Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation

Abstract

We study a class of conformable time-fractional stochastic equation T α , t a u ( x , t ) = σ ( u ( x , t ) ) W ˙ t , x ∈ R , t ∈ [ a , T ] , T < ∞ , 0 < α < 1 . The initial condition u ( x , 0 ) = u 0 ( x ) , x ∈ R is a non-random function assumed to be non-negative and bounded, T α , t a is a conformable time-fractional derivative, σ : R → R is Lipschitz continuous and W ˙ t a generalized derivative of Wiener process. Some precise condition for the existence and uniqueness of a solution of the class of equation is given and we also give an upper bound estimate on the growth moment of the solution. Unlike the growth moment of stochastic fractional heat equation with Riemann–Liouville or Caputo–Dzhrbashyan fractional derivative which grows in time like t c 1 exp ( c 2 t ) , c 1 , c 2 > 0 ; our result also shows that the energy of the solution (the second moment) grows exponentially in time for t ∈ [ a , T ] , T < ∞ but with at most c 1 exp ( c 2 ( t − a ) 2 α − 1 ) for some constants c 1 , and c 2 .