thereby providing a more general and flexible framework that is adapted to additive noise, multiplicative noise, and transport-type noise. As applications, our results cover a range of SPDEs, including the stochastic reaction-diffusion equations, stochastic 1D viscous Burgers equation, stochastic 2D Navier--Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 3D hyper-dissipative Navier--Stokes equations.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yong Liu, Bin Tang, Rangrang Zhang. 2026-04-24. Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients. https://arxiv.org/abs/2604.22461

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Generalized Edgeworth expansions for integer-valued additive functionals of uniformly elliptic Markov chains

We obtain asymptotic expansions for probabilities $\bbP(S_N=k)$ of partial sums of uniformly bounded integer-valued functionals $\DS S_N=\sum_{n=1}^N f_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansions involve products of polynomials and trigonometric polynomials, and they hold without additional assumptions. As an application of the explicit formulas of the trigonometric polynomials, we relate existence of the standard Edgeworth expansions of order $r$ to the rate of equidistributions of $S_N$ modulo $m$ for small positive integers $m.$

math.PR↗

arXiv2026

Permutations from Random Walk

Xavier and Yushi run a "random race" as follows. An atomless probability distribution $μ$ on the real line is chosen. The runners begin at zero. At time $i$ Xavier draws $\mathbf{X}_i$ from $μ$ and advances that distance, while Yushi advances by an independent drawing $\mathbf{Y}_i$. After $n$ such moves, what is the probability that Yushi led all the way? That the answer (namely, $4^{-n}\binom{2n}{n}$) is independent of $μ$ follows from a classical theorem of Darling, stating that for symmetric atomless increments, the distribution of each individual rank in the permutation obtained by ranking the partial sums is independent of the step law. We give a self-contained proof and extend the result to the permutations generated by partial sums of uniformly random signed permutations of any fixed, finite, generic set of reals. For atomless increments with mean zero and finite variance, without assuming symmetry, we show that random-walk permutations approach a random object that we call the "Wiener permuton," whose expected pattern densities equal the probabilities of the corresponding permutations generated by finite random walks with centered Laplace increments. Finally, we exhibit an infinite family of constructions whose limiting permutons interpolate between the Wiener permuton and the recursive separable permuton; each has the same intensity permuton, providing a single two-dimensional extension of the classical arcsine law for all of them.

math.PR↗

arXiv2026

On the uniqueness of quasi-stationary distributions for population models with spatial structure

Subcritical population processes are attracted to extinction and do not have non-trivial stationary distributions, which prompts the study of quasi-stationary distributions (QSDs) instead. In contrast to what generally happens for stationary distributions, QSDs may not be unique, even under irreducibility conditions. The general conditions for uniqueness of QSDs are not always easy to check. For the branching process, besides the quasi-limiting distribution there are many other QSDs. In this paper, we investigate whether adding little extra information to the continuous-time branching process is enough to obtain uniqueness. We consider the branching process with genealogy and branching random walks, and show that they have a unique QSD.

math.PR↗