Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.01725

Uniqueness of SRB measures for stochastic differential equations

Zeng Lian·Jianhua Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

For a dissipation stochastic differential equation (abbreviated as SDE), exponential ergodicity has been investigated a lot in a wide range of settings. In this paper, we prove that the Markovian measure $μ$ determined by the unique stationary measure is also the unique invariant measure which satisfies the entropy formula. Consequently, the existence of an SRB measure is completely determined by the sign of the sample top Lyapunov exponent of $μ$: if $χ_μ^1>0$, then $μ$ is the unique SRB measure, whereas if $χ_μ^1\leq0$, then the SDE admits no SRB measure. As an application, we establish a dichotomy of SRB measures for the 2D additive-noise stochastic Hopf normal form within a shear parameter domain.

Explore related subjects

math.DSmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zeng Lian, Jianhua Zhang. 2026-11-01. Uniqueness of SRB measures for stochastic differential equations. https://arxiv.org/abs/2611.01725

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

KEEP EXPLORING

Related papers

arXiv2026

Hilbert's 16th problem through Morse theory

We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound turns out to be a polynomial of degree $4$ in $n$. More specifically, if $H(n)$ indicates the maximum number of limit cycles among planar, differential, polynomial systems of degree $n$, then \begin{gather} H(n)\le \dfrac52 n^4-\dfrac{23}2 n^3+ \dfrac{43}2n^2-\dfrac{37}2n+7\,\,\,\, \mbox{if $n$ is even, and} \nonumber H(n)\le \dfrac52 n^4-\dfrac{23}2 n^3+ \dfrac{41}2n^2-\dfrac{33}2n+6\,\,\,\, \mbox{if $n$ is odd}.\nonumber \end{gather} For quadratic systems, we find $H(2)=4$. Our proof is entirely variational and utilizes in a fundamental way tools and facts from Morse theory, to the point that no particular expertise in dynamical systems is necessary or required. It is particularly interesting to compare our results with the ones for the same problem recently announced and made public by OpenAI as part of the solution of hundreds of important mathematical problems.

math.DS↗
arXiv2026

On the Dynamics of Invariant Graphs for Dissipative Twist Maps

We study the persistence and internal dynamics of invariant graphs for dissipative twist maps. Explicit $C^1$ persistence and parameter estimates allow every real rotation number to be realized by varying either translation parameter, while arbitrarily small zero-mean analytic perturbations at fixed irrational horizontal parameters can yield analytic graphs with dynamics conjugate to rational rotations. With a fixed Brjuno horizontal parameter and fixed-strip analyticity, normally attracting $C^r$ graphs persist under perturbations of size $(1-λ)^γ$, for any $γ>0$ and prescribed finite $r$, as $λ\to1^-$. At the critical Lipschitz bound, the invariant graphs can carry Denjoy dynamics and have countably many nondifferentiable points, with perturbations smooth outside a null Cantor set and tending to zero in every Hölder norm of order less than one.

math.DS↗
arXiv2026

Non-stationary normal coordinates on neighborhoods of Pesin stable manifolds

We construct non-stationary normal coordinates in a neighborhood of Pesin stable manifolds \cite{Pesin}. This construction is a natural extension, via jets in the normal directions, of the non-stationary normal coordinates on stable manifolds introduced by Guysinsky and Katok. We emphasize that this extension provides a useful framework for describing the fractal geometric structures of stable and unstable foliations in smooth dynamical systems.

math.DS↗