In the previous work by the second author, it was shown that the polynomial Brownian noise may suppress the potential explosion of the nonlinear system without the linear growth condition or the one-sided linear growth condition and the linear Brownian noise may stabilise this suppressed system. This article is a continuation of our previous paper and considers the asymptotic properties of the suppressed system. These asymptotic properties show that the suppressed system by polynomial Brownian noise will grow with at most polynomial speed.