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<p>Criteria for local asymptotic <i>p</i>-th mean stability of trivial equilibria of scalar, nonlinear stochastic differential equations (SDEs) with infinite memory, driven by standard Wiener processes, are derived and verified by Banach’s contraction mapping principle (CMP) on the weak Banach space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {W}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">W</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> of adapted, continuous time stochastic processes with finite <i>p</i>-th moments. For this purpose, we carry over basic ideas of T. Burton (2006) for ODEs with memory to SDEs with infinite memory. The verified criteria for existence and stability of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-solutions heavily depend on diverse system-parameters. Fairly simple examples with exponentially and polynomially decaying memory support our efficient findings. The method of successive iterations by Banach-Caccioppoli is able to provide approximations to the locally p-th mean stable solutions, including a criterion to control its <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-error for any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(p \ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Local P-th Mean Stability of Scalar SDEs with Infinite Memory By Contraction Mapping Principle on Weak Banach Spaces

  • 期刊论文

  • 发表日期：

    2026年2月11日

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  • Henri Schurz

摘要

Criteria for local asymptotic p-th mean stability of trivial equilibria of scalar, nonlinear stochastic differential equations (SDEs) with infinite memory, driven by standard Wiener processes, are derived and verified by Banach’s contraction mapping principle (CMP) on the weak Banach space \(\mathbb {W}_p\) W p of adapted, continuous time stochastic processes with finite p-th moments. For this purpose, we carry over basic ideas of T. Burton (2006) for ODEs with memory to SDEs with infinite memory. The verified criteria for existence and stability of \(L^p\) L p -solutions heavily depend on diverse system-parameters. Fairly simple examples with exponentially and polynomially decaying memory support our efficient findings. The method of successive iterations by Banach-Caccioppoli is able to provide approximations to the locally p-th mean stable solutions, including a criterion to control its \(L^p\) L p -error for any \(p \ge 1\) p ≥ 1 .