Stationary solutions of linear ODEs with a randomly perturbed system matrix and additive noise

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1 Stationary solutions of linear ODEs with a randomly perturbed system matrix and additive noise H.-J. Starkloff a, R. Wunderlich b a Martin-Luther-Universität Halle Wittenberg, Fachbereich für Mathematik und Informatik, 699 Halle (Saale), Germany b Westsächsische Hochschule Zwickau (FH), Fachgruppe Mathematik, PSF 237, 856 Zwickau, Germany Abstract The paper considers systems of linear first-order ODEs with a randomly perturbed system matrix and stationary additive noise. For the description of the long-term behavior of such systems it is necessary to study their stationary solutions. We deal with conditions for the existence of stationary solutions as well as with their representations and the computation of their moment functions. Assuming small perturbations of the system matrix we apply perturbation techniques to find series representations of the stationary solutions and give asymptotic expansions for their first- and second-order moment functions. We illustrate the findings with a numerical example of a scalar ODE, for which the moment functions of the stationary solution still can be computed explicitly. This allows the assessment of the goodness of the approximations found from the derived asymptotic expansions. Keywords: stationary solution, randomly perturbed system matrix, perturbation method, asymptotic expansions, correlation function MSC2 classification scheme numbers: 6G, 93E3

2 258 H.-J. Starkloff, R. Wunderlich Introduction The present paper considers systems of first-order ODEs ż(t,ω) A(ω)z(t,ω) + f(t,ω) (.) for the random function z z(t,ω) on R Ω with values in C n, n N, which is defined on a probability space (Ω, G,P), where G denotes a suitable σ-algebra of subsets of Ω on which a probability measure P is defined. Further, A denotes an n n matrix with random complex entries. The inhomogeneous term contains the random excitation function f(t,ω) defined on R Ω with values in C n. It is assumed that f is strict- and wide-sense stationary, pathwise and mean-square continuous and is independent of A. In technical applications the random inhomogeneous term f is called additive noise. The present paper deals with conditions for the existence of stationary solutions z to (.) as well as their representation and with the computation of their moment functions to given A and f. A random function ξ : R Ω C m, m N, is said to be stationary (in the strict sense) if for every sequence t,...,t N R, N N, the joint distribution of the random vectors ξ(t + τ),...,ξ(t N + τ) is independent of τ R. Further, a random function z(t,ω) is called a stationary solution of (.) if z pathwise satisfies Eq. (.) and if (z,f) is a stationary random function, i.e., z and f are stationarily related. The above problem arises e.g. in the investigation of the long-term behaviour of the response of discrete vibration systems with permanently acting random external excitations (see Soong, Grigoriu [6], Preumont [6] and [7, 3, 4]). Moreover equations of this type arise as result of the semi-discretization of some kinds of partial differential equations (PDEs) with respect to the spatial variables using finite difference or finite element methods. For PDEs describing random heat propagation in heterogeneous media (see e.g. [4]) the random matrix A represents a random spatially varying heat conductivity while the random process f represents random external heat fluxes on the boundary, heat sources or ambient temperatures. For PDEs describing random vibrations of continuous vibration systems we refer to [, ]. In this case the matrix A represents a random spatially varying bending stiffness and f describes external excitations. Especially in the mathematical modelling of vibration phenomena the modal analysis and the use of so-called modal coordinates leads to equations of type (.) with complex state variables and parameters. Therefore we consider this general case throughout this paper and mention that the real-valued case is contained as a special case. In the case of a non-random matrix A the stability of this matrix, i.e., all eigenvalues of A possess strictly negative real parts, guarantees that a unique stationary solution exists and solutions to initial value problems for Eq. (.) converge for t to this stationary solution (see Arnold, Wihstutz [2], Bunke [3], p. 45ff). For the computation of moment functions of the stationary solution there exist numerous methods, see e.g. Soong, Grigoriu [6], Preumont [6], [7, 3, 4]. In general, these methods can not be applied to the present case of a randomly perturbed matrix.

3 Stationary solutions of linear ODEs with a randomly perturbed system matrix 259 The paper is organized as follows. Section 2 gives an explicit analytic representation of the stationary solution of Equation (.) and proves its uniqueness. This representation is the starting point of the computation of first- and second-order moment functions for the stationary solution in Section 3. Decomposing the random parameters involved in (.) into their respective means and centered fluctuation terms the stationary solution can be decomposed in a similar way. Using this decomposition Subsection 3. gives general representations of the considered moment functions. It turns out that an explicit evaluation of the derived formulas is possible only in a very limited number of special cases. Therefore we present in Subsection 3.2 an approximate computation applying perturbation techniques and derive the leading as well as first (non-zero) correction terms of the corresponding expansions. These expansion terms are evaluated explicitly in Section 4. Thereby we restrict to a class of Equation (.) for which the correlation function of the stationary solution in the case of an unperturbed matrix A is exponentially decaying. Finally Section 5 presents some numerical results for the case of a scalar Equation (.) with real random parameters. This is one of the rare cases where the moment functions can be computed explicitly and we can study the goodness of the approximations using the perturbation techniques. Throughout the paper we use the following notation. For a vector x C n and a matrix M C n n we denote by x the norm of x and by M some matrix norm of M which is compatible with the chosen vector norm, i.e., there holds the relation Mx M x for all x C n. If not specified otherwise the vector norm. can be chosen arbitrarily. With x and M we denote the adjoint vector and matrix. The expectation w.r.t. the probability measure P is denoted by E {.}, the conditional expectation by E {..} and the conditional probability by P{..}. For random vectors ξ (ω), ξ 2 (ω) with values in C n the covariance matrix is denoted by cov(ξ,ξ 2 ) E {(ξ E {ξ })(ξ 2 E {ξ 2 }) } and for a stationary random vector function y(t,ω), t R, with values in C n the correlation function is defined as R yy (τ) cov(y(t),y(t + τ)), τ R. 2 Stationary solution In this section we give an explicit analytic representation of the stationary solution of Equation (.) and prove its uniqueness. This representation is the starting point of the computation of first- and second-order moment functions for the stationary solution in Section 3. First we provide the corresponding result for the case of a non-random matrix A which is known from the literature. (see e.g. Bunke [3], Theorem 3.5, p. 45ff; Arnold, Wihstutz [2]) Workshop Stochastische Analysis

