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Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1625–1646 | Cite as

Sensitivity to Small Delays of Pathwise Stability for Stochastic Retarded Evolution Equations

Open Access
Article

Abstract

In this paper, we shall study the almost sure pathwise exponential stability property for a class of stochastic functional differential equations with delays, possibly, in the highest-order derivative terms driven by multiplicative noise. Instead of establishing a moment exponential stability as the first step and then proceeding to investigate the pathwise stability of the system under consideration, we shall develop a direct approach for this problem. As a consequence, we can show that some systems, which are not exponential momently stable, have the exponential stability not sensitive to small delays in the almost sure sense.

Keywords

Stability sensitive to small delays Pathwise exponential stability Stochastic delay evolution equation 

Mathematics Subject Classification (2010)

60H15 60G15 60H05 

1 Introduction

We start with a simple one-dimensional linear differential equation
$$\begin{aligned} {\left\{ \begin{array}{ll} \text{ d }y(t) = (\alpha +\beta ) y(t)\hbox {d}t, &{} t\ge 0,\\ y(0)=y_0\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$
(1.1)
where \(y(0) = y_0\in {\mathbb {R}}\), and \(\alpha \), \(\beta \in {\mathbb {R}}\). Since \(y(t) = y_0 e^{(\alpha +\beta )t}\) for all \(t\ge 0\), it is easy to see that if \(\alpha +\beta <0\), the null solution of (1.1) is exponentially stable.
Let \(r>0\) and consider a time delay version of (1.1) in the form
$$\begin{aligned} \hbox {d}y(t) = \alpha y(t)\hbox {d}t + \beta y(t-r)\hbox {d}t,\qquad t\ge 0. \end{aligned}$$
(1.2)
Because of the time delay feature in (1.2), one need set up proper initial data different from those in (1.1) to make (1.2) well-defined. For instance, let \(y(0)=\phi _0\in {\mathbb {R}}\) and \(y(t)=\phi _1(t)\) for \(t\in [-r, 0]\) where \(\phi _1\in L^2([-r, 0], {\mathbb {R}})\). It is well known (see, e.g., [2] or [14]) that under the condition \(\alpha +\beta <0\), there exists \(r_0>0\) such that the null solution of (1.2) is exponentially stable for all \(r\in (0, r_0)\), while for \(r>r_0\), the null solution is exponentially unstable. In other words, exponential stability at this moment is not sensitive to small delays, a result which actually remains true in any finite dimensional context (see, e.g., Chap. 7 in [2]).
Now let us consider a linear one-dimensional Itô equation
$$\begin{aligned} \hbox {d}y(t) = (\alpha +\beta ) y(t)\hbox {d}t + \sigma y(t)\hbox {d}w(t),\qquad t\ge 0, \end{aligned}$$
(1.3)
where w(t), \(t\ge 0\), is a standard real Brownian motion, \(y(0) = y_0\in {\mathbb {R}}\), and \(\alpha \), \(\beta \) and \(\sigma \) are real constants. A direct computation shows that the solution process y(t), \(t\ge 0\), of (1.3) is explicitly given by
$$\begin{aligned} y(t) = \exp \big \{\sigma w(t) + (\alpha +\beta ) t - \sigma ^2t/2\big \}y_0,\qquad t\ge 0. \end{aligned}$$
(1.4)
Hence, by virtue of the well-known law of the iterated logarithm (cf. Revuz and Yor [13]), it is easy to see that the asymptotically exponential growth rate of solution y is given by
$$\begin{aligned} \limsup _{t\rightarrow \infty }{\frac{\log |y(t)|}{t}} = \alpha +\beta - {\frac{\sigma ^2}{2}}\qquad a.s. \end{aligned}$$
(1.5)
Therefore, the null solution of (1.3) has global exponential stability in the almost sure sense if and only if \(\alpha +\beta < \sigma ^2/2\).
On the other hand, consider a time delay version of (1.3) in the following form
$$\begin{aligned} \hbox {d}y(t) =\alpha y(t)\hbox {d}t + \beta y(t-r)\hbox {d}t + \sigma y(t)\hbox {d}w(t),\qquad t\ge 0, \end{aligned}$$
(1.6)
where \(r>0\). For pathwise stability of Eq. (1.6), it turns out that, in addition to the condition \(\alpha +\beta < \sigma ^2/2\), the null solution of Eq.  (1.6) could secure the same exponential stability in the almost sure sense as long as the delay parameter \(r>0\) is sufficiently small. In fact, it is shown in Bierkens [3] that if \(r>0\) is so small that
$$\begin{aligned} |\beta |e^{3\sigma ^2 r/2} + \alpha < \sigma ^2/2, \end{aligned}$$
then the null solution of (1.6) is exponentially stable almost surely. Thus, stability is not sensitive in this case to small delays. It is also worth mentioning that as a corollary of more general results for nonlinear stochastic systems, Appleby and Mao [1] showed that if \(r>0\) is such a small number that
$$\begin{aligned} 2e^{\sigma ^2 r}(|\beta |+\alpha )\Phi (\sigma \sqrt{r})< \sigma ^2/2, \end{aligned}$$
then the null solution of (1.6) has the same pathwise exponential stability. Here \(\Phi \) is the cumulative distribution function of a standard normal random variable.
In an infinite dimensional setting, the situation is quite different. In particular, it was observed by Datko et al. [6] (see also [5]) that small delays may destroy stability for a partial differential equation. More precisely, consider a deterministic differential equation in a Hilbert space H,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t) = A y(t)\hbox {d}t + By(t-r)\hbox {d}t,\qquad t\ge 0,\\ y(0)=\phi _0\in H, \,\,y(t)=\phi _1(t),\,\, t\in [-r, 0],\,\, \phi _1\in L^2([-r, 0], H), \end{array}\right. } \end{aligned}$$
(1.7)
where A generates a \(C_0\)-semigroup and B is a linear operator in H. If the spectrum \(\sigma (A)\) of A is unbounded along an imaginary line, it was shown (see Theorem 7.4 in [2]) that for any \(r_0>0\), one can always find an operator \(B\in {\mathscr {L}}(H)\), the family of all bounded linear operators on H, and \(r\in (0, r_0)\) such that \(A+B\) generates an exponentially stable semigroup and meanwhile the system (1.7) is not exponentially stable. From this observation, we can see that the unboundedness of the spectrum set of A along imaginary axes may cause trouble in the stability analysis of (1.7). Therefore, one need make additional assumptions on the semigroup generated by A. Note that in finite dimensional spaces, the spectrum set of A is always bounded. In fact, we have the following result which clearly implies the fact that for finite dimensional equations, the exponential stability cannot be destroyed by small delays.

Theorem 1.1

Assume that A generates a norm continuous \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), i.e., \(e^{\cdot A}:\, [0, \infty )\rightarrow {\mathscr {L}}(H)\) is continuous, and the semigroup generated by \(A+B\) is exponentially stable in H. Then, there exists a constant \(r_0>0\) such that the system (1.7) is exponentially stable for all \(r\in (0, r_0)\).

Proof

See, e.g., the proofs of Theorem 7.5 in [2]. \(\square \)

In this work, we want to consider the sensitivity problem of pathwise stability to small delays for stochastic retarded evolution equations in Hilbert spaces. Based on the above statements, especially Theorem 1.1, let us first consider a concrete example of linear stochastic partial differential equations to motivate our theory,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t, \xi ) =\displaystyle \frac{\partial ^2 }{\partial \xi ^2}y(t, \xi )\hbox {d}t +\alpha _0 y(t, \xi )\hbox {d}t + \sigma y(t, \xi )\hbox {d}w(t),\quad t\ge 0,\quad \xi \in [0, \pi ],\\ y(t, 0)=y(t, \pi )=0,\quad t\ge 0,\\ y(0, \cdot )=\phi _0(\cdot )\in H = L^2(0, \pi ), \end{array}\right. }\nonumber \\ \end{aligned}$$
(1.8)
where \(\alpha _0,\,\sigma \in {\mathbb {R}}\) and w is a standard real Brownian motion.
It is not difficult to get, e.g., by Example 2.1, that if
$$\begin{aligned} \alpha _0 -1< \sigma ^2/2, \end{aligned}$$
(1.9)
the null solution of (1.8) is exponentially stable almost surely. Further, let \(r>0\) and consider a time delay version of (1.8) in the form
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t, \xi ) =\displaystyle \frac{\partial ^2 }{\partial \xi ^2}y(t, \xi ) +\alpha _0 y(t-r, \xi )\hbox {d}t + \sigma y(t, \xi )\hbox {d}w(t),\quad t\ge 0,\quad \xi \in [0, \pi ],\\ y(t, 0)=y(t, \pi )=0,\quad t\ge 0,\\ y(0, \cdot )=\phi _0(\cdot )\in L^2(0, \pi ),\,\, y_0(\cdot , \cdot ) = \phi _1(\cdot , \cdot )\in L^2([-r, 0]; L^2(0, \pi )), \end{array}\right. }\nonumber \\ \end{aligned}$$
(1.10)
where \((\phi _0, \phi _1)\) is some properly given initial datum. Here, the problem we want to address is whether, under the condition \(\alpha _0 -1<\sigma ^2/2\), the null solution of Eq. (1.10) secures the same pathwise exponential stability, at least for sufficiently small delay parameter \(r>0\).
Note that this problem cannot be addressed by the usual method dealing with pathwise exponential stability, i.e., first considering moment stability and then establishing pathwise exponential stability. To illustrate this, let us consider a simple example of one-dimensional stochastic differential equation,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t) =\beta y(t)\hbox {d}t + \sigma y(t)\hbox {d}w(t),\qquad t\ge 0,\\ y(0)=y_0\in {\mathbb {R}}, \end{array}\right. } \end{aligned}$$
where \(\beta ,\,\sigma \in {\mathbb {R}}\). It is known that
$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{\log |y(t)|}{t} = \beta - {\frac{\sigma ^2}{2}}\qquad a.s. \end{aligned}$$
(1.11)
Thus, the null solution is exponentially stable if and only if \(\beta <\sigma ^2/2\). On the other hand, we know by a simple calculation that
$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{\log {\mathbb {E}}|y(t)|^2}{t} = 2\beta +\sigma ^2. \end{aligned}$$
Thus, the null solution is mean square exponentially stable if and only if \(\beta <-\sigma ^2/2\), a condition which is much more restrictive than \(\beta <\sigma ^2/2\).

