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<p>This paper addresses the tempered dichotomy for dynamical systems, generated by the random linear infinite-dimensional equations over measure-preserving dynamical systems <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\Omega , {\mathcal {F}}, P, \theta )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mi mathvariant="script">F</mi> <mo>,</mo> <mi>P</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> on the half-line <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\([0,\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We prove that the solutions of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\dot{u}=A(\theta _t)u+f, t\ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mo>,</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> belong to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\mathcal {L}}^\infty _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mn>1</mn> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(f\in {\mathcal {L}}^p_K\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msubsup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mi>K</mi> <mi>p</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> if and only if the homogenous system <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\dot{u}=A(\theta _t)u\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mi>t</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> has a tempered exponential dichotomy on the half-line <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathbb {R}_+\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </math></EquationSource> </InlineEquation>.</p>

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Tempered dichotomy of random dynamical systems on the half-line with unbounded generators

  • 期刊论文

  • 发表日期：

    2026年8月1日

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  • Ngo Thi Thanh Nga,
  • Nguyen Thi Phuong Mai,
  • Nguyen Huu Du

摘要

This paper addresses the tempered dichotomy for dynamical systems, generated by the random linear infinite-dimensional equations over measure-preserving dynamical systems \((\Omega , {\mathcal {F}}, P, \theta )\) ( Ω , F , P , θ ) on the half-line \([0,\infty )\) [ 0 , ∞ ) . We prove that the solutions of \(\dot{u}=A(\theta _t)u+f, t\ge 0\) u ˙ = A ( θ t ) u + f , t ≥ 0 belong to \({\mathcal {L}}^\infty _1\) L 1 ∞ for any \(f\in {\mathcal {L}}^p_K\) f ∈ L K p if and only if the homogenous system \(\dot{u}=A(\theta _t)u\) u ˙ = A ( θ t ) u has a tempered exponential dichotomy on the half-line \(\mathbb {R}_+\) R + .