4 26 H.-J. Starkloff, R. Wunderlich Lemma 2. Assuming a stationary, a.s. pathwise and mean-square continuous excitation function f and a non-random stable matrix A, i.e., all eigenvalues of A possess negative real parts, then the linear first-order system possesses the unique stationary solution z(t,ω) t ż(t,ω) Az(t,ω) + f(t,ω) e A(t u) f(u,ω)du in the a.s. pathwise as well as means-square sense. e Au f(t u,ω)du (2.) If the matrix A is random and stochastically independent of the excitation function f a similar result can be derived (see also Khasminskij [5]). Theorem 2.2 Assuming a stationary, a.s. pathwise and mean-square continuous excitation function f and a random matrix A with E { A 2 } <, which is a.s. stable and independent of f, then the first-order system (.) possesses the unique stationary solution z(t,ω) in the a.s. pathwise as well as means-square sense. e A(ω)u f(t u,ω)du (2.2) Proof. The assumptions on A and f ensure that the integral in (2.2) exists and is well-defined in the a.s. pathwise as well as mean-square sense. Obviously, z(t,ω) given in (2.2) satisfies Eq. (.), i.e., z is a solution in the a.s. pathwise as well as means-square sense. In order to prove that z(t,ω) is a stationary solution of Eq. (.), one has to show that for arbitrary N N, t,...,t N R and arbitrary Borel sets B,...,B N of C 2n the probability of the events C s G defined by C s : N { } (z(ti + s,ω),f(t i + s,ω)) B i i N { ( i does not depend on s, i.e., P(C s ) P(C ). ) } e A(ω)u f(t i + s u,ω)du,f(t i + s,ω) B i, s R, Denote by F A the (n n-dimensional) distribution function of the random matrix A. Then for the probability of the events C s it can be derived P(C s ) E {P{C s A}} P{C s A M}dF A (M). C n n

5 Stationary solutions of linear ODEs with a randomly perturbed system matrix 26 Using that A is independent of f and that e Mu f(t u,ω)du is a stationary solution of (.) for a fixed matrix A(ω) M it follows P{C s A M} which implies P(C s ) ( N { ( P i ( N { ( P i P{C A M}, ) } ) e Mu f(t i + s u,ω)du,f(t i + s,ω) B i ) } ) e Mu f(t i u,ω)du,f(t i,ω) B i C n n P{C A M}dF A (M) E {P{C A}} P(C ) and proves the stationarity of z. In the evaluation of the conditional expectation in the above derivation the random matrix A has been replaced by M which is possible due to the assumed independence of A and f (see e.g. Shiryaev [5], 7, p. 22). It remains to prove the uniqueness of the solution. To this end two stationary solutions z and z 2 of Eq. (.) are considered and it is proven that the difference r z z 2 vanishes a.s. for all t R. Consider initial value problems for functions x on [t, ) Ω, t R, given by ẋ(t,ω) A(ω)x(t,ω) + f(t,ω), x(t,ω) x (ω), possessing for almost all ω Ω the unique solution x(t,ω) e A(ω)(t t) x (ω) + e A(ω)(t u) f(u,ω)du. t t Since z and z 2 satisfy Eq. (.) for all t R they coincide on [t, ) with the solutions of the above initial value problem with initial values x (ω) z (t,ω) and x (ω) z 2 (t,ω), respectively. For the difference r z z 2 it holds for t [t, ) r(t,ω) z (t,ω) z 2 (t,ω) e A(ω)(t t ) (z (t,ω) z 2 (t,ω)). Since t can be chosen arbitrarily it holds There hold the following assertions r(t) e A(t s) r(s), for all s,t R with s t. (2.3) A(t s) a.s.. e for s, since A is assumed to be a.s. stable. Workshop Stochastische Analysis

6 262 H.-J. Starkloff, R. Wunderlich 2. r(s) is stochastically bounded, i.e., supp( r(s) > c) for c, which follows s R from the relation P( r(s) > c) P( z (s) z 2 (s) > c) P( z (s) + z 2 (s) > c) ( P z (s) > c ) ( + P z 2 (s) > c ) ( 2 2 P z () > c ) ( + P z 2 () > c ) s R, 2 2 c where the stationarity of z and z 2 has been used. Hence supp( r(s) > c) s R for c. 3. e A(t s) r(s) P for s, since for any random function M : R Ω C n n with M(s) > a.s. for arbitrary c > and ε > it holds P ( M(s) r(s) > ε ) P ( M(s) r(s) > ε ) ({ ε } { ε }) P r(s) > M(s) M(s) c + ({ ε } { ε }) P r(s) > M(s) M(s) > c ( ε ) P M(s) c + P( r(s) > c). Since r(s) is stochastically bounded it holds δ > c c δ such that P( r(s) > c δ ) < δ 2. Moreover with M(s) e A(t s) it follows ε >, c δ > s such that s s and it yields ( ε ) P M(s) c δ ( P M(s) ε ) ( P e A(t s) ε ) δ c δ c δ 2 ε >, δ > s such that s s it holds P ( e A(t s) r(s) > ε ) < δ which proves the above assertion. Relation (2.3) and assertion 3) imply r(t) a.s. t R, i.e., z and z 2 coincide which proves the uniqueness of the stationary solution.

7 Stationary solutions of linear ODEs with a randomly perturbed system matrix Moment functions of the stationary solution 3. Decomposition of the stationary solution For the subsequent investigation of stationary solutions of Eq. (.) we require that the assumptions of Theorem 2.2 are fulfilled and we consider the decomposition of the inhomogeneous term f(t,ω) f + f(t,ω) into the non-random constant mean f and the random fluctuation term f(t,ω). This random process is stationary, pathwise and mean-square continuous, its correlation function coincides with the correlation function of f. Substituting the decomposition f(t,ω) f + f(t,ω) of the excitation function into the representation (2.2) of the stationary solution z of system (.) the following representation of z can be derived z(t,ω) ẑ(ω) + z(t,ω) where ẑ(ω) and z(t, ω) e A(ω)u f du A (ω) f e A(ω)u f(t u,ω)du. Here, the random vector ẑ can be considered as the response of the system to the mean excitation f and the random process z as the response to the random fluctuations of the excitation f. Theorem 2.2 implies that z is the stationary solution of the system z(t,ω) A(ω) z(t,ω) + f(t,ω). (3.) Lemma 3. Under the assumptions of Theorem 2.2 it holds E { z(t)}. Proof. It holds E { z(t)} } E {e Au f(t u) du. For the integrand it can be derived } E {e Au f(t u) E { e Au} } E { f(t u) since A and f are independent and f has zero mean. This implies E { z(t)}. The next theorem states a decomposition of first- and second-order moment functions of the stationary solution z to system (.) which is helpful for the subsequent computation of moment functions of z. Workshop Stochastische Analysis

8 264 H.-J. Starkloff, R. Wunderlich Theorem 3.2 Let A possess finite second-order moments and let the assumptions of Theorem 2.2 be fulfilled. Then for the mean and the correlation function of the stationary solution z of system (.) it holds E {z(t)} E {ẑ} E { A } f (3.2) and R zz (τ) cov(ẑ,ẑ) + R ez ez (τ). (3.3) Proof. Using the decomposition z ẑ + z introduced above it follows E {z(t)} E {ẑ} + E { z(t)}. Since A possesses finite second-order moments E {A } exists and for the first term it holds E {ẑ} E {A } f while Lemma 3. implies that the second term vanishes. This proves Eq. (3.2). To prove Eq. (3.3) again the decomposition z ẑ + z is used. It holds R zz (τ) cov(z(t),z(t + τ)) cov(ẑ,ẑ) + cov( z(t), z(t + τ)) + cov( z(t),ẑ) + cov(ẑ, z(t + τ)) cov(ẑ,ẑ) + R ez ez (τ), since the two last covariances vanish because of cov( z(t),ẑ) E { z(t) ẑ } E { z(t)}e{ẑ } EE { z(t)ẑ A} E {ẑ } E {E { z(t) A} ẑ } { } } E e Au E { f(t u) A du ẑ. Here it has been used that ẑ A f is A-measurable and that f is centered and independent of A. Analogously cov(ẑ, z(t + τ)) can be proven. The above theorem shows that the influence of the random parameters A and f on the moments of z can be separated. The mean Ez depends only on the distribution of A via E {A } and linearly on the mean excitation f. For a centered excitation f, i.e., for f, the stationary solution is centered, too. The correlation function R zz (τ) can be decomposed into the covariance of ẑ and the correlation function of z. The first term cov(ẑ,ẑ) is invariant w.r.t. τ and depends only on the distribution of A and on f. It vanishes in case of a centered excitation f. The second term R ez ez (τ) depends on the distributions of A and the fluctuating part f of the excitation but not on the mean excitation f. The explicit computation of the mean of z requires the computation of E {A }, i.e., the mean of the inverse of A. In the multi-dimensional case this is often very cumbersome