In this work, we shall employ a different method to address the pathwise sensitivity problem to small delays. In particular, as a by-product of the general theory established later on, we shall answer this question affirmatively for (1.10).

2 Pathwise Exponential Stability

Let H, K be two Hilbert spaces with their norms \(\Vert \cdot \Vert _H\), \(\Vert \cdot \Vert _K\) and inner products \(\langle \cdot , \cdot \rangle _H\), \(\langle \cdot , \cdot \rangle _K\), respectively. We denote by \({\mathscr {L}}(K, H)\) the space of all bounded and linear operators from K into H. If \(H=K\), we simply denote \({\mathscr {L}}(H, H)\) by \({\mathscr {L}}(H)\). Let \(\{\Omega , {\mathscr {F}}, {\mathbb {P}}\}\) be a complete probability space equipped with some filtration \(\{{\mathscr {F}}_{t}\}_{t\ge 0}\). Let \(\{W_Q(t),\,t\ge 0\}\) denote a K-valued \(\{{\mathscr {F}}_{t}\}_{t\ge 0}\)-Wiener process defined on \(\{\Omega , {\mathscr {F}}, {\mathbb {P}}\}\) with covariance operator Q satisfying that
$$\begin{aligned} \mathbb {E}\langle W_Q(t), u\rangle _K\langle W_Q(s), v\rangle _K = (t\wedge s)\langle Qu, v\rangle _K\,\,\,\,\hbox {for all}\,\,\,\,\, u,\,\,v\in K, \end{aligned}$$
where Q is a positive, self-adjoint and trace class operator, \(Tr(Q)<\infty \), on K. In particular, we shall call such \(W_Q(t)\), \(t\ge 0\), a K-valued Q-Wiener process with respect to \(\{{\mathscr {F}}_{t}\}_{t\ge 0}\). We introduce the subspace \(K_Q=\hbox {Ran}\,Q^{1/2}\), the range of \(Q^{1/2}\), of K and let \({\mathscr {L}}_2={\mathscr {L}}_2(K_Q, H)\) denote the space of all Hilbert–Schmidt operators from \(K_Q\) into H.
Consider the following linear stochastic evolution equation
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t) =Ay(t)\hbox {d}t + By(t)\hbox {d}W(t),\,\,\,\,t\ge 0,\\ y(0)=y_0\in H, \end{array}\right. } \end{aligned}$$
(2.1)
where \(A:\, {\mathscr {D}}(A)\subset H\rightarrow H\) generates a \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), \(B\in {\mathscr {L}}(H, {\mathscr {L}}_2)\), and W(t), \(t\ge 0\), is a K-valued Q-Wiener process.

Lemma 2.1

Assume that A generates an exponentially stable \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), such that
$$\begin{aligned} \Vert e^{tA}\Vert \le Me^{-\lambda t},\qquad t\ge 0, \end{aligned}$$
for \(M\ge 1\), \(\lambda >0\) with \(M^2\Vert B\Vert ^2 <2\lambda \) where \(\Vert B\Vert := \Vert B\Vert _{{\mathscr {L}}(H,\, {\mathscr {L}}_2)}\). Then, for any self-adjoint, nonnegative operator \(L\in {\mathscr {L}}(H)\), there exists a unique nonnegative, self-adjoint operator \(P\in {\mathscr {L}}(H)\) such that
$$\begin{aligned} \langle Ax, Px\rangle _H + \langle Px, Ax\rangle _H + \langle \Delta (P)x, x\rangle _H =-\langle Lx, x\rangle _H\qquad \hbox {for any}\qquad x\in {\mathscr {D}}(A),\nonumber \\ \end{aligned}$$
(2.2)
where \(\Delta (P)\in {\mathscr {L}}(H)\) is the unique operator defined by the relation
$$\begin{aligned} \langle x, \Delta (P)y\rangle _H :=Tr\big \{P B(x)Q^{1/2}(B(y)Q^{1/2})^*\big \},\qquad x,\,\, y\in H,\,\,\,\,\,\, P\in {\mathscr {L}}(H).\nonumber \\ \end{aligned}$$
(2.3)
In this case, the operator P satisfies
$$\begin{aligned} P = \int ^\infty _0 e^{tA^*}(\Delta (P) +L)e^{tA}\mathrm{d}t. \end{aligned}$$
(2.4)

Proof

Define a series of operators \(\{P(j)\}_{j\ge 0}\) in H by
$$\begin{aligned} P(0)=0,\qquad P({j+1}) =\int ^\infty _0 e^{tA^*}(\Delta (P(j)) +L)e^{tA}\hbox {d}t,\qquad j\in {\mathbb {N}}_+. \end{aligned}$$
(2.5)
It is immediate to see that this series is a contraction since
$$\begin{aligned} \begin{aligned} \Vert P({j+1})-P(j)\Vert&= \Big \Vert \int ^\infty _0 e^{tA^*}\Delta (P(j) -P({j-1}))e^{tA}\hbox {d}t\Big \Vert \\&\le M^2\Vert B\Vert ^2\int ^\infty _0 e^{-2\lambda t}\hbox {d}t \Vert P(j)-P({j-1})\Vert \\&= \frac{M^2\Vert B\Vert ^2}{2\lambda }\Vert P(j)-P({j-1})\Vert \end{aligned} \end{aligned}$$
and \(M^2\Vert B\Vert ^2/2\lambda <1\). Hence, there exists a unique fixed point \(P\in {\mathscr {L}}(H)\) so that, letting \(j\rightarrow \infty \) in (2.5), we obtain the relation (2.4).
To show the equality (2.2), let us consider the following backward linear operator differential equation
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {\frac{d}{\hbox {d}t}}\langle P(t)x, x\rangle _H + \langle Ax, P(t)x\rangle _H +\langle P(t)x, Ax\rangle _H + \langle [L + \Delta (P(t))]x, x\rangle _H =0,\\ \qquad \qquad \qquad \qquad \qquad \qquad \,\,\quad \quad \qquad x\in {\mathscr {D}}(A),\qquad 0\le t\le T, \\ P(T) =G, \end{array}\right. }\nonumber \\ \end{aligned}$$
(2.6)
where \(G\in {\mathscr {L}}(H)\) is a nonnegative and self-adjoint operator in H and \(T\ge 0\), and the associated integral equation
$$\begin{aligned} \begin{aligned} P(t) x =&\,\,\displaystyle \int ^T_t e^{(s-t)A^*}[L + \Delta (P(s))]e^{(s-t)A}x\,\hbox {d}s\\&\,\, + e^{(T-t)A^*}Ge^{(T-t)A}x,\qquad \forall \, x\in H,\qquad 0\le t\le T. \end{aligned} \end{aligned}$$
(2.7)
We shall show that Eqs.,(2.6) and (2.7) are actually equivalent. Indeed, first suppose that P(t), \(t\ge 0\), satisfies (2.7). Then, by differentiating \(\langle P(t)x, x\rangle _H\), \(x\in {\mathscr {D}}(A)\), we obtain (2.6) immediately.
Conversely, suppose P(t), \(t\ge 0\), satisfies (2.6). Let \(x\in {\mathscr {D}}(A)\) and \(0 < s\le t\), and then \(\langle P(t)e^{(t-s)A}x, e^{(t-s)A}x\rangle _H\) is differentiable in t and
$$\begin{aligned} \begin{aligned} {\frac{d}{\hbox {d}t}}\langle P(t)e^{(t-s)A}x, e^{(t-s)A}x\rangle _H =&\, -\langle Ae^{(t-s)A}x, P(t)e^{(t-s)A}x\rangle _H - \langle P(t)e^{(t-s)A}x, \\&Ae^{(t-s)A}x\rangle _H\\&\, + \langle P(t)e^{(t-s)A}x, Ae^{(t-s)A}x\rangle _H + \langle A e^{(t-s)A}x, \\&P(t)e^{(t-s)A}x\rangle _H\\&\, -\langle [L + \Delta (P(t))]e^{(t-s)A}x, e^{(t-s)A}x\rangle _H\\ =&\,-\langle [L +\Delta (P(t))]e^{(t-s)A}x, e^{(t-s)A}x\rangle _H. \end{aligned}\nonumber \\ \end{aligned}$$
(2.8)
By integrating (2.8) in t from s to T, we then obtain for any \(0\le s\le T\) that
$$\begin{aligned} \begin{aligned} \langle P(s)x, x\rangle _H = \int ^T_s \langle [L + \Delta (P(u))]e^{(u-s)A}x, e^{(u-s)A}x\rangle _H\,\hbox {d}u\qquad \forall \,x\in {\mathscr {D}}(A). \end{aligned}\nonumber \\ \end{aligned}$$
(2.9)
Since \({\mathscr {D}}(A)\) is dense in H, the relation (2.7) follows easily from (2.9). Last, since the semigroup \(e^{tA}\), \(t\ge 0\), is exponentially stable and P is the unique solution of (2.4), one can see that the family of nonnegative, self-adjoint operators \(P(t)\equiv P\in {\mathscr {L}}(H)\) is the unique solution of (2.7) with \(t=0\), \(T= +\infty \) and \(G=P\). Therefore, P satisfies (2.6) with \(G=P\) which is exactly the equality (2.2). Thus, the proof is complete. \(\square \)

