<p>This paper is concerned with the existence of mean random attractors and invariant measures for fractional stochastic lattice systems driven by nonlinear noise. We firstly establish the global existence and uniqueness of solutions, and then prove the existence and uniqueness of weak pullback mean random attractors of the fractional stochastic lattice systems in the Bochner space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1442_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{2}(\Omega ,\ell ^{2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Under certain conditions, we establish the tightness of a family of distributions of solutions by using the uniform estimates on the tails of solutions and then show the existence of invariant measures of the system. We also discuss the limiting behavior of invariant measures of fractional stochastic lattice systems driven by nonlinear noise as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1442_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Finally, under further assumptions on the nonlinear terms, we show that the system has a unique, ergodic, mixing, and stable invariant probability measure in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1442_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell ^{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>.</p>

Mean Attractors and Invariant Measures for Fractional Stochastic Lattice Systems Driven by Nonlinear Noise

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发表日期：

2025年8月28日

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Pengyu Chen,
Ailin Bai,
Xuping Zhang

摘要

This paper is concerned with the existence of mean random attractors and invariant measures for fractional stochastic lattice systems driven by nonlinear noise. We firstly establish the global existence and uniqueness of solutions, and then prove the existence and uniqueness of weak pullback mean random attractors of the fractional stochastic lattice systems in the Bochner space $L^{2}(\Omega ,\ell ^{2})$ $L^{2} (Ω, ℓ^{2})$ . Under certain conditions, we establish the tightness of a family of distributions of solutions by using the uniform estimates on the tails of solutions and then show the existence of invariant measures of the system. We also discuss the limiting behavior of invariant measures of fractional stochastic lattice systems driven by nonlinear noise as $\epsilon \rightarrow 0$ $ϵ \to 0$ . Finally, under further assumptions on the nonlinear terms, we show that the system has a unique, ergodic, mixing, and stable invariant probability measure in $\ell ^{2}$ $ℓ^{2}$ .