9 Stationary solutions of linear ODEs with a randomly perturbed system matrix 265 or even impossible. The same holds for the computation of { cov(ẑ,ẑ) E A f f A } Eẑ (Eẑ). For the correlation function of z it holds { R ez ez (τ) E { z(t) z (t + τ)} E E { e Au R ef e f (τ + u v)e A v } dudv, e Au f(t } u) f (t + τ v)e A v dudv where the independence of A and f had been used. It can be seen that the evaluation R ez ez (τ) requires only second-order moments of f but the complete distribution of A, since it is involved in the matrix exponentials. In general, an explicit computation of expectations of these matrix exponentials fails. Remark 3.3 Another way for the computation of the above correlation function makes use of conditional expectations w.r.t. the random matrix A. This approach might be useful for the computation of approximations of the correlation function R ez ez (τ) based on Monte-Carlo simulations. It holds R ez ez (τ) EE { z(t) z (t + τ) A } { e Au f(t } u) f (t + τ v)e A v dudv A M df A (M) C n n E C n n E { C n n R M ez ez(τ)df A (M) e M u f(t } u) f (t + τ v)e M v dudv df A (M) where F A (.) denotes the distribution function of the random matrix A and { Rez M ez(τ) E e M u f(t } u) f (t + τ v)e M v dudv e M u R ef e f (τ + u v)e M v dudv denotes the correlation function of the stationary solution of z M z + f, i.e., a system with a non-random matrix M. In the evaluation of the conditional expectation in the above derivation the random matrix A has been replaced by M which is possible due to the assumed independence of A and f (see e.g. Shiryaev [5], 7, p. 22). For special choices of the correlation function of the fluctuating part f of the excitation (e.g. weakly or exponential correlated) it is possible to simplify the above double integral and to find explicit representations or power series expansions (see Soong, Grigoriu [6], [7, 8, 2] ). Workshop Stochastische Analysis

10 266 H.-J. Starkloff, R. Wunderlich In many cases only partial information about the distribution of the random parameters (e.g. only first- and second-order moments) is available. Therefore, in the next section approximate representations of the moments of z in terms of first- and second-order moments of A and f are derived by applying perturbation methods. 3.2 Perturbation methods This section deals with the approximative computation of the mean and the correlation function of the stationary solution z of Eq. (.). The main idea of the approximation is the decomposition of the random matrix A(ω) into its constant mean  and a random fluctuating part which is scaled by a non-negative perturbation parameter η, i.e., it is set A(ω)  + ηc(ω), with η. We consider the random matrix A as a perturbation of the constant matrix Â. The desired moments of z are expanded in powers of η and for sufficiently small values of η the appropriately truncated power series can be used as approximations of the exact moments functions. If in Eq. (.) the random matrix A is replaced by its mean while the excitation term f remains unchanged the system ẏ(t,ω) Ây(t,om) + f(t,ω) (3.4) arises. We call the above system the unperturbed system since it contains the unperturbed matrix Â. From Lemma 2. it is known that y(t,ω) e b Au f(t u,ω)du is the unique stationary solution of Eq. (3.4) since A and consequently  E {A} is a stable matrix. If in addition to the random matrix A(ω) also the random inhomogeneous term f(t,ω) is replaced by its constant mean, we get the so-called averaged system ẋ(t) Âx(t) + f. (3.5) This is a non-random system possessing the trivial stationary solution x(t)  f which is non-random and does not depend on t. We note that a non-random function is stationary only if it is a constant function. The mean of stationary solution y of the unperturbed system (3.4) coincides with the solution of the averaged problem (3.5), i.e., it holds E {y(t)} e b Au E {f(t u)}du  f x.

11 Stationary solutions of linear ODEs with a randomly perturbed system matrix 267 The so-called averaging problem arises if one compares the solution x of the above averaged system with the average of the stationary solution of the original system (.). We will come back to this problem in Remark 3.5 below. For the existence and uniqueness of the stationary solution z the matrix A(ω)  + ηc(ω) is supposed to be stable and for the existence of first- and second-order moments of z finite second-order moments of the inverse matrix A (ω) are required. It is noted, that the latter condition implies the existence of first-order moments of A (ω). In order to check these conditions in terms of the perturbation parameter η it is assumed that  EA is a stable matrix which implies the existence of Â. Moreover it is assumed that the matrix C is bounded, i.e., there exists a positive real number c such that C(ω) c a.s.. Then for sufficiently small η > the matrix A(ω)  + ηc(ω) is stable and its inverse possesses finite second-order moments. Define η S : sup{η > :  + ηc(ω) is a.s. stable} η M : sup{η > : E { (  + ηc) 2 } < } then for η < η S the matrix A is stable and for η < η M its inverse possesses finite second-order moments. It is noted that the a.s. boundedness of the matrix C implies the existence of first- and second-order moments of C as well as of A  + ηc Moments of ẑ For the evaluation of Formulas (3.2) and (3.3) the mean and the covariance of ẑ A f are required. Substituting the representation A  + ηc the inverse of A can be represented using a Neumann series A which is convergent for η < ( + ηc) (Â(I + ηâ C)) (  C) p η p Â, (3.6) p  C a.s.. Let η N : sup{η > : η <  C(ω) a.s.}, then the inequality  C  C c  leads to the lower bound η N c A b for the radius of convergence of the Neumann series. The next theorem provides expansions of the first- and second-order moments of ẑ including the leading terms and the first correction terms which are of order η 2. Workshop Stochastische Analysis

12 268 H.-J. Starkloff, R. Wunderlich Theorem 3.4 For η < min{η S,η M } there hold the following expansions for η ( { ) E {ẑ} I +  E C C }η 2 x + o(η 2 ) (3.7) where x  f. and cov(ẑ,ẑ)  E {Cxx C } η 2 + o(η 2 ) (3.8) Proof. The Neumann series expansion (3.6) for A yields for η ẑ A f (I  Cη + ( C) 2 η 2 + o(η 2 )) f ) (I  Cη + ( C) 2 η 2 x + o(η 2 ) (3.9) For η < min{η S,η M } the moments E {ẑ} and cov(ẑ,ẑ) are well-defined and from Eq. (3.9) it follows ( { ) E {ẑ} I  E {C} η +  E C C }η 2 x + o(η 2 ) ( { } ) I +  E C C η 2 x + o(η 2 ) since E {C}. This proves (3.7). The proof of Eq. (3.8) uses the relation cov(ẑ,ẑ) E {(ẑ E {ẑ})(ẑ E {ẑ}) } and the expansion ( ) ẑ E {ẑ} I  Cη + o(η) x (x + o(η))  Cxη + o(η) which follows from (3.7) and (3.9). It results cov(ẑ,ẑ)  E {Cxx C } η 2 + o(η 2 ). Remark 3.5 With the help of the above theorem there can be given an answer (in an approximative sense) to the so-called averaging problem which consists in the computation of the difference between the average of the stationary solution E {z(t)} and the stationary solution x of the averaged equation (3.5). While for systems with a nonrandom linear operator both quantities coincide this is in general not the case for systems containing random operators or nonlinearities. It holds { } E {z(t)} x E {ẑ} x  E C C η 2 x + o(η 2 ).