Lemma 2.2

Suppose that A generates a \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), such that
$$\begin{aligned} \Vert e^{tA}\Vert \le Me^{-\lambda t},\qquad t\ge 0, \end{aligned}$$
for \(M>0\), \(\lambda >0\) with \(M^2\Vert B\Vert ^2<2\lambda \). Let \(A_n\) be the Yosida approximation of A, i.e., \(A_n= nR(n, A)A\), \(R(n, A)=(nI -A)^{-1}\), \(n\in \rho (A)\), the resolvent set of A, and \(L\in {\mathscr {L}}(H)\) be a nonnegative, self-adjoint operator. For \(n\in \rho (A)\), let \(P_n\ge 0\) denote the unique solution to Lyapunov equation
$$\begin{aligned} A^*_nP_n + P_nA_n + \Delta (P_n)=-L. \end{aligned}$$
(2.10)
Then, as \(n\rightarrow \infty \),
$$\begin{aligned} \langle x, P_n y\rangle _H \rightarrow \langle x, Py\rangle _H\qquad \hbox {for any}\qquad x,\,\,y\in H, \end{aligned}$$
where \(P\in {\mathscr {L}}(H)\) is the unique nonnegative solution to
$$\begin{aligned} A^*P + PA + \Delta (P)= -L. \end{aligned}$$
(2.11)

Proof

Recall that (see (A.13) in [4])
$$\begin{aligned} \Vert e^{tA_n}\Vert \le Me^{-\frac{n\lambda t}{n-\lambda }},\qquad n>\lambda ,\qquad t\ge 0. \end{aligned}$$
Thus, the exponential stability of \(e^{tA}\) implies the same property of \(e^{tA_n}\) for \(n\in \rho (A)\). By virtue of Lemma 2.1, for any self-adjoint, nonnegative operator \(L\in {\mathscr {L}}(H)\), there exists a unique solution \(P_n\in {\mathscr {L}}(H)\) to (2.10) for \(n\in \rho (A)\).
For each \(n\in {\mathbb {N}}\), we construct a sequence
$$\begin{aligned} P_n(0) :=0\qquad \hbox {and}\qquad P_n({j+1}) := \int ^\infty _0 e^{tA^*_n}(\Delta (P_n(j)) +L)e^{tA_n}\hbox {d}t,\qquad j\in {\mathbb {N}}, \end{aligned}$$
and in a similar way, let
$$\begin{aligned} P(0) :=0\qquad \hbox {and}\qquad P({j+1}) := \int ^\infty _0 e^{tA^*}(\Delta (P(j)) +L)e^{tA}\hbox {d}t,\qquad j\in {\mathbb {N}}. \end{aligned}$$
We show that for all \(x,\,y\in H\) and \(j\in {\mathbb {N}}\), there is the relation \(\langle x, (P_n(j)-P(j)) y\rangle _H\rightarrow 0\) as \(n\rightarrow \infty \). Indeed, we can see this by induction. For \(j=0\), the claim holds trivially. Suppose now that the claim holds for \(j=k-1\), \(k\ge 1\). Let \(x,\,y\in H\), and then for \(j=k\),
$$\begin{aligned} \begin{aligned} |\langle&x, (P_n(k)-P(k))y\rangle _H| \\ =&\, \Big |\int ^\infty _0 \langle e^{tA_n}x, (\Delta (P_n({k-1})) +L)e^{tA_n}y\rangle _H\hbox {d}t \\&- \int ^\infty _0 \langle e^{tA}x, (\Delta (P({k-1})) +L)e^{tA}y\rangle _H\hbox {d}t\Big |\\ \le&\, \int ^\infty _0 |\langle e^{tA_n}x, \Delta (P_n({k-1})-P({k-1}))e^{tA_n}y\rangle _H|\hbox {d}t\\&\, + \Big |\Big \langle x, \int ^\infty _0 e^{tA^*_n}(\Delta (P({k-1}))+L)e^{tA_n}y\hbox {d}t \\&- \int ^\infty _0 e^{tA^*}(\Delta (P({k-1})) +L)e^{tA} y\hbox {d}t\Big \rangle _H\Big |\\ =:&\, I_1(n) + I_2(n). \end{aligned} \end{aligned}$$
(2.12)
By induction hypothesis, we have
$$\begin{aligned} \langle x, (P_n({k-1})-P({k-1}))y\rangle _H \rightarrow 0,\qquad \forall \, x,\,\,y\in H,\qquad \hbox { as}\,\,n\rightarrow \infty . \end{aligned}$$
This fact further implies that \(I_1(n)\rightarrow 0\) as \(n\rightarrow \infty \) by Dominated Convergence Theorem. For \(I_2(n)\), we may similarly have that
$$\begin{aligned} \begin{aligned}&\Big |\Big \langle x, \int ^\infty _0 \Big (e^{tA^*_n}(\Delta (P({k-1})) +L)e^{tA_n}y -e^{tA^*}(\Delta (P({k-1})) +L)e^{tA}y\Big )\hbox {d}t\Big \rangle _H\Big |\\&= \Big |\int ^\infty _0 \langle e^{tA_n}x, (\Delta (P({k-1})) +L)(e^{tA_n}-e^{tA})y\rangle _H\\&\,\,\,\,\,\,\, + \langle (e^{tA_n}-e^{tA})x, (\Delta (P({k-1})) +L)e^{tA}y\rangle _H\,\hbox {d}t\Big |\\&\le \int ^\infty _0 \Vert e^{tA_n}\Vert \cdot \Vert x\Vert _H \Vert \Delta (P({k-1})) +L\Vert \cdot \Vert (e^{tA_n}-e^{tA})y\Vert _H\mathrm{d}t\\&\,\,\,\,\,\,\, + \int ^\infty _0 \Vert (e^{tA_n}-e^{tA})x\Vert _H \Vert \Delta (P({k-1})) +L\Vert \cdot \Vert e^{tA}\Vert \cdot \Vert y\Vert _H\hbox {d}t\\&\rightarrow 0\qquad \hbox {as}\,\,\,n\rightarrow \infty , \end{aligned} \end{aligned}$$
by virtue of Dominated Convergence Theorem. Hence, the claim is proved by induction.
Last, by the proofs of Lemma 2.1 we have that \(P(j)\rightarrow P\) and \(P_n(j)\rightarrow P_n\), uniformly in n, as \(j\rightarrow \infty \) in the norm topology of \({\mathscr {L}}(H)\). Therefore, we obtain that
$$\begin{aligned} \langle x, (P_n-P)y\rangle _H\rightarrow 0\qquad \hbox {as}\qquad n\rightarrow \infty \qquad \hbox {for any}\qquad x,\,y\in H \end{aligned}$$
by using the relation
$$\begin{aligned} \begin{aligned} |\langle x, (P_n-P)y\rangle _H| \le&\,\, |\langle x, (P_n-P_n(j))y\rangle _H| +|\langle x, (P_n(j)-P(j))y\rangle _H|\\&\,\, + |\langle x, (P(j)-P)y\rangle _H|. \end{aligned} \end{aligned}$$
The proof is thus complete. \(\square \)
Now consider the following linear stochastic evolution equation on the Hilbert space H,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t) =Ay(t)\hbox {d}t + \displaystyle \sum ^n_{i=1} B_iy(t)\hbox {d}w_i(t),\,\,\,\,t\ge 0,\\ y(0)=y_0\in H, \end{array}\right. } \end{aligned}$$
(2.13)
where \(B_i\in {\mathscr {L}}(H)\) and \(w_i(t)\), \(t\ge 0\), \(i=1,\,\ldots ,\, n\), are a group of independent, standard real-valued Brownian motions. To proceed further, we first recall two useful lemmas which are important in their own right. The first lemma is a strong law of large numbers for continuous local martingales (see, e.g., [9] or Theorem 1.3.4 in [12]), and the proof of the second is referred to that of Proposition 2.1.4 in [10].

