Measure theory and square-mean pseudo almost periodic and automorphic process: application to stochastic evolution equations
This paper deals with the existence and uniqueness of square-mean \(\mu\)-pseudo almost periodic and automorphic mild solution to the following nonlinear stochastic evolution equation in a Hilbert space \(H\): \[ dx(t)=Ax(t)dt+h(t,x(t))dt+\theta(t,x(t))dW(t) \text{ for all } t\in \mathbb{R}\quad (1) \] where \(A:D(A)\subset H\) is the infinitesimal generator of a \(C_0\)-semigroup \((T(t))_{t\geq 0}\) on \(H\), \(h,\theta: \,\mathbb{R}\times L^2(P,H)\to L^2(P,H)\) are two stochastic processes and \(W(t)\) is a two-side and standard one-dimensional Brownian motion defined on the filtered probability space \((\Omega,\mathcal{F},P,\mathcal{F}_t)\) with \(\mathcal{F}_t=\sigma\{W(u)-W(v)|\, u,v\leq t\}.\) The authors start by proving that the linear stochastic evolution equation associated to (1) has square-mean \(\mu\)-pseudo almost periodic and automorphic mild solution. Then by means of Banach fixed point theorem they obtain the same results for (1). The proof is achieved by means of a new concept of square-mean pseudo almost periodic and automorphic process using the measure theory. Actually, the spaces of \(\mu\)-pseudo almost periodic and automorphic processes, is defined by the authors by means of the \(\mu\)-ergodic process. Many interesting results like completeness and composition theorems in those space are also established. An example is also given to illustrate the results.
scientific article; zbMATH DE number 6835868
scientific article