13 Stationary solutions of linear ODEs with a randomly perturbed system matrix Correlation function of z The evaluation of Formula (3.3) for the correlation function of z requires the computation of the correlation function of z which is the stationary solution of system (3.). To find an expansion of the latter correlation function in powers of η the function z is represented as a power series z(t,ω) p ζ(t,ω)η p (3.) p with respect to the parameter η. The coefficients ζ, ζ,... can be found by substituting series (3.) into Eq. (3.) and equating the coefficients of the powers of η. First, this procedure is carried out formally. A verification of the results is given afterwards. The substitution of series (3.) into Eq. (3.) gives p ζ(t,ω)η p ( + ηc(ω)) p ζ(t,ω)η p + f(t,ω). p p For the coefficients p ζ it results an infinite sequence of linear first-order systems ζ  ζ + f p ζ  p ζ + C p ζ, p. According to Lemma 2. for the above linear systems with the non-random matrix  stationary solutions can be found recursively as follows ζ(t,ω) p ζ(t,ω) e b Au f(t u,ω)du (3.) e b Au C(ω) p ζ(t u,ω)du, p, which imply the following explicit representation of p ζ in terms of ζ p ζ(t,ω) R p + p k ( ) Au e b k C(ω) ζ(t u... u p,ω)du...du p, p. After the investigation of the coefficients of the perturbation series conditions for the convergence of series (3.) will be determined. We use the assumption on the stability of A which implies the stability of Â, i.e., it possesses eigenvalues with strictly negative Au real parts, only. Then positive real numbers λ and v exist such that e b v e λ u for all u. Further, it will be used that the random matrix C is a.s. bounded and that the function f is pathwise continuous on R. As an additional condition we impose the pathwise boundedness of f. Workshop Stochastische Analysis

14 27 H.-J. Starkloff, R. Wunderlich Theorem 3.6 Let the following assumptions be fulfilled. the matrix  is stable, there exist positive real numbers λ and v such that Au e b v e λu for u, 2. f is stationary, with a.s. continuous and bounded paths, there exists a positive random variable f (ω) such that f(t,ω) f (ω), t R, a.s., 3. there exists a positive real number c such that C(ω) c a.s.. Then the series p ζ(t) η p with coefficients p ζ(t) given in (3.) converges almost surely p with respect to ω and uniformly with respect to t R for η < η P : λ v c. Proof. In a first step by means of mathematical induction it is proven that q ζ(t) is bounded by q ζ(t) v f λ ( ) q v c t R, q,,...,a.s.. (3.2) We start with q where it holds ζ(t) Au e b f(t u)du and ζ(t) Au e b λ f(t u) du f e b Au du t R,a.s.. (3.3) Using assumption it follows Au e b du v e λ u du v λ. (3.4) Applying inequalities (3.3) and (3.4) it results ζ(t) v f λ v f λ ( ) v c t R,a.s.. Now assuming the assertion (3.2) is valid for q p the assertion for q p + will be proven. For p+ ζ(t), it follows p+ ζ(t) e Au b C p ζ(t u) du Au e b C p ζ(t u) du, (3.5) t R, a.s.. Using relation (3.2) for p ζ, relation (3.4) and C c it follows p+ ζ(t) v λ c vf λ λ ( ) p v c v ( ) p+ f v c t R,a.s. λ λ λ

15 Stationary solutions of linear ODEs with a randomly perturbed system matrix 27 and the assertion (3.2) is proven. From inequality (3.2) it results for the perturbation series p ζ(t) η p p ζ(t) η p v f ( ) p v c η t R,a.s.. λ λ p p Since the majorizing series converges for v c η λ <, a sufficient condition for the convergence of perturbation series (3.) is η < η P λ p v c. Remark 3.7 If in addition to the stability of the matrix  (see assumption in Theorem 3.6) the diagonalizability of  is supposed as an additional technical condition then the positive numbers λ and v can be further specified. Let there exists a representation  VΛV with a matrix Λ diag(λ,...,λ n ), containing the eigenvalues of  on its diagonal and a regular matrix V. Because of the stability of  the eigenvalues satisfy the relation Re [λ i ] <, i,...,n. The above representation of  yields Au e b V e Λu V V V e Λu. It holds cond(v) V V where cond(.) denotes the condition number of a matrix. If moreover the matrix norm is set to be column-sum, spectral or row-sum norm denoted by.,. 2 or., respectively, then it holds for the diagonal matrix e Λu e Λu,2, max e λ iu e max Re[λ i ] u min Re[λ i ] u i e i. i Au Hence the positive numbers λ and v involved in the inequality e b v e λu,,2, for all u R, can be chosen as λ min i Re [λ i ] and v cond,2, (V). For other matrix norms the number v has to be modified. If the matrix V can be chosen as a unitary matrix then in case of the spectral norm it holds v cond 2 (V) V 2 V 2. Assumption 2 of Theorem 3.6 on the pathwise boundedness of the random process f excludes e.g. Gaussian processes from the consideration. On the other hand the positive random variable f (ω) which bounds f(t,ω) is not involved in the definition of the bound η P for the radius of convergence. Next we show that assumption 2 can be relaxed by requiring the boundedness of f(t) in the mean-square sense. This property is fulfilled for stationary mean-square continuous random processes. Then it is possible to prove a similar convergence statement in the mean-square sense (see Theorem 3.8 below). The Workshop Stochastische Analysis

16 272 H.-J. Starkloff, R. Wunderlich class of mean-square bounded processes contains the pathwise bounded processes but also Gaussian processes. Let Q Q(Ω, G,P; C n ) be the space of random vectors with values in C n and finite second-order moments equipped with the the norm ξ Q (E { ξ 2} ) 2, where ξ Q is a random vector and. denotes some norm in C n. The space Q is known to be complete. For the convergence of a sequence (ξ m ) in the mean-square sense it is necessary and sufficient that (ξ m ) is fundamental. Moreover if ξ m Q c m with c m R, m N and c m < then ξ m converges in the mean-square sense. m m Now we formulate the announced convergence theorem. Theorem 3.8 Let the following assumptions be fulfilled. the matrix  is stable, there exist positive real numbers λ and v such that Au e b v e λu for u, 2. f is stationary, mean-square and pathwise continuous on R and there exists a positive real number f such that f(t) f, t R, Q 3. there exists a positive real number c such that C(ω) c a.s.. p Then the series p ζ(t) η p with coefficients p ζ(t) given in (3.) converges in the meansquare sense with respect to ω and uniformly with respect to t R for η < η P : λ v c. Proof. As in the proof of Theorem 3.6 we use the mathematical induction to prove that q ζ(t) Q is bounded by q ζ(t) Q v f λ ( ) q v c t R, q,,.... (3.6) λ For a non-random matrix M and a random vector ξ Q there holds the relation Mξ 2 Q E { Mξ 2} E { M 2 ξ 2} M 2 E { ξ 2} M 2 ξ 2 Q, hence we have Mξ Q M ξ Q. For q the above relation yields ζ(t) Q Au e b f(t u) du f Q e b Au du t R.

