Abstract
We consider the following stochastic fractional differential equation D-C(0+)alpha,rho phi(t) = kappa theta(t, phi(t)) (w)over dot (t), 0 < t <= T, where phi(0) = phi 0 is the initial function, D-C(0+)alpha,rho is the Caputo-Katugampola fractional differential operator of orders 0 < alpha <= 1, rho > 0, the function theta : [0,T] x R -> R is Lipschitz continuous on the second variable, (w)over dot(t) denotes the generalized derivative of the Wiener process w(t) and kappa > 0 represents the noise level. The main result of the paper focuses on the energy growth bound and the asymptotic behaviour of the random solution. Furthermore, we employ Banach fixed point theorem to establish the existence and uniqueness result of the mild solution.