A theorem on discrete-time scale in the qualitative theory of stochastic evolution equation.

Preda, Ciprian Ion ; Mosincat, Razvan Octavian ; Bobitan, Nicolae 等

1. INTRODUCTION

The problem of existence of semiflows for stochastic evolution equations is a non-trivial one, mainly due to the well-known fact that finite-dimensional methods for setting (even continuous) stochastic flows break down in the infinite-dimensional context of stochastic evolution equations. In particular, Kolmogorov's continuity theorem fails for random fields parametrized by infinite-dimensional Hilbert spaces.

For the case of linear stochastic evolution equations with finite-dimensional noise, a stochastic semiflow (i.e. a random evolution operator) was obtained by Bensoussan & Flandoli (1995). Recently, Mohammed, Zhang & Zhao (2006) detect the existence of stochastic cocycles generated by mild solutions of a large class of semilinear stochastic evolution equations. Continuing their work, we obtain a theorem on discrete-time scale that addresses concerns with respect to the asymptotic behavior of the stochastic cocycles.

Our approach follows the well-established line of results initiated by Perron (1930) and Li (1934) for the deterministic case, which concerns the problem of stability of a deterministic system x'(t) = A(t)x(t) and its connection with the existence of bounded solutions of the inhomogeneous system x'(t) = A(t)x(t)+ f(t).

2. STOCHASTIC COCYCLES

Let (X, [parallel]*[parallel]) be a Banach space and let B(X) be the Banach algebra of all linear and bounded operators acting from X into X. By ([OMEGA], F, [{[F.sub.t]}.sub.t[greater than or equal to]0], P) we denote a complete filtered probability space (i.e. ([OMEGA], F, P) is a complete probability space, [{[F.sub.t]}.sub.t[greater than or equal to]0] is an increasing families of [sigma]-algebras, [F.sub.0] contains all P-null sets of F and [F.sub.t] = [[intersection].sub.s[greater than or equal to]t][F.sub.s] for every t [greater than or equal to] 0).

Definition 2.1. A stochastic semiflow on [OMEGA] is a random field [phi]: [R.sub.+] X [OMEGA] [right arrow] [OMEGA] such that

([s.sub.1]) [phi](0, [omega]) = [omega], for all [omega] [member of] [OMEGA];

([s.sub.2]) [phi](t + s, [omega]) = [phi](t, [phi](s, [omega])), for all t, s [greater than or equal to] 0, [omega] [member of] [OMEGA].

Example 2.2. Let X be a separable real Hilbert space and consider [OMEGA] as the space (with the compact open topology) of all continuous paths [omega]: [R.sub.+] [right arrow] X with [omega](0) = 0. Let [F.sub.t] be the [sigma]-algebra generated by the set {[omega] - [omega](u): u [less than or equal to] t}, for every t [greater than or equal to] 0, and let F be the Borel [sigma]-algebra on [OMEGA]. If P is the Wiener measure on [OMEGA], then the quadruplet ([OMEGA], F, [{[F.sub.t]}.sub.t[greater than or equal to]0], P) is the canonical complete filtered probability space with the Wiener motion W(t,[omega]) = [omega] (t) for all t [greater than or equal to] 0 and [omega] [member of] [OMEGA].

The map

[phi]: [R.sub.+] X [OMEGA] [right arrow] [OMEGA], [phi](t, [omega])(s) = [omega](t + s) - [omega](t) (1)

defines a stochastic semiflow on [OMEGA].

Definition 2.3. Let [phi]: [R.sub.+] X [OMEGA] [right arrow] [OMEGA] be a stochastic semiflow on [OMEGA]. The mapping [PHI]: [R.sub.+] X [OMEGA] [right arrow] B(X) is said to be a stochastic cocycle (over the semiflow [phi]) if it satisfies ([c.sub.1]) [PHI](0, [omega]) = I (the identity on X), for all [omega] [member of] [OMEGA]; ([c.sub.2]) [PHI](t + s, [omega]) = [PHI](t, [phi](s, [omega]))[PHI](s, [omega]), for all t, s [greater than or equal to] 0, and [omega] [member of] [OMEGA].

If in addition, there exist M, [lambda] > 0 such that ([c.sub.3]) E[[parallel][PHI](t,*)x[parallel].sup.2] [less than or equal to]M[e.sup.[lambda](t-s)]E[[parallel] [PHI](s,*)x[parallel].sup.2], for all t, s [greater than or equal to] 0, and x [member of] X, then [PHI] is a stochastic cocycle with exponential growth in mean square.

Example 2.4. Consider again the complete filtered probability space introduced in Example 2.2 and let [{W(t)}.sub.t[greater than or equal to]0] be an X-valued Brownian motion with a separable covariance Hilbert space H.

As usual, 23(H, X) is the Banach space of all bounded linear operators from H into X, while [??](H, X) [subset] B(H, X) is the subspace of all Hilbert-Schmidt operators S: H [right arrow] X endowed with the norm

|S| = [([[infinity].summation over (k=1)][[parallel]S([e.sub.k])[parallel].sup.2]).sup.1/2] (2)

where [{[e.sub.k]}.sub.k[greater than or equal to]1] is a complete orthonormal system on H.

Next, consider the linear stochastic evolution equation

du(t,x,*) = Au(t,x,*)dt + Bu(t,x,*)dW(t) (3)

where A:D(A) [subset] X [right arrow] X is the infinitesimal generator of a strongly continuous semigroup [{T(t)}.sub.t[greater than or equal to]0], and B: X [right arrow] [??](H, X) is a bounded linear operator.

Assume that B can be extended to a bounded linear operator B: X [right arrow] B(X) (denoted by the same symbol B), and that the series [[SIGMA].sup.[infinity].sub.k=1] [[parallel][B.sup.2.sub.k][parallel].sub.B(X)] converge, where [B.sub.k] is the bounded linear operator on X defined by [B.sub.k](x) = (Bx)([e.sub.k]), x [member of] X, k [greater than or equal to] 1.

A mild solution of the above stochastic evolution equation is given by the family of [{[F.sub.t]}.sub.t[greater than or equal to]0]-adapted processes u(*,x,*): [R.sub.+] X [OMEGA] [right arrow] X, x [member of] X, satisfying the stochastic integral equation:

u(t,x,*) = T(t)x + [[integral].sup.t.sub.0] T(t - s)Bu(x,x,*)dW(s) (4)

Then, the mapping [PHI]: [R.sub.+] X [OMEGA] [right arrow] B(X) defined by [PHI](t, [omega])x = u(t, x, [omega]) (5)

is a stochastic cocycle over the semiflow [phi] (see Example 2.2.)

3. EXPONENTIAL STABILITY IN MEAN SQUARE OF STOCHASTIC COCYCLES

In this section, we investigate a type of asymptotic behavior of a stochastic cocycle [PHI], namely the exponential stability in mean square.

Definition 3.1. The stochastic cocycle [PHI]: [R.sub.+] X [OMEGA] [right arrow] B(X) is said to be exponentially stable in mean square if there exist two positive constants N, v such that

E[[parallel][PHI](t, *)x[parallel].sup.2] [less than or equal to] N[e.sup.- v(t-s)]E[[parallel][PHI] (s, *)x[parallel].sup.2] (6)

for all t [greater than or equal to] s [greater than or equal to] 0 and x [member of] X.

Define C as the space of all X-valued stochastic processes [alpha] with

[sup.sub.n[member of]N](E[[parallel][alpha](n)(*)[parallel].sup.2]) < [infinity]. (7)

Next, for [alpha] [member of] C set

([GAMMA][alpha])(n)(*) = [n.summation over (j=0)] [PHI] (n - k, [phi](k,*))[alpha](*), (8)

for every n [member of] N.

