Strong and mean square convergence of linear drift-implicit Theta methods Y to X governed by systems of non-linear stochastic differential equations (SDEs) with monotone drift and degenerate state-dependent noise driven by standard Wiener processes are studied. For this purpose, we begin with collecting basic properties of strong solutions X and systematically investigate uniform boundedness and asymptotic stability of moments (Lyapunov-type functionals) of their both solutions X and approximations Y. Under some polynomial growth conditions, the one-sided Lipschitz-continuity of the drift with non-increasing nonlinearities and uniform Lipschitz-continuity of diffusion coefficients, we can verify uniform boundedness of moments, ergodicity, exponential p-th mean stability, contractivity and rates of mean square convergence of Y to X with implicitness parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\theta \ge 0.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>≥</mo> <mn>0.5</mn> </mrow> </math></EquationSource> </InlineEquation>, even on infinite time-intervals. While applying a fairly general convergence theorem (2006) on Hilbert spaces, we justify the global rate <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$r_g = 0.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mi>g</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </math></EquationSource> </InlineEquation> of mean square convergence of those methods for systems with state-dependent noise on infinite time-horizons. Three simple examples with power-law non-linearities support our findings (among them, a simplified 1D Ginzburg–Landau model from field theory, systems with dissipative power-laws and non-linear stochastic heat equations), demonstrating some potential applicability of our findings. Our Theta methods with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\theta \ge 0.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>≥</mo> <mn>0.5</mn> </mrow> </math></EquationSource> </InlineEquation> can control the long-term behavior of numerical dynamics by their moment-dissipativity and are fairly easily implementable.

Long-term convergence and stability of linear-implicit theta methods for moment-dissipative stochastic differential equations with multiplicative noise

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2026年5月25日

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Henri Schurz

摘要

Strong and mean square convergence of linear drift-implicit Theta methods Y to X governed by systems of non-linear stochastic differential equations (SDEs) with monotone drift and degenerate state-dependent noise driven by standard Wiener processes are studied. For this purpose, we begin with collecting basic properties of strong solutions X and systematically investigate uniform boundedness and asymptotic stability of moments (Lyapunov-type functionals) of their both solutions X and approximations Y. Under some polynomial growth conditions, the one-sided Lipschitz-continuity of the drift with non-increasing nonlinearities and uniform Lipschitz-continuity of diffusion coefficients, we can verify uniform boundedness of moments, ergodicity, exponential p-th mean stability, contractivity and rates of mean square convergence of Y to X with implicitness parameters $\theta \ge 0.5$ $θ \geq 0.5$ , even on infinite time-intervals. While applying a fairly general convergence theorem (2006) on Hilbert spaces, we justify the global rate $r_g = 0.5$ $r_{g} = 0.5$ of mean square convergence of those methods for systems with state-dependent noise on infinite time-horizons. Three simple examples with power-law non-linearities support our findings (among them, a simplified 1D Ginzburg–Landau model from field theory, systems with dissipative power-laws and non-linear stochastic heat equations), demonstrating some potential applicability of our findings. Our Theta methods with $\theta \ge 0.5$ $θ \geq 0.5$ can control the long-term behavior of numerical dynamics by their moment-dissipativity and are fairly easily implementable.