In this section we collect some notions, conceptions, and lemmas on stochastic process and linear evolution system which will be used throughout the whole paper.
Let be a probability space on which an increasing and right continuous family of complete sub--algebras of is defined. and are two separable Hilbert space. Suppose that is a given -valued Wiener Process with a finite trace nuclear covariance operator . Let , , be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over . Set
where are nonnegative real numbers, and is a complete orthonormal basis in . Let be an operator defined by with finite trace . Then the above -valued stochastic process is called a -Wiener process.
Let and define
If , then is called a -Hilbert-Schmidt operator, and let denote, the space of all -Hilbert-Schmidt operators .
In the next section the following lemma (see [1, Lemma 7.2]) plays an important role.
For any and for arbitrary -valued predictable process , , one has
for some constant .
Now we turn to state some notations and basic facts from the theory of linear evolution system.
Throughout this paper, is a family of linear operators defined on Hilbert space , and for this family we always impose on the following restrictions.
The domain of is dense in and independent of ; is closed linear operator.
For each , the resolvent exists for all with Re and there exists so that .
There exists and such that for all .
Under these assumptions, the family generates a unique linear evolution system, or called linear evolution operators , and there exists a family of bounded linear operators with such that has the representation
where denotes the analytic semigroup having infinitesimal generator (note that Assumption guarantees that generates an analytic semigroup on ).
For the linear evolution system , the following properties are well known:
(a), the space of bounded linear transformations on , whenever and maps into as . For each , the mapping is continuous jointly in and ;
(b) for ;
(d) for .
We also have the following inequalities:
Furthermore, Assumptions imply that for each the integral
exists for each . The operator defined by (2.6) is a bounded linear operator and yields . Thus, we can define the fractional power as
which is a closed linear operator with dense in and for . becomes a Banach space endowed with the norm , which is denoted by .
The following estimates and Lemma 2.3 are from ([23, Part II]):
for and , and
for some , where and indicate their dependence on the constants , .
Assume that hold. If , , , then, for any , ,
For more details about the theory of linear evolution system, operator semigroups, and fraction powers of operators, we can refer to [23–25].
In the sequel, we denote for brevity that for some , and , the space of all continuous functions from into . Suppose that , , is a continuous -adapted, -valued stochastic process, we can associate with another process , , by setting , . Then we say that the process is generated by the process . Let , , denote the space of all -measurable functions which belong to ; that is, , , is the space of all -measurable -valued functions with the norm .
Now we end this section by stating the following result which is fundamental to the work of this note and can be proved by the similar method as that of [1, Proposition 4.15].
Let , be a predictable, -adapted process. If , , for arbitrary , and , , then there holds