Two Classes Of Stochastic Models Driven By G-brownian Motion

This paper aim to study two classes of stochastic models driven by G-Brownian motion (G-BM, in short). The paper is divided into two sections. In the first section,we use the Banach contraction mapping principle to prove the existence and uniqueness of square-mean pseudo almost automorphic mild solutions of the following stochastic evolution equations driven by G-Brownian motion(G-SEE, in short):dXt=A(t)Xtdt+F(t,Xt)dt+G(t,Xt)d<B>t+H(t,Xt)dBt,t∈R,(1)Here, A(t) : D(A(t)) (?) LG2(Ω)→LG2(Ω) satisfies the so-called Acquistapace-Terrani conditions (for more details, see Chang et al. [6]), moreover, it is a set of densely defined and linear closed operators, Bt is a two-sided standard one-dimensional G-Brownian motion, F, G and H : R × LG2(Ω) → LG2() are jointly continuous functions.In the second section, we are devoted to study the stability of a class of impulsive stochastic Cohen-Grossberg neural networks driven by G-BM with the following form:where x(t) is an Rn-valued stochastic process describing the state variables at time t,which corresponds to the n neurons of the network. Furthermore, H(x(t)) is the am-plification function at timet, C(t,x(t)) represents appropriately behaved function which is related to t and on the state process x(t), A(t) denotes the strength of the neuron interconnections in the networks at time t, F(x(t)) denotes activation function at time t,φ(t,x(t))is a continuous function,σ(t,x(t))is a diffusion coefficient,Bt is an n-dimensional G-Brownian motion, and (B, B)t=(<Bi, Bj>t)i,jn=i is the mutual variation process of Bt. At the same -time, denote fixed moments tk the impulsive moments, which satisfied t1<t2<t3 <… and lim k→∞ tk = ∞. Moreover, pk(x(t-)) and qk(x(t-))represent impulsive perturbations at time tk,where x(t-) is the left limit of x(t).By means of the mathematical induction method and G-Lyapunov function, we get the trivial solution of the system (2) is p-th moment stability with a decay function λ(t)of orderr η and quasi sure stability with a decay function λ(t) of order n-ρ/p.