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Xia Pan

Publications and source records attributed to Xia Pan.

5 recordsLinked to original sources
arXiv2026

Well-posedness, mean attractors and invariant measures of stochastic discrete long-wave-short-wave resonance equations driven by locally Lipschitz nonlinear noise

This paper is devoted to investigating the random dynamics of stochastic discrete long-wave-short-wave resonance equations, which are characterized by the following features: $(1)$ the equations contain locally Lipschitz nonlinear coupling terms $u_mv_m$ and $(B(|u(t)|^2))_m$ for $m\in \mathbb{Z}$; $(2)$ the nonlinear coefficients of noises satisfy local Lipschitz conditions; and $(3)$ the system couples real and complex equations and is infinite-dimensional. These inherent structural properties prevent the analysis from being carried out in a standard Bochner product space of the same order and make it difficult to directly verify the tightness of the distribution family of solutions. To address these challenges, we adopt a higher-order Bochner product space $L^4(\Omega,\ell_c^2)\times L^2(\Omega,\ell^2)$ as the phase space and employ the technique of uniform tail-end estimates. The main results include: establishing the global well-posedness of the nonautonomous stochastic discrete long-wave-short-wave resonance equations driven by nonlinear noise in $L^4(\Omega,\ell_c^2)\times L^2(\Omega,\ell^2)$; based on this, defining the mean random dynamical system and proving the existence and uniqueness of weak $\mathscr{D}$-pullback mean random attractors. When the external forcing terms are independent of time and sample, we investigate the existence of invariant measures for the corresponding autonomous system and examine the limiting behavior of the invariant measure as the noise intensity tends to zero.

math.PR↗
arXiv2017

On the convergence of the time average for skew-product structure and multiple ergodic system

In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation and observation function are neglected in some measure-theoretical sense. The theorems extend the classical results which have been established for continuous dynamical systems or continuous observation functions. Meanwhile, on the torus $\mathbb{T}^{d}$ with special rotation, we prove the pointwise convergence of multiple ergodic average $\disp \f 1 N \sum_{n=0}^{N-1} f_{1}(R_α^{n}x)f_{2}(R_α^{2n}x)$ in $\mathbb{T}^{d}$.

math.DS↗
arXiv2017

Ergodic behaviour of nonconventional ergodic averages for commuting transformations

Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to the space average, of diagonal measures is obtained and we give different result according to the classification of transformations. Additionally, on the torus with special rotation. we can not only get the convergence in T.Tao's paper for every point in Td, but also get a beautiful result for ergodic behaviour.

math.DS↗