17 Stationary solutions of linear ODEs with a randomly perturbed system matrix 273 Applying inequality (3.4) to the integral on the right hand side it results ζ(t) Q v f λ v f λ ( ) v c t R. Now assuming the assertion (3.2) is valid for q p the assertion for q p + will be proven. For p+ ζ(t), it follows p+ ζ(t) Q e Au b C p ζ(t u) du e Au b C p ζ(t u) du Q Q Au e b C p ζ(t u) Q du Au e b ( E { C 2 p ζ(t u) 2}) 2 du Au e b c p ζ(t u) Q du t R. Using relation (3.6) for p ζ and relation (3.4) it follows p+ ζ(t) Q v λ c vf λ and the assertion (3.6) is proven. λ ( ) p v c v ( ) p+ f v c t R From inequality (3.6) it results for the perturbation series p ζ(t) η p p ζ(t) Q η p v f ( ) p v c η t R. λ p Q p λ p λ Since the majorizing series converges for v c η λ <, a sufficient condition for the meansquare convergence of perturbation series (3.) is η < η P λ The a.s. convergence of the series ζ(t) p ζ(t) η p for η < η P which follows from p λ λ v c. Theorem 3.6 is the key result of the proof of the following theorem. Theorem 3.9 Let f(t,ω) be a stationary and an a.s. pathwise as well as mean-square continuous random function. Further, let A(ω)  + ηc(ω) be a random matrix which is independent of f and  E {A} is assumed to be a stable matrix. Finally, let the assumptions of Theorem 3.6 be fulfilled and let η S > be such that for η < η S the matrix A(ω) is a.s. stable. Then for η < η S Equation (3.) z(t,ω) A(ω) z(t,ω) + f(t,ω) Workshop Stochastische Analysis

18 274 H.-J. Starkloff, R. Wunderlich possesses the unique stationary solution z(t,ω) e A(ω)u f(t u,ω)du which admits for η < min{η S,η P } a representation as perturbation series z(t) p ζ(t) η p with coefficients p ζ given in (3.). p Proof. First, we prove that for η < η P the perturbation series ζ(t) p p ζ(t)η p is a stationary process and stationarily related to f. Following the lines of the proof of Theorem 3 in our paper [9] by means of mathematical induction it can be deduced that for all N,,... the processes f, ζ,..., N ζ as well as the processes f and N p ζ η p are stationarily related. From Theorem 3.6 it is known that for η < η P the series N p ζ(t) η p converges for N almost surely with respect to ω and uniformly with respect to t R. Consequently, the limit ζ(t) p ζ(t) η p is stationarily related to f. p Now, it suffices to prove that ζ satisfies Equation (3.) for η < min{η S,η P } since due to Theorem 2.2, Eq. (3.) possesses a unique stationary solution. Therefore, if ζ satisfies (3.) then it coincides with the stationary solution z given above. In order to show that ζ satisfies Eq. (3.) it is first proven that the representation ζ(t) p ζ(t) η p is valid for η < η P. To this end the uniform convergence of the formal p ( differentiated series d p ζ(t) η p) for η < min{η dt S,η P } is checked. Using representation p (3.) of the coefficients p ζ formal differentiation leads to ( d ) p ζ(t) η p dt p p ζ(t) η p p  ζ + f + p p p ( p ζ(t) + C p ζ(t)) η p ( + ηc) p ζ(t)η p + f(t). (3.7) The uniform convergence of the series on the right hand side for η < min{η S,η P } follows immediately from Theorem 3.6. Moreover, from the above relation it follows ζ(t) Aζ + f, i.e., ζ satisfies Eq. (3.) for η < min{η S,η P }. p

19 Stationary solutions of linear ODEs with a randomly perturbed system matrix 275 The series expansion of z given in Theorem 3.9 can be used to find expansions of the correlation function of z in powers of η. Using z ζ + ζη + 2 ζη 2 + o(η 2 ) leads to the following result. Theorem 3. Let E { A 2 } <. Then under the assumptions of Theorem 3.9 the correlation function of z(t) e Au f(t u)du for η < ηs possesses the expansion for η R ez ez (τ) Rζ ζ(τ) + (R2ζ ζ(τ) + Rζ ζ(τ) + Rζ 2 ζ(τ))η 2 + o(η 2 ), uniformly for all τ R where R2ζ ζ(τ) Rζ ζ(τ) { } Au e b Au E Ce b 2 C Rζ ζ(τ + u + u 2 )du du 2, e b Au E { CRζ ζ(τ + u u 2 )C } e b A u 2 du du 2 and Rζ 2 ζ(τ) R 2ζ ζ ( τ). Proof. The assumption E { A 2 } < ensures that the correlation function R ez ez (τ) exists and is well-defined. From the perturbation series representation of z given in Theorem 3.9 it follows z(t) ζ(t) + ζ(t)η + 2 ζ(t)η 2 + o(η 2 ) uniformly for all t R and R ez ez (τ) Rζ ζ(τ) + ( Rζ ζ(τ) + Rζ ζ(τ) ) η + ( R2ζ ζ(τ) + Rζ ζ(τ) + Rζ 2 ζ(τ) ) η 2 + o(η 2 ) (3.8) uniformly for all τ R. Recalling representations (3.) for ζ, ζ and 2 ζ, i.e., ζ(t) ζ(t) 2 ζ(t) e b Au f(t u)du, e b Au C ζ(t u)du and e b Au C ζ(t u)du e b Au Ce b Au 2 C ζ(t u u 2 )du du 2, and using the independence of C and f, } E {C} and E { f(t) it follows E { ζ(t) } E { ζ(t) } E {2 ζ(t) } } e Au b E { f(t u) du, e b Au E {C} E { ζ(t u) } du and { } Au e b Au E Ce b 2 C E { ζ(t u u 2 ) } du du 2. Workshop Stochastische Analysis

