扰动非线性Volterra方程的增长性和涨落
Growth and fluctuation in perturbed nonlinear Volterra equations
摘要：
我们给出了受外力扰动的确定性和随机非线性Volterra方程的无界解的增长率和波动大小的精确界。该方程是次线性的状态的大值，在这个意义上的状态依赖性是可以忽略的相对于线性函数。如果强迫项的适当泛函在无穷远处有一个极限$L$，则当$L=0$时微分方程的解渐近地表现为基本的非强迫方程，当$L=+\infty$时，微分方程的解渐近地表现为强迫项，并且对于$L\in (0,\infty)$的值，继承了强迫项和基本微分方程的性质。我们的方法进行在一个自然的方式随机方程加性噪声，我们处理布朗和Lévy噪声的说明性情况。
英文摘要：
We develop precise bounds on the growth rates and fluctuation sizes of unbounded solutions of deterministic and stochastic nonlinear Volterra equations perturbed by external forces. The equation is sublinear for large values of the state, in the sense that the state--dependence is negligible relative to linear functions. If an appropriate functional of the forcing term has a limit $L$ at infinity, the solution of the differential equation behaves asymptotically like the underlying unforced equation when $L=0$, like the forcing term when $L=+\infty$, and inherits properties of both the forcing term and underlying differential equation for values of $L\in (0,\infty)$. Our approach carries over in a natural way to stochastic equations with additive noise and we treat the illustrative cases of Brownian and Lévy noise.
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