Lemma 2.3

Let M(t), \(t\ge 0\), be a real-valued, continuous local martingale with \(M(0)=0\). If its quadratic variation [M] satisfies that
$$\begin{aligned} \limsup _{t\rightarrow \infty } \frac{[M](t)}{t}<\infty \qquad a.s. \end{aligned}$$
then there is
$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{M(t)}{t}=0\qquad a.s. \end{aligned}$$

Lemma 2.4

Suppose that A generates a \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), on the Hilbert space H and \(A^*\) is its adjoint operator. There exists a real number \(\alpha \in {\mathbb {R}}\) such that
$$\begin{aligned} Re\langle x, \,Ax\rangle _H \le \alpha \Vert x\Vert ^2_H\qquad \hbox { for all}\qquad x\in {\mathscr {D}}(A), \end{aligned}$$
(2.14)
and
$$\begin{aligned} Re\langle x, \,A^*x\rangle _H \le \alpha \Vert x\Vert ^2_H\qquad \hbox { for all}\qquad x\in {\mathscr {D}}(A^*), \end{aligned}$$
(2.15)
if and only if A generates a pseudo-contraction \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), in the sense that
$$\begin{aligned} \Vert e^{tA}\Vert \le e^{\alpha t}\qquad \hbox {for all}\qquad t\ge 0. \end{aligned}$$
(2.16)

Proposition 2.1

Let \(B=(B_1,\, \ldots ,\, B_n)\). Suppose that there exist \(\beta = (\beta _1,\,\ldots ,\, \beta _n) \in {\mathbb {R}}^n\), \(\lambda >0\) and \(M\ge 1\) such that the \(C_0\)-semigroup \(e^{t(A+\beta \cdot B)}\) generated by \(A +\beta \cdot B\) satisfies
$$\begin{aligned} \Vert e^{t(A+\beta \cdot B)}\Vert \le M e^{-\lambda t},\qquad t\ge 0, \end{aligned}$$
and
$$\begin{aligned} \frac{1}{4}\sum ^n_{i=1} \beta ^2_i + \frac{M^2}{2}\sum ^n_{i=1}\Vert B_i\Vert ^2< \lambda , \end{aligned}$$
(2.17)
then the mild solution of (2.13) is almost sure exponentially stable with
$$\begin{aligned} \limsup _{t\rightarrow \infty } \frac{1}{t}\log \Vert y(t)\Vert _H\le -\Big (\lambda - \frac{1}{4}\sum ^n_{i=1} \beta ^2_i - \frac{M^2}{2}\sum ^n_{i=1}\Vert B_i\Vert ^2\Big )\qquad a.s. \end{aligned}$$

Proof

Choose arbitrarily
$$\begin{aligned} \gamma \in \Big (0, \lambda - \frac{1}{4}\sum ^n_{i=1} \beta ^2_i - \frac{M^2}{2}\sum ^n_{i=1}\Vert B_i\Vert ^2\Big ). \end{aligned}$$
Let us put \(\Vert \beta \Vert ^2 := \sum ^n_{i=1}\beta ^2_i\) and \(\Lambda = A + \beta \cdot B + \frac{\Vert \beta \Vert ^2}{4} + \gamma \), and then \(\Lambda \) is the generator of a \(C_0\)-semigroup T(t) which satisfies
$$\begin{aligned} \Vert T(t)\Vert \le Me^{- (\lambda - \Vert \beta \Vert ^2/4 -\gamma ) t},\qquad t\ge 0. \end{aligned}$$
For each \(m\in {\mathbb {N}}\), let \(\Lambda _m :=A_m +\beta \cdot B + \frac{\Vert \beta \Vert ^2}{4} +\gamma \) where \(A_m\) is the Yosida approximation of A. Without loss of generality, it is assumed that for all \(m\ge 1\), we have
$$\begin{aligned} \Vert T_m(t)\Vert \le Me^{- (\lambda - \Vert \beta \Vert ^2/4 -\gamma ) t},\qquad t\ge 0. \end{aligned}$$
Since semigroups \(T(\cdot )\) and \(T_m(\cdot )\), \(m\ge 1\), are exponentially stable, by virtue of (2.17) and Lemma 2.1, one can find for \(m\in {\mathbb {N}}\) and \(\Delta (I) \in {\mathscr {L}}^+(H)\) a unique solution \(Q_m\in {\mathscr {L}}(H)\) to the following Lyapunov equations
$$\begin{aligned} \Lambda ^*_m Q_m +Q_m\Lambda _m +\Delta (Q_m) = -\Delta (I). \end{aligned}$$
(2.18)
In other words, for each \(m\ge 1\), \(P_m = Q_m+I\in {\mathscr {L}}(H)\) is the unique solution to the Lyapunov equation
$$\begin{aligned} \Lambda ^*_m (P_m-I) +(P_m-I)\Lambda _m +\Delta (P_m-I) = -\Delta (I). \end{aligned}$$
(2.19)
In an analogous manner, let \(P\in {\mathscr {L}}(H)\) be the unique solution to
$$\begin{aligned} \Lambda ^*(P -I) +(P-I)\Lambda + \Delta (P-I) =-\Delta (I). \end{aligned}$$
Then, by virtue of Lemmas 2.1 and 2.2, it is easy to show
$$\begin{aligned} P_m\ge I,\,\,\, m\in {\mathbb {N}},\,\,\, P\ge I\qquad \hbox {and}\qquad \langle P_mx, y\rangle _H \rightarrow \langle Px, y\rangle _H,\qquad \hbox {as}\,\,\,\,m\rightarrow \infty ,\nonumber \\ \end{aligned}$$
(2.20)
for all \(x,\,y\in H.\) By the definition of \(\Lambda _m\), \(m\ge 1\), and Lemma 2.4, it follows that
$$\begin{aligned} \langle (\Lambda _m +\Lambda ^*_m)x, x\rangle _H\le 0\qquad \hbox {for any}\qquad x\in H. \end{aligned}$$
Hence, by a direct computation we find from relation (2.19) that
$$\begin{aligned}&(A_m +\beta \cdot B)^*P_m + P_m(A_m + \beta \cdot B) + \Delta (P_m) + \frac{\Vert \beta \Vert ^2}{2} P_m +2\gamma P_m \le 0 \\&\quad \hbox {for each}\qquad m\in {\mathbb {N}}. \end{aligned}$$
In other words, for any \(m\in {\mathbb {N}}\) and \(x\in H\), we have
$$\begin{aligned} \begin{aligned} \langle P_m A_mx, x\rangle _H\, +&\,\langle A_mx, P_m x\rangle _H + \langle \Delta (P_m)x, x\rangle _H +2\gamma \langle P_m x, x\rangle _H\\&\le -\sum ^n_{i=1}\beta _i\langle P_mx, B_ix\rangle _H -\sum ^n_{i=1}\beta _i\langle B_ix, P_mx\rangle _H \\&-\frac{\Vert \beta \Vert ^2}{2}\langle P_m x, x\rangle _H, \end{aligned} \end{aligned}$$
(2.21)
which further implies that for any \(m\in {\mathbb {N}}\) and \(x\in H\),
$$\begin{aligned} \begin{aligned} \langle&P_mx, x\rangle _H\Big [\langle P_mx, A_mx\rangle _H +\langle A_mx, P_mx\rangle _H + \langle \Delta (P_m)x, x\rangle _H + 2\gamma \langle P_mx, x\rangle _H\Big ]\\&\le -\sum ^n_{i=1}\beta _i\langle P_mx, x\rangle _H \langle P_mx, B_ix\rangle _H -\sum ^n_{i=1}\beta _i\langle P_mx, x\rangle _H \langle B_ix, P_mx\rangle _H \\&-\frac{\Vert \beta \Vert ^2}{2}\langle P_m x, x\rangle ^2_H\\&\le 2\sum ^n_{i=1}\langle P_mx, B_i x\rangle ^2_H. \end{aligned} \end{aligned}$$
(2.22)
On the other hand, let \(y_m\) denote the mild solution to the equation
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y_m(t) = A_my_m(t)\hbox {d}t + \displaystyle \sum ^n_{i=1} B_i y_m(t)\hbox {d}w_i(t), \,\,\,\, t\ge 0, \\ y_m(0)=y_0, \end{array}\right. } \end{aligned}$$
(2.23)
where \(m\in {\mathbb {N}}\). Then, there exists a sequence, still denote it by \(y_m\in H\) (see [4]), such that
$$\begin{aligned} \sup _{t\in [0, T]}\Vert y_{m}(t) -y(t)\Vert _H \rightarrow 0\qquad \hbox {as}\,\,\,\,\, m\rightarrow \infty \qquad a.s. \end{aligned}$$
(2.24)
If \(y_0=0\), then \({\mathbb {P}}(y(t, 0)=0)=1\) for \(t\ge 0\), and the desired estimate holds trivially. Now suppose that \(y_0\not = 0\), then by uniqueness of the solutions and strict positivity of \(P_m\), \({\mathbb {P}}(\langle P_my_m(t), y_m(t)\rangle _H=0)=0\) for all \(t\ge 0\) (or, the reader is referred to a proof similar to that of Lemma 2.1 in [11]). Letting \(\theta _m(t) = \langle P_my_m(t), y_m(t)\rangle _H\), \(t\ge 0\), and applying Itô’s formula to \(\log \theta _m(t)\), we obtain by (2.22) that for \(t\ge 0\),
$$\begin{aligned}&\log \langle P_my_m(t), y_m(t)\rangle _H\nonumber \\&\quad = \log \langle P_my_0, y_0\rangle _H + \int ^t_0 \Big \{\frac{1}{\theta _m(s)}\Big [\langle P_my_m(s), A_my_m(s)\rangle _H\nonumber \\&\qquad + \langle A_my_m(s), P_my_m(s)\rangle _H + \langle \Delta (P_m)y_m(s), y_m(s)\rangle _H\Big ]\nonumber \\&\qquad - \frac{2}{\theta ^2_m(s)} \sum ^n_{i=1}\langle P_my_m(s), B_iy_m(s)\rangle _H^2 \Big \}ds \nonumber \\&\qquad + \int ^t_0\frac{2}{\theta _m(s)}\sum ^n_{i=1}\langle P_my_m(s), B_iy_m(s)\hbox {d}w_i(s)\rangle _H\nonumber \\&\quad \le \log \langle P_my_0, y_0\rangle _H -2\gamma t + 2\int ^t_0 \sum ^n_{i=1}\Big \langle \frac{P_my_m(s)}{\langle P_my_m(s), y_m(s)\rangle _H}, B_iy_m(s)\hbox {d}w_i(s)\Big \rangle _H. \end{aligned}$$
(2.25)
Letting \(m\rightarrow \infty \) and using (2.20) and (2.24), we have from (2.25) that
$$\begin{aligned} \begin{aligned}&\log \langle Py(t), y(t)\rangle _H\le \log \langle Py_0, y_0\rangle _H -2\gamma t \\&+ 2\sum ^n_{i=1}\int ^t_0 \Big \langle \frac{Py(s)}{\langle Py(s), y(s)\rangle _H}, By(s)\hbox {d}w_i(s)\Big \rangle _H. \end{aligned} \end{aligned}$$
(2.26)
On the other hand, it is easy to see that the stochastic integral
$$\begin{aligned} \int ^t_0 \Big \langle \frac{2Py(s)}{\langle Py(s), y(s)\rangle _H}, B_iy(s)\hbox {d}w_i(s)\Big \rangle _H,\qquad t\ge 0,\qquad i=1,\,\ldots ,\,n, \end{aligned}$$
is a real continuous local martingale, and since \(\langle Px, x\rangle _H \ge \Vert x\Vert ^2_H\) for any \(x\in H\), it follows that
$$\begin{aligned} \sup _{x\not = 0,\, x\in H}\frac{\langle Px, B_ix\rangle _H^2}{\langle Px, x\rangle ^2_H} \le \sup _{x\not = 0,\, x\in H}\frac{\Vert P\Vert ^2\Vert B_i\Vert ^2\Vert x\Vert ^4_H}{\Vert x\Vert ^4_H} <\infty . \end{aligned}$$
Thus, one can employ Lemma 2.3 to get that for each \(i\in \{1,\,\ldots ,\,n\}\),
$$\begin{aligned} \frac{1}{t}\int ^t_0 \Big \langle \frac{2Py(s)}{\langle Py(s), y(s)\rangle _H}, B_iy(s)\hbox {d}w_i(s)\Big \rangle _H \rightarrow 0\qquad \hbox {as}\qquad t\rightarrow \infty \qquad a.s. \end{aligned}$$
which, in addition to (2.26), immediately implies that
$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t}\log \Vert y(t)\Vert ^2_H \le \limsup _{t\rightarrow \infty }\frac{1}{t}\log \langle Py(t), y(t)\rangle _H \le -2\gamma \qquad a.s. \end{aligned}$$
Last, letting \(\gamma \rightarrow \lambda - \frac{1}{4}\sum ^n_{i=1} \beta ^2_i - \frac{M^2}{2}\sum ^n_{i=1}\Vert B_i\Vert ^2\), we obtain the desired result. \(\square \)