20 276 H.-J. Starkloff, R. Wunderlich Then for the correlation functions involved in (3.8) in the coefficient of η it can be derived Rζ ζ(τ) E { ζ(t) ζ (t + τ) } { } E e Au b C ζ(t u)du ζ (t + τ) e b Au E {C} E { ζ(t u) ζ (t + τ) } du and analogously Rζ ζ(τ) while for the correlation functions involved in the coefficient of η 2 it holds R2ζ ζ(τ) E {2 ζ(t) ζ (t + τ) } { } Au E e b Au Ce b 2 C ζ(t u u 2 )du du 2 ζ (t + τ) and e b Au E { } Au Ce b 2 C { } Au e b Au E Ce b 2 C Rζ ζ(τ) E { ζ(t) ζ (t + τ) } { ( E e Au b C ζ(t u)du Since for t,t 2 R it follows E { ζ(t u u 2 ) ζ (t + τ) } du du 2 Rζ ζ(τ + u + u 2 )du du 2 ) } e Au b C ζ(t + τ u)du e b Au E { C ζ(t u ) ζ (t + τ u 2 )C } e b A u 2 du du 2. E { C ζ(t ) ζ (t 2 )C } E E { C ζ(t ) ζ (t 2 )C C } Rζ ζ(τ) E { CE { ζ(t ) ζ (t 2 ) C } C } E { CE { ζ(t ) ζ (t 2 ) } C } E { CRζ ζ(t 2 t )C } e b Au E { CRζ ζ(τ + u u 2 )C } e b A u 2 du du 2. Remark 3. If the distribution of the random matrix C is symmetric w.r.t. then it can be shown, that there vanish all terms of the expansion for the correlation function corresponding to odd powers of η. In this case the remainder term in the above theorem is of order o(η 3 ) instead of o(η 2 ).

21 Stationary solutions of linear ODEs with a randomly perturbed system matrix 277 The above theorem allows an approximate computation of the correlation function of z in terms of the second-order moments of the random matrix C and the correlation function of ζ which is the stationary solution of the linear system ζ  ζ + f. Here, the system matrix  is non-random and standard procedures for the computation of the correlation function of ζ can be applied. The correlation function of ζ coincides with the correlation function of the stationary solution y of system (3.4) since the excitation terms differ only by the constant vector f. 4 Computation of expansion terms This section deals with procedures for an efficient computation of the expansion terms for the mean and the correlation function of the stationary solution z. Based on the decomposition z(t,ω) ẑ + z(t,ω) Theorem 3.2 shows that Ez(t) Eẑ and R zz (τ) cov(ẑ,ẑ) + R ez ez (τ). For the moments on the right hand sides Theorems 3.4 and 3. provide the leading and the first non-zero correction terms of expansions in powers of η where the representation A(ω)  + ηc(ω) has been used. While the correction terms of the expansions for Eẑ and cov(ẑ,ẑ) allow a straightforward computation as linear combinations of the covariances of the entries of C the situation in the case of R ez ez (τ) is more complicated. Here, the computation of the correction term requires the evaluation of double integrals containing the covariances of C, the correlation function of f and matrix exponentials of the form e b Au. A numerically efficient computation of these double integrals is possible for the special case of a diagonal matrix Â, i.e., Â Λ diag(λ,...,λ n ), then it holds e b Au diag(e λ u,...,e λnu ). In the general case Eq. (.) can be transformed into a system with a diagonal matrix provided  is diagonalizable. Using the substitution z Vz where V is such that  VΛV from Eq. (.) it follows that z satisfies ż V ( + ηc)vz + V f (Λ + ηc )z + f, (4.) where C V CV and f V f. The moments of C and f are obtained from the corresponding moments of C and f, it holds Ef V Ef, R f f (τ) V R ff (τ)v, EC E {C (C ) } and V E {CVV C }V while the moments of z can be found from the moments of z by Ez V Ez and R zz (τ) V R z z (τ) V. Therefore the subsequent computation of expansion terms can be restricted to the case of a diagonal matrix Â. Workshop Stochastische Analysis

22 278 H.-J. Starkloff, R. Wunderlich Corollary 4. Let the assumptions of Theorem 2.2 applied to Eq. (4.) ż (Λ + ηc)z + f, be fulfilled. Then for η < min{η S,η M } there hold the following expansions for η ( { n } Ez Eẑ I + E {C ik C kj } η )x 2 + o(η 2 ) λ i λ k ij k { n and cov(ẑ, ẑ) E { } } C ik C jl xk x l η 2 + o(η 2 ), λ i λ j ij where x Λ f. k,l Proof. Applying Theorem 3.4 to Eq. (4.) it follows Ez Eẑ ( I + Λ E { CΛ C } η 2) x + o(η 2 ) and cov(ẑ,ẑ) Λ E {Cxx C }Λ η 2 + o(η 2 ), from which the assertions follow immediately. For the correlation function of z Theorem 3. gives for η < η S and τ R the expansion for η R ez ez (τ) Rζ ζ(τ) + (R2ζ ζ(τ) + Rζ ζ(τ) + Rζ 2 ζ(τ))η 2 + o(η 2 ), where R2ζ ζ(τ) e Λu E { Ce Λu 2 C } Rζ ζ(τ + u + u 2 )du du 2, Rζ ζ(τ) e Λu E { CRζ ζ(τ + u u 2 )C } e Λ u 2 du du 2 and Rζ 2 ζ(τ) R 2ζ ζ ( τ). For the explicit computation of the expansion terms it is necessary to prescribe a certain form of the correlation function R ζ ζ (τ). Here it is assumed that it holds R ζ ζ (τ) m Q r e Πrτ for τ ρ r (τ) with ρ r (τ) e Π rτ Q r for τ < r (4.2) where m N and for r,...,m the Q r are some non-negative definite and Hermitian n n-matrices and Π r diag(π r,...,π rn ) with Re [π ri ] <.

23 Stationary solutions of linear ODEs with a randomly perturbed system matrix 279 This type of correlation function arises e.g. if the correlation function of f is chosen to be R ef e f (τ) Le Γτ, τ, where L is some non-negative definite and Hermitian matrix and Γ diag(γ,...,γ n ) with Re [γ i ] <, i...,n. Then for τ it can be derived R ζ ζ (τ) Q e Λ τ + Q 2 e Γτ, with some matrices Q and Q 2 depending on L,Λ and Γ whose sum Q +Q 2 forms the covariance matrix of ζ. It is noticed that for a δ-correlated excitation f with the correlation function R ef e f (τ) L δ(τ) one obtains for τ R ζ ζ (τ) Q e Λ τ with Q { Lij λ i + λ j }i,j,...,n and for a weakly correlated excitation the above exponential type correlation function arises in the terms of the expansion of Rζ ζ(τ) in powers of the correlation length (see [7] and [7, 8]). The subsequent evaluations of the entries of the matrix-valued functions R2ζ ζ(τ), Rζ ζ(τ) and Rζ 2 ζ(τ) are given for τ. For negative τ the property R ξ ξ 2 (τ) R ξ 2 ξ ( τ) can be used.. It holds for i,j,...,n R2 ζ i ζ j (τ) where J,rijkl (τ) l m n e λ iu { { E Ce Λu 2 C } } m ρ il rlj (τ + u + u 2, )du du 2 n l r k,l e λ iu r n E {C ik C kl } e λ ku 2 m k r n E {C ik C kl }J,rijkl (τ), e λ iu +λ k u 2 ρ rlj (τ + u + u 2 )du du 2, ρ rlj (τ + u + u 2 )du du 2 R ζ i 2 ζ j (τ) R2 ζ j ζ i ( τ) m n E { } C jk C kl J,rjikl ( τ) r k,l Workshop Stochastische Analysis

