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 is zero, like the forcing term when L is infinite and inherits properties of both the forcing term and underlying differential equation for finite positive values of L. Our approach carries over in a natural way to stochastic equations with additive noise, and we treat the illustrative cases of Brownian and Levy noise.

中文翻译：

---

扰动非线性Volterra方程的增长和波动

我们对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制定了精确的界限。对于较大的状态值，该方程是次线性的，因为状态相关性相对于线性函数可以忽略不计。如果强制项的适当泛函在无穷大处具有极限 L，则微分方程的解在 L 为零时表现得像潜在的非受迫方程一样渐近，就像在 L 无穷大时的强制项一样，并且继承了强制项和L 的有限正值的基本微分方程。我们的方法以自然的方式延续到具有加性噪声的随机方程，并且我们处理了布朗和列维噪声的说明性案例。