Example 2.1

Consider the following linear stochastic differential equation in H,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t) = Ay(t)\hbox {d}t + \sigma y(t)\hbox {d}w(t),\qquad t\ge 0,\\ y(0)=y_0\in H, \end{array}\right. } \end{aligned}$$
(2.27)
where \(w(\cdot )\) is a standard real Brownian motion and A generates a \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), satisfying \(\Vert e^{tA}\Vert \le e^{\alpha t}\) for some \(\alpha \in {\mathbb {R}}\) and \(\sigma \in {\mathbb {R}}\). We claim that if \(\alpha < \sigma ^2/2\), the mild solution y to (2.27) is exponentially stable almost surely.
To see this, it suffices by Proposition 2.1 to find \(\lambda >0\) and \(\beta \in {\mathbb {R}}\) such that
$$\begin{aligned} -\lambda >\alpha +\beta \sigma \qquad \hbox {and}\qquad \sigma ^2 +\frac{\beta ^2}{2} -2\lambda <0. \end{aligned}$$
(2.28)
However, this is immediate since we can take \(\beta = -2\sigma \) and \(\lambda = 3\sigma ^2/2 +\varepsilon \) with \(\varepsilon >0\) small enough by using the condition \(\alpha < \sigma ^2/2\) and the mild solution y is pathwise exponentially stable with
$$\begin{aligned} \limsup _{t\rightarrow \infty }\frac{1}{t}\log \Vert y(t)\Vert _H \le \alpha - \frac{\sigma ^2}{2}\qquad a.s. \end{aligned}$$

3 Stochastic Delay Evolution Equations

Let H be a real Hilbert space and V be another Hilbert space such that V is dense in H and the inclusion map \(V\hookrightarrow H\) is continuous. The norms and inner products of H, V are denoted by \(\Vert \cdot \Vert _H\), \(\Vert \cdot \Vert _V\) and \(\langle \cdot , \cdot \rangle _H\), \(\langle \cdot , \cdot \rangle _V\), respectively. By identifying the dual of H with H, we may assume that
$$\begin{aligned} V\hookrightarrow H\hookrightarrow V^*\qquad \hbox {and}\qquad \Vert x\Vert ^2_{H}\le \nu \Vert x\Vert ^2_{V},\,\,\,\,\,\,x\in V, \end{aligned}$$
(3.1)
for some constant \(\nu \ge 1\). Moreover, we have for any \(T\ge 0\) that
$$\begin{aligned} L^2([0, T]; V)\cap W^{1, 2}([0, T]; V^*)\subset C([0, T]; H) \end{aligned}$$
where \(W^{1, 2}([0, T]; V^*)\) is the classic Sobolev space consisting of all functions \(y\in L^2([0, T]; V^*)\) such that y and its first-order distributional derivatives are in \(L^2([0, T]; V^*)\) and C([0, T]; H) is the space of all continuous functions from [0, T] into H, respectively. The duality pair between V and \(V^*\) is denoted by \(\langle \!\langle \cdot , \cdot \rangle \!\rangle _{V, V^*}\). Let \(a:\, V\times V\rightarrow {\mathbb {R}}\) be a bounded bilinear form satisfying Gårding’s inequality
$$\begin{aligned} a(x, x)\le -\alpha \Vert x\Vert ^2_V+\lambda \Vert x\Vert ^2_H,\qquad \forall \, x\in V, \end{aligned}$$
(3.2)
for some \(\alpha >0\) and \(\lambda \in {\mathbb {R}}\). Let A be the operator associated with this form by
$$\begin{aligned} a(x, y)=\langle \!\langle x, Ay\rangle \!\rangle _{V, V^*},\qquad x,\,\,y\in V. \end{aligned}$$
Then, \(A\in {\mathscr {L}}(V, V^*)\), the family of all bounded and linear operators from V to \(V^*\). The realization of A in H, which is the restriction of A to the domain \({\mathscr {D}}(A) =\{x\in V:\, Ax\in H\}\), is also denoted by A. Hence, \(\langle x, Ay\rangle _H = \langle \!\langle x, Ay\rangle \!\rangle _{V, V^*}\) for all \(x\in V\) and \(y\in V\) with \(Ay\in H\). It is well known that A generates an analytic \(C_0\)-semigroup \(e^{tA}\), \(t\ge 0\), in \(V^*\) such that \(e^{tA}:\, V^*\rightarrow V\) for each \(t>0\). In this work, we always suppose, for simplicity, that \(\lambda =0\) in (3.2) and A generates an exponentially stable \(C_0\)-semigroup with \(\Vert e^{tA}\Vert \le e^{-\alpha t}\) for all \(t\ge 0\).
Let \(r>0\), \(T\ge 0\) and \({\mathcal {H}} = H\times L^2([-r, 0]; V)\). We first consider the following deterministic retarded differential equation in space \(V^*\),
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t) = Ay(t)\hbox {d}t + A_1y(t-r)\hbox {d}t + \displaystyle \int ^0_{-r} A_2(\theta ) y(t+\theta )\hbox {d}\theta \hbox {d}t,\qquad t\in [0, T],\\ y(0)=\phi _0,\,\,y(\theta ) =\phi _1(\theta ),\,\,\theta \in [-r, 0],\,\,(\phi _0, \phi _1)\in {\mathcal {H}}, \end{array}\right. }\nonumber \\ \end{aligned}$$
(3.3)
where \(A_1\in {\mathscr {L}}(V, V^*)\) and \(A_2(\cdot )\in L^2([-r, 0]; {\mathscr {L}}(V, V^*))\). In particular, a function \(y\in L^2([-r, T]; V)\cap W^{1, 2}([0, T]; V^*)\) which satisfies (3.3) is called a strong solution of (3.3) in \([-r, T]\). It has been shown (see [7] or [8]) that there exists a unique solution y of (3.3) such that
$$\begin{aligned} y\in L^2([0, T]; V)\cap W^{1, 2}([0, T]; V^*)\subset C([0, T]; H). \end{aligned}$$
In association with this solution y, we may define a family of operators \({\mathcal {T}}(t)\), \(t\ge 0\), on \({\mathcal {H}}\) by
$$\begin{aligned} {\mathcal {T}}(t)\phi = (y(t; \phi ), y_t(\phi )),\qquad t\ge 0,\,\,\,\,\phi \in {\mathcal {H}}. \end{aligned}$$
(3.4)
Then, it has been shown (see [7]) that \({\mathcal {T}}(t)\), \(t\ge 0\), is actually a \(C_0\)-semigroup on \({\mathcal {H}}\). Let \({\mathcal {A}}\) be the infinitesimal generator of \({\mathcal {T}}(t)\) or \(e^{t{\mathcal {A}}}\), \(t\ge 0\), and then \({\mathcal {A}}\) is completely described by the following theorem.

Theorem 3.1

The operator \({\mathcal {A}}\) is given by
$$\begin{aligned} \begin{aligned} {\mathscr {D}}({\mathcal {A}}) = \Big \{\phi =(\phi _0, \phi _1):\,\,&\phi _1 \in W^{1, 2}([-r, 0]; V), \phi _0=\phi _1(0),\\&A\phi _0+ A_1\phi _1(-r)+\displaystyle \int ^0_{-r}A_2(\theta )\phi _1(\theta )\hbox {d}\theta \in H\Big \}, \end{aligned} \end{aligned}$$
(3.5)
and for any \(\phi =(\phi _0, \phi _1)\in {\mathscr {D}}({\mathcal {A}})\),
$$\begin{aligned} {\mathcal {A}}\phi = \Big ( A\phi _0 + A_1\phi _1(-r) + \displaystyle \int ^0_{-r}A_2(\theta )\phi _1(\theta )\hbox {d}\theta , \frac{\hbox {d}\phi _1(\theta )}{\hbox {d}\theta }\Big ). \end{aligned}$$
(3.6)
Note that \(C_0\)-semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), allows us to transfer the exponential stability problem of the time delay system (3.3) to that one of a non-time delay system. To see this, we can rewrite (3.3) as a Cauchy problem,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}Y(t) = {\mathcal {A}}Y(t)\hbox {d}t,\qquad t\ge 0,\\ Y(0)=\phi \in {\mathcal {H}}, \end{array}\right. } \end{aligned}$$
(3.7)
where \(Y(t)=(y(t), y(t+\cdot ))\) is the lift-up function of y(t), \(t\ge 0\). To show the exponential stability of the null solution of (3.3) in H, it suffices to study the corresponding exponential stability of system (3.7). Indeed, assume that the null solution of (3.7) is exponentially stable, and then there exist constants \(M>0\) and \(\mu >0\) such that
$$\begin{aligned} \begin{aligned} \Vert y(t, \phi )\Vert ^2_H&\le \Vert y(t, \phi )\Vert ^2_H + \int ^0_{-r} \Vert y(t+\theta , \phi )\Vert ^2_{V}\hbox {d}\theta \\&= \Vert Y(t, \phi )\Vert ^2_{\mathcal {H}}\\&\le Me^{-\mu t}\Vert \phi \Vert ^2_{\mathcal {H}},\,\,\,\,t\ge 0. \end{aligned} \end{aligned}$$
Hence, it is natural to consider exponential stability of the non-time delay system (3.7) rather than time delay one (3.3). To establish the exponential stability of the \(C_0\)-semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), one may want to develop a dissipative operator program by using Lemma 2.4. However, the following example shows some difficulties in association with this scheme.
Let \(\alpha > 0\), \(\beta \in {\mathbb {R}}\) and consider a real-valued differential equation
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}u(t) = -\alpha u(t)\hbox {d}t + \beta u(t-1)\hbox {d}t,\qquad t\ge 0,\\ u(\theta )=1, \,\theta \in [-1, 0]. \end{array}\right. } \end{aligned}$$
(3.8)
If \(\beta =0\), the solution \(u(t)= e^{-\alpha t}\), \(t\ge 0\), which is exponentially stable. Otherwise, it is clear that \({\mathcal {H}}= {\mathbb {R}}\times L^2([-1, 0]; {\mathbb {R}})\) equipped with the usual inner product. If we take the above viewpoint to consider system (3.7), then it turns out in this case that for any number \(a>0\) and \(\phi \in {\mathscr {D}}({\mathcal {A}})\),
$$\begin{aligned} \begin{aligned} \langle {\mathcal {A}}\phi , \phi \rangle _{{\mathcal {H}}} + a\Vert \phi \Vert ^2_{{\mathcal {H}}}=&\,\, (a-\alpha )\phi _0^2 + \beta \phi _0\phi _1(-1) + \frac{1}{2}\phi ^2_1(0)\\&\,\, -\frac{1}{2}\phi ^2_1(-r) + a\int ^0_{-r} \phi ^2_1(\theta )\hbox {d}\theta . \end{aligned} \end{aligned}$$
It is obvious that one can always find a nonzero element \(\phi \in {\mathscr {D}}({\mathcal {A}})\) with \(\phi _0=\phi _1(0)=\phi _1(-r)=0\) such that
$$\begin{aligned} \langle {\mathcal {A}}\phi , \phi \rangle _{{\mathcal {H}}} + a\Vert \phi \Vert ^2_{{\mathcal {H}}}>0. \end{aligned}$$
In other words, it is impossible, according to Lemma 2.4, to find a constant \(a>0\) such that \(\Vert e^{t{\mathcal {A}}}\Vert \le e^{-a t}\), \(t\ge 0\) in spite of the fact that \(\Vert e^{tA}\Vert \le e^{-\alpha t}\), \(t\ge 0\). To avoid this difficulty, we can introduce an equivalent inner product \((\cdot , \cdot )_{\mathcal {H}}\) or norm \(|\cdot |\) to the canonical one \(\langle \cdot , \cdot \rangle _{\mathcal {H}}\) or \(\Vert \cdot \Vert \) on \({\mathcal {H}}\) so as that the semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), becomes exponentially stable under the norm \(\Vert \cdot \Vert \).

Theorem 3.2

Suppose that \(\langle \!\langle x, Ax\rangle \!\rangle _{V, V^*}\le -\alpha \Vert x\Vert ^2_V\) for all \(x\in V\) and some \(\alpha >0\). If, further,
$$\begin{aligned} \lambda< \alpha -\Big (\big (|\eta |(0) - |\eta |(-r)\big )\int ^0_{-r} e^{-2\lambda \theta }d|\eta |(\theta )\Big )^{1/2}\qquad \hbox {for some}\qquad 0<\lambda <\alpha ,\nonumber \\ \end{aligned}$$
(3.9)
where \(|\eta |(\theta )\) is the total variation on \([-r, \theta ]\), \(\theta \in [-r, 0]\), of the bounded variation function \(\eta \) given by
$$\begin{aligned} \eta (\tau ) = - \mathbf{1}_{(-\infty , -r]}(\tau )A_1 - \int ^0_{\tau }A_2(\theta )\hbox {d}\theta ,\qquad \tau \in [-r, 0], \end{aligned}$$
then there exists a constant \(M>0\) such that
$$\begin{aligned} \Vert e^{t{\mathcal {A}}}\Vert \le Me^{-\frac{\lambda }{\nu } t}\qquad \hbox {for all}\qquad t\ge 0 \end{aligned}$$
where \(\nu \) is the constant given in (3.1).

Proof

We show that there exists an equivalent inner product \((\cdot , \cdot )_{\mathcal {H}}\) on \({\mathcal {H}}= H\times L^2([-r, 0]; V)\) to the canonical one \(\langle \cdot , \cdot \rangle _{\mathcal {H}}\) on \({\mathcal {H}}\) such that
$$\begin{aligned} ({\mathcal {A}}\phi , \phi )_{\mathcal {H}}\le -\lambda \nu ^{-1}\Vert \phi \Vert _{\mathcal {H}}\qquad \hbox {for all}\qquad \phi \in {\mathscr {D}}({\mathcal {A}}). \end{aligned}$$
Here \((\cdot , \cdot )_{\mathcal {H}}\) is defined by
$$\begin{aligned} (\phi , \psi )_{\mathcal {H}}:= \langle \phi _0, \psi _0\rangle _H + \int ^0_{-r} \gamma (\theta )\langle \phi _1(\theta ), \psi _1(\theta )\rangle _{V} \hbox {d}\theta ,\qquad \phi ,\,\,\psi \in {\mathcal {H}},\qquad \end{aligned}$$
(3.10)
where \(\gamma :\, [-r, 0]\rightarrow {\mathbb {R}}_+\) is given by
$$\begin{aligned} \gamma (\theta )= e^{2\lambda \theta }\Big [\alpha -\lambda -\frac{|\eta |(0) - |\eta |(-r)}{\alpha -\lambda }\int ^0_\theta e^{-2\lambda \tau } d|\eta |(\tau )\Big ],\qquad \theta \in [-r, 0].\nonumber \\ \end{aligned}$$
(3.11)
First, note that under the conditions of Theorem 3.2, bilinear form (3.10) does define an inner product. Indeed, both (3.9) and (3.11) imply the lower boundedness of \(\gamma (\cdot )\),
$$\begin{aligned} \gamma (\theta )\ge & {} e^{-2\lambda r}\Big [\alpha -\lambda -\frac{|\eta |(0)- |\eta (-r)|}{\alpha -\lambda }\int ^0_{-r} e^{-2\lambda \tau } d|\eta |(\tau )\Big ] \\&\hbox {for any}\qquad \theta \in [-r, 0]. \end{aligned}$$
This implies that \((\cdot , \cdot )_{\mathcal {H}}\) defines an inner product on \({\mathcal {H}}\). On the other hand, it is easy to see that for any \(\phi \in {\mathcal {H}}\),
$$\begin{aligned} \begin{aligned} (\phi , \phi )_{\mathcal {H}}&= \Vert \phi _0\Vert ^2_H + \int ^0_{-r} \gamma (\theta )\Vert \phi _1(\theta )\Vert ^2_{V}\hbox {d}\theta \\&\le [1 + e^{2\lambda r}(\alpha -\lambda )]\Big (\Vert \phi _0\Vert ^2_H + \int ^0_{-r}\Vert \phi _1(\theta )\Vert ^2_{V}\hbox {d}\theta \Big ) \end{aligned} \end{aligned}$$
which implies that the inner product \((\cdot , \cdot )_{\mathcal {H}}\) is also equivalent to the canonical inner product \(\langle \cdot , \cdot \rangle _{\mathcal {H}}\) on \({\mathcal {H}}\). On the other hand,
$$\begin{aligned} \begin{aligned} (\phi , ({\mathcal {A}}+\lambda \nu ^{-1})\phi )_{\mathcal {H}} =&\,\, \Big \langle \phi _0, A\phi _0 +\lambda \nu ^{-1}\phi _0 + A_1\phi _1(-r) + \int ^0_{-r} A_2(\theta ) \phi _1(\theta )\hbox {d}\theta \Big \rangle _H\\&\,\, + \int ^0_{-r}\gamma (\theta )\langle \phi _1(\theta ), \dot{\phi }_1(\theta ) + \lambda \nu ^{-1} \phi _1(\theta )\rangle _{V}\hbox {d}\theta . \end{aligned} \end{aligned}$$
Since \(\langle x, y\rangle _H = \langle \!\langle x, y\rangle \!\rangle _{V, V^*}\) for any \(x\in V\), \(y\in H\), and
$$\begin{aligned} \langle \!\langle x, Ax\rangle \!\rangle _{V, V^*} \le -\alpha \Vert x\Vert ^2_V\qquad \hbox { for any}\qquad x\in V, \end{aligned}$$
it follows for any \(\phi \in {\mathscr {D}}({\mathcal {A}})\) that
$$\begin{aligned} \begin{aligned}&(\phi , ({\mathcal {A}}+\lambda \nu ^{-1})\phi )_{\mathcal {H}}\\&\quad \le \langle \langle \phi _0, A\phi _0\rangle \rangle _{V, V^*} + \Big \langle \Big \langle \phi _0, A_1\phi _1(-r) + \int ^0_{-r} A_2(\theta ) \phi _1(\theta )\hbox {d}\theta \Big \rangle \Big \rangle _{V, V^*}\\&\qquad +\, \int ^0_{-r} \gamma (\theta )\langle \dot{\phi }_1(\theta ), \phi _1(\theta )\rangle _{V} \hbox {d}\theta +\lambda \Vert \phi _0\Vert ^2_V +\lambda \int ^0_{-r} \gamma (\theta )\Vert \phi _1(\theta )\Vert ^2_{V} \hbox {d}\theta \qquad \qquad \\&\quad \le \,\,(\lambda -\alpha )\Vert \phi _0\Vert ^2_V + \Vert \phi _0\Vert _V\int ^0_{-r} \Vert \phi _1(\theta )\Vert _{V} d|\eta |(\theta )\\&\qquad +\,\int ^0_{-r}\gamma (\theta )\Big (\frac{1}{2}\frac{d}{\hbox {d}\theta }\Vert \phi _1(\theta )\Vert ^2_{V} +\lambda \Vert \phi _1(\theta )\Vert ^2_{V}\Big )\hbox {d}\theta . \end{aligned} \end{aligned}$$
(3.12)
By using integration by parts, one can further derive from (3.12) that for \(\phi \in {\mathscr {D}}({\mathcal {A}})\),
$$\begin{aligned} \begin{aligned} (\phi , ({\mathcal {A}}+\lambda \nu ^{-1})\phi )_{\mathcal {H}}&\le (\lambda -\alpha )\Vert \phi _0\Vert ^2_V +\Vert \phi _0\Vert _V\int ^0_{-r} \Vert \phi _1(\theta )\Vert _{V} d|\eta |(\theta ) \\&\quad \, + \frac{1}{2}(\alpha -\lambda )\Vert \phi _0\Vert ^2_V\\&\quad \, - \frac{1}{2}\gamma (-r) \Vert \phi _1(-r)\Vert ^2_{V} - \frac{|\eta |(0)-|\eta |(-r)}{2(\alpha -\lambda )} \int ^0_{-r} \Vert \phi _1(\theta )\Vert ^2_{V} d|\eta |(\theta )\qquad \qquad \\&\le -\frac{1}{2}(\alpha -\lambda )\Vert \phi _0\Vert ^2_V + \Vert \phi _0\Vert _{V}\int ^0_{-r} \Vert \phi _1(\theta )\Vert _{V} d|\eta |(\theta )\\&\quad \, - \frac{|\eta |(0)-|\eta |(-r)}{2(\alpha -\lambda )}\int ^0_{-r} \Vert \phi _1(\theta )\Vert ^2_{V} d|\eta |(\theta ). \end{aligned} \end{aligned}$$
(3.13)
If \(\Vert \phi _0\Vert _V=0\) or \(|\eta |(0)=0\), i.e., \(\eta \) is constant, it is immediate from (3.13) that
$$\begin{aligned} (\phi , ({\mathcal {A}}+\lambda \nu ^{-1} )\phi )_{\mathcal {H}}\le 0\qquad \hbox { for all}\qquad \phi \in {\mathscr {D}}({\mathcal {A}}). \end{aligned}$$
If \(\Vert \phi _0\Vert _V\not =0\) and \(|\eta |(0)>0\) for \(\phi \in {\mathscr {D}}({\mathcal {A}})\), we have from (3.13) that
$$\begin{aligned} \begin{aligned} (\phi , ({\mathcal {A}}+\lambda \nu ^{-1} )\phi )_{\mathcal {H}} \le&\,\, \Vert \phi _0\Vert ^2_V\int ^0_{-r} \Big [ -\frac{\alpha -\lambda }{2(|\eta |(0)-|\eta |(-r))} + \frac{\Vert \phi _1(\theta )\Vert _{V}}{\Vert \phi _0\Vert _V}\\&\,\, - \frac{|\eta |(0)-|\eta |(-r)}{2(\alpha -\lambda )}\frac{\Vert \phi _1(\theta )\Vert _{V}^2}{\Vert \phi _0\Vert _V^2}\Big ] d|\eta |(\theta )\\ =&\,\, \frac{\Vert \phi _0\Vert ^2_V}{2} \cdot \frac{|\eta |(0)-|\eta |(-r)}{\lambda -\alpha }\int ^0_{-r} \Big (\frac{\Vert \phi _1(\theta )\Vert _{V}}{\Vert \phi _0\Vert _V} - \frac{\alpha -\lambda }{|\eta |(0)-|\eta |(-r)}\Big )^2 d|\eta |(\theta )\\ \le&\,\, 0. \end{aligned} \end{aligned}$$
Hence, it follows that \((\phi , ({\mathcal {A}}+\lambda \nu ^{-1} )\phi )_{\mathcal {H}}\le 0\) for all \(\phi \in {\mathscr {D}}({\mathcal {A}})\). Due to the equivalence between \(\langle \cdot , \cdot \rangle _{\mathcal {H}}\) and \((\cdot , \cdot )_{\mathcal {H}}\), this further implies, in addition to Lemma 2.4, that \(C_0\)-semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), is exponentially stable. The proof is complete now. \(\square \)