24 28 H.-J. Starkloff, R. Wunderlich and R ζ i ζ j (τ) where J 2,rijkl (τ) e λ iu m r k,l { e λ iu [E C }] m ρ r (τ + u u 2 )C e λ ju 2 du du 2 r ij n E { } m C ik C jl ρ rkl (τ + u u 2 ) e λ ju 2 du du 2 k,l r n E { } C ik C jl J2,rijkl (τ), e λ iu +λ j u 2 ρ rkl (τ + u u 2 )du du 2. Using that the entries of ρ r (τ) {ρ rij (τ)} i,j,...,n given in (4.2) can be represented as Q rij e π rjτ for τ ρ rij (τ) Q rji e π riτ for τ <. the terms J and J 2 can be computed explicitly. First, J is evaluated, it holds J,rijkl (τ) e λ iu +λ k u 2 ρ rlj (τ + u + u 2 )du du 2 Q rlj e λ iu +λ k u 2 +π rj (τ+u +u 2 ) du du 2 Q rlj e π rjτ e (λ i+π rj )u du e (λ k+π rj )u 2 du 2, J,rijkl (τ) Q rlj (λ i + π rj )(λ k + π rj ) eπ rjτ. (4.3) Here, the stability of Λ, i.e., Re [λ i ] <, Re [πri i ] < for all r,i and the property τ + u + u 2 for τ has been used. Next, J,rijkl ( τ) is evaluated. In this case it is necessary to split the integral according

25 Stationary solutions of linear ODEs with a randomly perturbed system matrix 28 to the sign of the argument of ρ. The substitution v u and w τ +u +u 2 leads to J,rijkl ( τ) e λ iv+λ k (τ+w v) ρ rlj (w)dw dv v τ e λ kτ e (λ i λ k )v e λ kw ρ rlj (w)dw dv where I,rijkl (τ) v τ e λ kτ (I,rijkl (τ) + I 2,rijkl (τ) + I 3,rijkl (τ)) τ e (λ i λ k )v e λ kw ρ rlj (w)dw dv I 2,rijkl (τ) I 3,rijkl (τ) τ v τ e (λ i λ k )v e (λ i λ k )v e λ kw ρ rlj (w)dw dv e λ kw ρ rlj (w)dw dv. τ v τ For I it follows by using the substitution w w and ρ rlj ( w) ρ rjl (w) Q rjl e π rl w I,rijkl (τ) τ τ v e (λ i λ k )v e λ kw ρ rjl (w )dw dv τ Q rjl e (λ i λ k )v τ v e (π rl λ k )w dw dv Q rjl τ e (λ i λ k )v G(τ v, π rl,λ k )dv with G(s, α, β) For λ k π rl then it holds I,rijkl (τ) s Q rjl π rl λ k e (α β)v dv τ Q [ rjl e (π rl λ k )τ π rl λ k { α β( e (α β)s ) for α β s for α β. e (λ i λ k )v ( e (π rl λ k )(τ v) ) dv τ e (λ i π rl )v dv τ ] e (λ i λ k )v dv (4.4) Q rjl π rl λ k [ e (π rl λ k )τ G(τ,λ i,π rl ) G(τ,λ i,λ k ) ]. Workshop Stochastische Analysis

26 282 H.-J. Starkloff, R. Wunderlich τ In case of λ k π rl it follows I,rijkl (τ) Q rjl e (λ i λ k )v (τ v)dv. If moreover λ k λ i then it holds I,rijkl (τ) Q rjl τ it follows [ τ ( I,rijkl (τ) Q rjl e (λ i λ k )τ ) λ i λ k τ (τ v)dv Q rjl τ 2 2 while for λ k λ i ] e (λ i λ k )v v dv [ τ ( Q rjl e (λ i λ k )τ ) e(λi λk)v v τ + λ i λ k λ i λ k λ i λ k τ ] e (λ i λ k )v dv Q [ rjl τ ( e (λ i λ k )τ ) e (λ i λ k )τ ( τ + e (λ i λ k )τ )] λ i λ k λ i λ k Q rjl (G(τ,λ i,λ k ) τ). λ i λ k Summarizing, the integral I can be represented as Q rjl ( π rl λ k e (π rl λ k )τ G(τ,λ i,π rl ) G(τ,λ i,λ k ) ) for λ k π rl Q I,rijkl (τ) rjl λ i λ k (G(τ,λ i,λ k ) τ) for λ k π rl, λ k λ i. τ Q 2 rjl for λ 2 k π rl λ i (4.5) The integral I 2 can be evaluated as follows I 2,rijkl (τ) τ e (λ i λ k )v Q rlj and for I 3 one obtaines I 3,rijkl (τ) e (λ i λ k )v τ e (λ i λ k )v e λ kw ρ rlj (w)dw dv e λ kw ρ rlj (w)dw dv e (λ k+π rj )w dw dv Q rlj λ k + π rj G(τ,λ i,λ k ) Q rlj τ v τ e (λ i λ k )v e (λ k+π rj )w dw dv τ Q rlj λ k + π rj τ v τ e (λ i λ k )v e (λ k+π rj )(v τ) dv Q rlje (λ k+π rj )τ Q rlj (λ k + π rj )(λ i + π rj ) e(λ i λ k )τ. λ k + π rj τ e (λ i+π rj )v dv

27 Stationary solutions of linear ODEs with a randomly perturbed system matrix 283 Summarizing, the integral J,rijkl ( τ) is given by J,rijkl ( τ) e λkτ (I,rijkl (τ) + I 2,rijkl (τ) + I 3,rijkl (τ)) ( e λ kτ I,rijkl (τ) + Q ( rlj e (λi λk)τ )) G(τ,λ i,λ k ), (4.6) λ k + π rj λ i + π rj where I is defined in (4.5) and G in (4.4). Finally, J 2,rijkl (τ) is evaluated, it holds J 2,rijkl (τ) e λ iu +λ j u 2 ρ rkl (τ + u u 2 )du du 2. As in the case of the evaluation of J for negative τ it is necessary to split the integral according to the sign of the argument of ρ. The substitution v u and w τ +u u 2 leads to J 2,rijkl (τ) v+τ e λ iv+λ j (τ+v w) ρ rkl (w)dw dv e λ jτ e (λ i+λ j )v v+τ e λ jw ρ rkl (w)dw dv where I 4,rijkl (τ) e λ jτ (I 4,rijkl (τ) + I 5,rijkl (τ)) e (λ i+λ j )v e λ jw ρ rkl (w)dw dv I 5,rijkl (τ) e (λ i+λ j )v v+τ e λ jw ρ rkl (w)dw dv. Using ρ rkl (w) ρ rlk ( w) Q rlk e π rkw for w < it follows e λ jw ρ rkl (w)dw Q rlk and I 4,rijkl (τ) e (λ i+λ j )v e (λ j+π rk )w dw Q rlk λ j + π rk e λ jw ρ rkl (w)dw dv Q rlk (λ i + λ j )(π rk + λ j ). Workshop Stochastische Analysis