Obs. In what follows, we will often use E[[parallel][alpha](n)[parallel].sup.2], E[[parallel]([GAMMA][alpha])(n)[parallel].sup.2] instead of E[[parallel][alpha](n)(*)[parallel].sup.2], E[[parallel]([GAMMA][alpha])(n)(*)[parallel].sup.2], respectively.

Condition A. There exists some positive constant K such that [sup.sub.n[member of]N](E[[parallel]([GAMMA][alpha])(n)[parallel].sup.2]) [less than or equal to] K [sup.sub.n[member of]N](E[[parallel][alpha](n)[parallel].sup.2]), (9) for every stochastic process [alpha] [member of] C.

Theorem 3.2. Let [PHI] be a stochastic cocycle with uniform exponential growth in mean square satisfying Condition A. Then, [PHI] is exponentially stable in mean square.

Proof. Let x [member of] X, s [greater than or equal to] 0 and set [n.sub.0] = s + 1, where [s] denotes the largest integer less than or equal with s. Consider the stochastic process

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

We have that [sup.sub.n[member of]N](E[[parallel][alpha](n)[parallel].sup.2])=1 and for each n [greater than or equal to] [n.sub.0],

([GAMMA][alpha])(n) = [(E[[parallel][PHI]([n.sub.0],*)x[parallel].sup.2].sup.1/2] [PHI](n,*)x. (11)

By the hypothesis, we obtain that E[[parallel]([GAMMA][alpha])(n)[parallel].sup.2] [less than or equal to] K, which implies that

E[[parallel][PHI](n,*)x[parallel].sup.2] [less than or equal to] K E[[parallel][PHI]([n.sub.0],*)x[parallel].sup.2], (12)

for each n [member of] N and x [member of] X.

Now, let m [member of] N and consider the stochastic process

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Clearly, [sup.sub.n[member of]N](E[[parallel][beta](n)[parallel].sup.2])[less than or equal to] K and

([GAMMA][beta])(j) = (j + 1) [(E[[parallel][PHI]([n.sub.0],*)x[parallel].sup.2].sup.1/2] [PHI] (j,* x), (14)

for every j [greater than or equal to] [n.sub.0]. It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

which implies

E[[parallel][PHI]([n.sub.0] + m,*)x[parallel].sup.2] [less than or equal to] 2 [K.sup.3]/m + 2 E[[parallel][PHI]([n.sub.0],*)x[parallel].sup.2]. (16)

Therefore, there exist k [member of] N and [eta] [member of] (0,1) such that

E[[parallel][PHI]([n.sub.0] + k,*)x[parallel].sup.2] [less than or equal to] [eta] E[[parallel][PHI]([n.sub.0],*)x[parallel].sup.2] , (17)

for all [n.sub.0] [member of] N and x [member of] X.

Let t [greater than or equal to] 0 and take n = [t-s/k]. Then, we can write down

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Set now v := - 1/k ln [eta] and N := [M.sup.3][e.sup.[lambda](k+2)][[eta].sup.-1] to obtain

E[[parallel][PHI](t, * )x[parallel].sup.2] [less than or equal to] N[e.sup.- v(t-s)]E[[parallel][PHI](s, *)x[parallel].sup.2], (19)

for all t [greater than or equal to] s [greater than or equal to] 0 and x [member of] X.

4. CONCLUSIONS

Theorem 3.2 can be seen as a sufficient condition for the exponential stability in mean square of the mild solutions of a stochastic evolution equation, such as (3). This condition is an extension of the so-called notion of admissibility, firstly used by Perron (1930) in the case of deterministic differential equations.

A possible extension of this paper concerns Condition A. We are looking to lessen this assumption by defining:

Condition A'. The stochastic process Ta belongs to C, for every stochastic process a EC.

If we replace Condition A by Coondition A' in Theorem 3.2, does the conclusion remain valid?

5. REFERENCES

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