## Representations of mild solutions of time-varying linear stochastic equations and the exponential stability of periodic systems.(English)Zbl 1072.60047

Chapter 1 contains preliminaries. Let $$(\Omega, F,{\mathcal F}_t$$, $$t\in [0,\infty), P)$$ be a stochastic basis and $$L^2_s(H)= L^2(\Omega,{\mathcal F}_s, P,H)$$ and consider the stochastic equation $dy(t)= A(t) y(t)\,dt+ \sum^m_{i=1} G_i(t) y(t)\,d\omega_i(t).\tag{1}$ The author associates to (1) the approximating system $dy_n(t)= A_m(t) y_n(t)\,dt+ \sum^m_{i=1} G_i(t) y_m(t)\,d\omega_i(t).$ Also, she considers the Lyapunov equation ${dQ(s)\over ds}+ A^*(s) Q(s)+ Q(s)A(s)+ \sum^m_{i=1} G^*_i(s) Q(s) G_i(s)= 0,\quad s\geq 0.\tag{2}$ $$Q(s)$$ [see G. Da Prato and A. Ichikawa, Syst. Control Lett. 9, 165–172 (1987; Zbl 0678.93051)] is a mild solution on the interval $$J\subset\mathbb{R}$$ of (2).
Chapter 2 treats “Differential equations on $${\mathcal H}_2$$”. Chapter 3 is entitled “The covariance operator of the mild solutions of linear stochastic differential equations and the Lyapunov equations”. This chapter contains the proof of the theorem: Let $$V$$ be another real separable Hilbert space and $$B\in L(H, V)$$. If $$y(t,s;\xi)$$, $$\xi\in L^2_s(H)$$, is the mild solution of (2) and $$Q(t,s,R)$$ is the unique mild solution of (2) with the final value $$Q(t)= R\geq 0$$, then
a) $$(E[y(t, s;\xi)\otimes y(t,s,\xi)])= \text{Tr\,}Q(t,s,u\otimes u)E(\xi\otimes\xi)$$ for all $$u\in H$$;
b) $$E\| By(t,s;\xi)\|= \text{Tr\,}Q(t,s,B^* B)E(\xi\otimes\xi)$$.
Chapter 4 is devoted to “The solution operators associated to the Lyapunov equations”. The next chapter “The uniform exponential stability of linear stochastic systems with periodic coefficients” contains the proof of the theorem: The following assertions are equivalent:
a) equation (1) is uniformly exponential stable;
b) $$\lim_{n\to\infty} E\| y(n,\tau,0; x)\|^2= 0$$ uniformly for $$x\in H$$, $$\| x\|= 1$$;
c) $$\rho(T(\tau, 0))< 1$$.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35B40 Asymptotic behavior of solutions to PDEs

Zbl 0678.93051
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