28 284 H.-J. Starkloff, R. Wunderlich For the evaluation of I 5 the substitution of ρ rkl (w) Q rkl e π rlw for w yields v+τ e λ jw ρ rkl (w)dw Q rkl For π rl λ j then it can be derived v+τ e (π rl λ j )w dw Q rkl π rl λ j ( e (π rl λ j )(v+τ) ) for π rl λ j Q rkl (v + τ) for π rl λ j. I 5,rijkl (τ) : e (λ i+λ j )v v+τ e λ jw ρ rkl (w)dw dv Q [ rkl e (π rl λ j )τ π rl λ j Q [ rkl π rl λ j λ i + λ j e (λ i+π rl )v dv λ i + π rl e (π rl λ j )τ ] ] e (λ i+λ j )v dv while for π rl λ j it holds I 5,rijkl (τ) Q rkl Summarizing it follows e (λ i+λ j )v (v + τ)dv Q rkl e (λ i+λ j )v v dv τ λ i + λ j ( Q rkl (λ i + λ j ) τ ) Q ( rkl ) τ. 2 λ i + λ j λ i + λ j λ i + λ j J 2,rijkl (τ) e λjτ (I 4,rijkl (τ) + I 5,rijkl (τ)) ( e λ Q ) jτ rlk (λ i + λ j )(π rk + λ j ) + I 5,rijkl(τ) ( ) Q rkl π rl λ j λ i +λ j λ i +π rl e (π rl λ j )τ for π rl λ j with I 5,rijkl (τ) ( ). Q rkl τ for π λ i +λ j λ i +λ rl λ j j (4.7) Corollary 4.2 Let the assumptions of Theorem 3. applied to z (Λ + ηc) z + f,

29 Stationary solutions of linear ODEs with a randomly perturbed system matrix 285 be fulfilled and R ζ ζ (τ) be of the form (4.2). Then the correlation function of the stationary solution z for η < η S and τ, i,j,...,n, possesses the expansion for η R ezi ez j (τ) m n [Q rij e πrjτ + η 2 r k,l ( E {C ik C kl } J,rijkl (τ) +E { } C jk C kl J,rjikl ( τ) + E { } )] C ik C jl J2,rijkl (τ) + o(η 2 ), where J and J 2 are given in (4.3), (4.6) and (4.7). 5 Numerical results In this section the results of the preceding sections are applied to the special case of a real and scalar equation (.) which arises for n and real-valued random parameters. Thus A(ω) a(ω) is a real-valued random variable and the excitation f(t,ω) is a scalar real-valued random process. As before the random parameters are decomposed, i.e., a(ω) â + ηc(ω) and f(t,ω) f + f(t,ω), with Ea â, Ef(t) f, a centered random variable c(ω) and a centered process f(t,ω). Then Eq. (.) reads as ż(t,ω) (â + ηc(ω)) z(t,ω) + f(t,ω). (5.) The random variable c is assumed to be a.s. bounded, i.e., c(ω) c where for convenience it is set c. The stability condition for A(ω) a(ω) â + ηc(ω) is satisfied for a(ω) < a.s., i.e., for η < η S ba c â. In this case Eq. (5.) possesses the unique stationary solution z(t, ω) with ẑ(ω) and z(t, ω) e a(ω)u f(t u,ω)du ẑ(ω) + z(t,ω) e a(ω)u f du a (ω) f e a(ω)u f(t u,ω)du. Due to Theorem 3.2 the mean and the correlation function of z are given by E {z(t)} E {ẑ} E { a } f and R zz (τ) D 2 ẑ + R ez ez (τ), (5.2) where D 2 ẑ cov(ẑ,ẑ) denotes the variance of ẑ. In the considered special case it is possible to compute these moments explicitly in terms of the mean and the correlation function of f and the distribution of c. In order to perform Workshop Stochastische Analysis

30 286 H.-J. Starkloff, R. Wunderlich numerical experiments which compare these exact moments with approximations from the perturbation approach it is assumed that c is uniformly distributed on [, ], then a is uniformly distributed on [â η,â+η] and we have the following probability density functions p c (s) { for s [, ] 2 else and p a (s) { 2η for s [â η,â + η] else Moreover it is assumed that f possesses the exponentially decaying correlation function R ff (τ) R ef e f (τ) σ 2 e γ τ, σ >,γ < 2â, where σ denotes the standard deviation of f and γ is a parameter describing the decay of the correlation function of f. It is noted that we impose instead of γ < the stronger condition γ < 2â in order to simplify the subsequent calculations. This condition ensures that γ does not coincide with the parameter a [â η,â+η] (2â, ) since η < η S â. First- and second-order moments of ẑ exist if E {a 2 } <. This condition is fulfilled for η < η M â η S since E { a 2} ba+η ba η ba+η s p a(s)ds 2 ba η (â + η) 2 p a(s)ds If a is uniformly distributed on [â η,â + η] for η < η M it holds Eẑ E { a } f f ba+η ba η s p a(s)ds f 2η (â + η) 2. ba+η ba η s ds f 2η ln â η â + η, (5.3) Eẑ 2 f 2 ba+η s p a(s)ds f 2 2η f 2 2η ba η 2 ba+η ba η s 2ds ( â η ) f 2 â + η â 2 η 2, and D 2 ẑ Eẑ 2 (Eẑ) 2 f 2( â 2 η â η 2 4η 2 ln2 â + η ). (5.4) For the correlation function of z it holds R ez ez (τ) R ez ez ( τ) and for τ it can be evaluated as follows { } R ez ez (τ) E { z(t) z(t + τ)} E e au f(t u )e au 2 f(t + τ u2 )du du 2 E { e a(u +u 2 ) } R ef e f (τ + u u 2 )du du 2,.

31 Stationary solutions of linear ODEs with a randomly perturbed system matrix 287 where the independence of a and f has been applied. Using the substitution v u, w τ + u u 2 and the assumption of an exponential decaying correlation function, i.e., R ef e f (τ) σ 2 e γ τ, it follows R ez ez (τ) v+τ v+τ ba+η E { e a(τ+2v w)} R ef e f (w)dw dv p a (x)e x(τ+2v w) σ 2 e γ w dxdw dv ba η σ 2 ba+η ba η p a (x)e xτ J(x)dx where J(x) : e 2xv e (γ+x)w dw + v+τ e (γ x)w dw dv [ e 2xv γ + x + ( e (γ x)(v+τ) )] dv γ x [ ] 2γ e 2xv γ 2 x + e(γ x)τ 2 γ x e(γ x)v dv γ x(γ 2 x 2 ) e(γ x)τ γ 2 x 2. Hence R ez ez (τ) σ 2 ba+η ba η [ p a (x) γ x(γ 2 x 2 ) exτ ] dx. γ 2 x 2eγτ If additionally a is uniformly distributed on [â η,â + η] we have R ez ez (τ) where I (τ) : σ2 2η γ ba+η ba η x(γ 2 x 2 ) exτ dx ba+η ba η dx γ 2 x 2eγτ σ2 2η [γi (τ) e γτ I 2 ] (5.5) ba+η ba η e xτ x(γ 2 x 2 ) dx and I 2 : ba+η ba η γ 2 x 2 dx. Workshop Stochastische Analysis