标题： 随机泛型方程多项式渐近行为的充分条件 显示英文标题

标题： Sufficient Conditions for Polynomial Asymptotic Behaviour of the Stochastic Pantograph Equation

摘要： 本文研究了带有乘性噪声的随机泛函微分方程解的渐近增长和衰减性质。 我们给出了参数的充分条件，使得解在$p$-阶矩和几乎处处意义上以多项式速率增长。 在更强的条件下，解在$p$-阶矩和几乎处处意义上以多项式速率衰减到零。 当无法获得多项式界时，我们展示了对于另一组参数，可以在$p$-阶矩和几乎处处意义上得到解的指数增长界。 对于具有多个延迟的泛函微分方程以及一般的有限维方程，建立了类似的结果。 显示英文摘要

摘要： This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in $p$-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in $p$-th mean and in the almost sure sense. When polynomial bounds cannot be achieved, we show for a different set of parameters that exponential growth bounds of solutions in $p$-th mean and an almost sure sense can be obtained. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations.