Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition

In this paper, we investigate the problem of suppression explosive solutions by noise for nonlinear deterministic differential system. Given a deterministic differential system (y) over dot(t) = f(y(t), t) with coefficients satisfying a more general one-sided polynomial growth condition, we introduce Brownian noise feedback and therefore stochastically perturb this system into the nonlinear stochastic differential system dx(t) = f(x(t), t)dt + vertical bar x(t)vertical bar(beta) Sigma x(t)dB(t). We show that appropriate beta, Sigma guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic systems may explode in a finite time. Under some weaker conditions, we reveal that the single noise vertical bar x(t)vertical bar(beta) Sigma x(t)dB(t) can also make almost every path of the solution of corresponding stochastically perturbed system grow at most polynomially. (C) 2012 Elsevier Ltd. All rights reserved.