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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

Citations:

Zbl 0678.93051
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