Linear differential systems with parameter excitation

This chapter intends to introduce to the general setting of linear stochastic systems

$$\dot Z_t = {\bf A}(t) Z_t$$

, where

$$({\bf A}(t))_{t \geq 0}$$

is a matrix valued stochastic process and

$$\dot Z_t \equiv {d\over{dt}} Z_t$$

denotes the derivative with respect to the “time” variable t. Such differential systems are called parametrically excited (perturbed) or real noise linear systems; in the engineering literature the terminology rheo linear system is also used. The system matrix is assumed to be a continuous mapping defined on the state space of a Markov process which serves as stochastic input for the system. Hence, the above matrix process is of the form

$${\bf{A}}(t) = {\bf A} {\big (}X_t{\big )}$$

, where

$$(X_t)_{t \geq 0}$$

is the input process. Note for preciseness that some authors further differentiate the nature of the noise by distinguishing “real noise” on the one hand which is defined on the two-sided time set ℝ and Markovian noise on the other hand which is a Markov process with time set ℝ₊; see Arnold and Kliemann [Ar-Kl 83, p.4]. In this book both terms will be used interchangeably in the latter sense. Linear real noise systems with state-dependent coefficient matrix A(X _t ) and Markovian input noise X _t as above have for example been investigated by Frisch [Fs 66]. In this reference it is argued heuristically how a Fokker-Planck equation might be obtained for (X _t , Z _t ) _t , if X _t is stationary. Real noise systems with Markovian input noise are also subject to the considerations of Kats and Krasovskii [Ka-Kv 60]; however, these authors consider the case that

$$(X_t)_{t \geq 0}$$

is a Markov chain with finite state space. Now the setting of our work also assumes that the input process has finitely many “states of preference”, but is a continuous process defined by a stochastic differential equation with respect to Brownian motion. These preferential ( “metastable” ) states correspond to certain time scales which will be made precise in the next chapter.