On Caputo–Katugampola Fractional Stochastic Differential Equation

We consider the following stochastic fractional differential equation C D 0 + α , ρ φ ( t ) = κ ϑ ( t , φ ( t ) ) w ˙ ( t ) , 0 < t ≤ T , where φ ( 0 ) = φ 0 is the initial function, C D 0 + α , ρ is the Caputo–Katugampola fractional differential operator of orders 0 < α ≤ 1 , ρ > 0 , the function ϑ : [ 0 , T ] × R → R is Lipschitz continuous on the second variable, w ˙ ( t ) denotes the generalized derivative of the Wiener process w ( t ) and κ > 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.