4 Stability of Stochastic Delay Systems

Let \(r>0\) and we consider the following stochastic retarded evolution equation in the real Hilbert space H,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t) = Ay(t)\hbox {d}t + A_1y(t-r)\hbox {d}t + \displaystyle \int ^0_{-r} A_2(\theta ) y(t+\theta )\hbox {d}\theta \hbox {d}t + By(t)\hbox {d}W(t),\quad t\ge 0,\\ y(0)=\phi _0,\,\,y(\theta ) =\phi _1(\theta ),\,\,\theta \in [-r, 0],\,\,(\phi _0, \phi _1)\in {\mathcal {H}} = H\times L^2([-r, 0]; V), \end{array}\right. }\nonumber \\ \end{aligned}$$
(4.1)
where \(B\in {\mathscr {L}}(H, {\mathscr {L}}_2(K_Q, H))\), \(A_1\in {\mathscr {L}}(V, V^*)\), \(A_2(\cdot )\in L^2([-r, 0]; {\mathscr {L}}(V, V^*))\) and \(W(\cdot )\) is a given Q-Wiener process.
Let \({\mathcal {A}}\) be the generator of the delay semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), on \({\mathcal {H}}= H\times L^2([-r, 0], V)\) with
$$\begin{aligned} {\mathcal {A}}\phi =\Big (A\phi _0 +A_1 \phi _1(-r) + \displaystyle \int ^0_{-r} A_2(\theta )\phi _1(\theta )\hbox {d}\theta , \frac{\hbox {d}\phi _1(\theta )}{{\hbox {d}\theta }}\Big )\qquad \hbox {for any}\,\,\,\,\phi \in {\mathscr {D}}({\mathcal {A}}).\end{aligned}$$
Further, define a linear operator \({\mathcal {B}}\in {\mathscr {L}}({\mathcal {H}}, {\mathscr {L}}_2(K_Q, {\mathcal {H}}))\) by
$$\begin{aligned} {\mathcal {B}}\phi = (B\phi _0, 0)\qquad \hbox { for any}\qquad \phi \in {\mathcal {H}}. \end{aligned}$$
(4.2)
Then, equation (4.1) can be lifted up into an equivalent stochastic evolution equation without delays,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}Y(t) = {\mathcal {A}}Y(t)\hbox {d}t + {\mathcal {B}}Y(t)\hbox {d}W(t),\,\,\,\,t\ge 0,\\ Y(0)=\phi ,\,\,\,\,\,\phi \in {\mathcal {H}}, \end{array}\right. } \end{aligned}$$
(4.3)
where \(Y(t) =(y(t), y_t)\), \(t\ge 0\). Note that we have the relation \(\Vert {\mathcal {B}}\Vert = \Vert B\Vert .\) Indeed, for any \(\phi \in {\mathcal {H}}\), it is easy to see that
$$\begin{aligned} \Vert {\mathcal {B}}\phi \Vert ^2_{\mathcal {H}}= \Vert (B\phi _0, 0)\Vert ^2_{\mathcal {H}} = \Vert B\phi _0\Vert ^2_H \le \Vert B\Vert ^2\Vert \phi _0\Vert ^2_H\le \Vert B\Vert ^2\Vert \phi \Vert ^2_{\mathcal {H}}, \end{aligned}$$
thus, \(\Vert {\mathcal {B}}\Vert \le \Vert B\Vert \). On the other hand, let \(\phi =(\phi _0, 0)\), \(\phi _0\in H\), then it follows that
$$\begin{aligned} \Vert B\phi _0\Vert ^2_H = \Vert {\mathcal {B}}\phi \Vert ^2_{\mathcal {H}} \le \Vert {\mathcal {B}}\Vert ^2\Vert \phi \Vert ^2_{\mathcal {H}} = \Vert {\mathcal {B}}\Vert ^2\Vert \phi _0\Vert ^2_H, \end{aligned}$$
which immediately implies that \(\Vert B\Vert \le \Vert {\mathcal {B}}\Vert .\) Hence, we finally obtain the equality \(\Vert B\Vert =\Vert {\mathcal {B}}\Vert \).
Now let \(K= {\mathbb {R}}\), \(W(t)=w(t)\), a standard real Brownian motion, \(A_2(\cdot )\equiv 0\), \(B\in {\mathscr {L}}(H)\), then we have for any \(\beta \in {\mathbb {R}}\) that
$$\begin{aligned} ({\mathcal {A}} +\beta {\mathcal {B}})\phi = \Big ((A+\beta B)\phi _0 + A_1\phi _1(-r), \frac{\hbox {d}\phi _1(\theta )}{\hbox {d}\theta }\Big ). \end{aligned}$$
Hence, by virtue of Proposition 2.1 and Theorem 3.2, we have that if there exists a number \(\beta \in {\mathbb {R}}\) such that \({A}+\beta {B}\) generates an exponentially stable \(C_0\)-semigroup and
$$\begin{aligned} \Vert e^{t(A+\beta B)}\Vert\le & {} Me^{-\alpha t},\quad \frac{\beta ^2}{2} + \Vert B\Vert ^2 \le 2\lambda ,\Vert A_1\Vert ^2e^{2\lambda r}< (\alpha - \lambda )^2\quad \hbox {for some}\quad \alpha>\lambda >0,\nonumber \\ \end{aligned}$$
(4.4)
then the mild solution Y of (4.3) (thus, y of (4.1)) is exponentially stable in the almost sure sense. In this case, if \(A_1\not = 0\), the delay parameter \(r>0\) satisfies
$$\begin{aligned} r< \frac{1}{\lambda }\ln \Big (\frac{\alpha -\lambda }{\Vert A_1\Vert }\Big ). \end{aligned}$$
(4.5)
Last, let us study an example in which the delay appears in the highest-order derivative term of the stochastic system to close this work. The example (1.10) in Section 1 can be also treated in a similar manner.

Example 4.1

First consider the following linear stochastic partial differential equation,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t, \xi ) =\displaystyle (1 +\alpha _0) \frac{\partial ^2}{\partial \xi ^2}y(t, \xi )\hbox {d}t + \sigma y(t, \xi )\hbox {d}w(t),\qquad t\ge 0,\qquad \xi \in [0, \pi ],\\ y(t, 0)=y(t, \pi )=0,\qquad t\ge 0,\\ y(0, \cdot )=\phi _0(\cdot )\in L^2(0, \pi ), \end{array}\right. } \end{aligned}$$
(4.6)
where \(\alpha _0\in {\mathbb {R}}\) and w is a standard real Brownian motion. By virtue of Example 2.1, it is known that if
$$\begin{aligned} -1-\alpha _0<\sigma ^2/2, \end{aligned}$$
the null solution of (4.6) is exponentially stable in the almost sure sense.
Now let \(r>0\) and consider the following linear stochastic delay partial differential equation,
$$\begin{aligned} {\left\{ \begin{array}{ll} \hbox {d}y(t, \xi ) =\displaystyle \frac{\partial ^2}{\partial \xi ^2}y(t, \xi )\hbox {d}t + \alpha _0 \frac{\partial ^2}{\partial \xi ^2}y(t-r, \xi )\hbox {d}t + \sigma y(t, \xi )\hbox {d}w(t),\quad t\ge 0,\quad \xi \in [0, \pi ],\\ y(t, 0)=y(t, \pi )=0,\quad t\ge 0,\\ y(0, \cdot )=\phi _0(\cdot )\in L^2(0, \pi ),\,\, y_0(\cdot , \cdot ) = \phi _1(\cdot , \cdot )\in L^2([-r, 0]; L^2(0, \pi )). \end{array}\right. } \end{aligned}$$
(4.7)
We show that under the condition \(-1-\alpha _0<\sigma ^2/2\), the pathwise exponential stability of the null solution of (4.7) is not sensitive to small delays \(r>0\).
Indeed, let \(V=H^1_0(0, \pi )\cap H^2(0, \pi )\), \(H= L^2(0, \pi )\) and \({\mathcal {A}}\) be the generator of the associated delay semigroup \(e^{t{\mathcal {A}}}\), \(t\ge 0\), in \({\mathcal {H}}= H\times L^2([-r, 0], V)\) with
$$\begin{aligned} {\mathcal {A}}\phi =(\Delta \phi _0 + \alpha _0\Delta \phi _1(-r), {\hbox {d}\phi _1(\theta )}/{\hbox {d}\theta })\qquad \hbox {for}\,\,\,\,\phi =(\phi _0, \phi _1)\in {\mathscr {D}}({\mathcal {A}}), \end{aligned}$$
where \(\Delta = \partial ^2/\partial \xi ^2\) is the standard Laplace operator. Once again, let \({\mathcal {B}}\in {\mathscr {L}}({\mathcal {H}})\) be given by \({\mathcal {B}}\phi = (\sigma \phi _0, 0)\) for \(\phi = (\phi _0, \phi _1)\in {\mathcal {H}}\). Then,
$$\begin{aligned} ({\mathcal {A}}+\beta {\mathcal {B}})\phi =(\Delta \phi _0 +\alpha _0\Delta \phi _1(-r) + \beta \sigma \phi _0, {\hbox {d}\phi _1(\theta )}/{\hbox {d}\theta }),\qquad \phi \in {\mathscr {D}}({\mathcal {A}}). \end{aligned}$$
By virtue of (4.4), we have that if there exists some number \(\beta \in {\mathbb {R}}\) such that for some \(\lambda >0\),
$$\begin{aligned} \lambda< 1-\beta \sigma ,\qquad \frac{\beta ^2}{2} + \sigma ^2\le 2\lambda ,\qquad \alpha ^2_0e^{2\lambda r}< (1-\beta \sigma -\lambda )^2, \end{aligned}$$
(4.8)
then the null solution of (4.7) is exponentially stable in the almost sure sense. It may be verified that \(\beta =-2\sigma \) and \(\lambda = \frac{3}{2}\sigma ^2 +\varepsilon \) with \(\varepsilon >0\) sufficiently small satisfy condition (4.8). In this case, the third inequality in (4.8) is actually reduced to
$$\begin{aligned} |\alpha _0|\exp (3\sigma ^2 r/2)-1 < \sigma ^2/2. \end{aligned}$$
If \(\alpha _0\not = 0\), \(\sigma \not = 0\), this condition is equivalent to
$$\begin{aligned} r< \frac{2}{3\sigma ^2}\ln \Big (\frac{\frac{1}{2}\sigma ^2+1}{|\alpha _0|}\Big ). \end{aligned}$$
(4.9)
In other words, in addition to the condition \(-1 -\alpha _0<\sigma ^2/2\) which ensures the pathwise exponential stability of Eq. (4.6), if we further assume that the relation (4.9) is true, then the null solution of (4.7) is exponentially stable in almost sure sense. Thus, the stability is not sensitive to small delays.

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of Mathematical SciencesTianjin Normal UniversityTianjinPeople’s Republic of China
  2. 2.Department of Mathematical Sciences, School of Physical SciencesThe University of LiverpoolLiverpoolUK

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