Almost sure and moment stability for linear ito equations

The asymptotic behavior of the linear stochastic differential equation in Rd $$dx = Ax dt + \mathop \Sigma \limits_{i = 1}^m B_i x o dW_i (t), x(O) = x_O \ne O,$$ is studied. It is known (see [2]) in these Proceedings) that the projection of the solution x(t;xo) onto the unit sphere has a unique invariant probability, while $$\lambda = \mathop {\lim }\limits_{t \to \infty } \tfrac{1}{t} \log |x(t;x_O )|$$ exists a.s. and is essentially independent of chance and of xo. Here we prove that $$g(p) = \mathop {\lim }\limits_{t \to \infty } \tfrac{1}{t} \log E|x(t;x_O )|^P , p \in R,$$ exists and is independent of xo. Further, g: R → R is convex and analytic with g(p)/p increasing (to γ, say) with g(O)=O and g' (O)=λ. The cases γ O.

[1]  H. Kunita.  Stochastic differential equations and stochastic flows of diffeomorphisms , 1984 .

[2]  M. Chappell.  Bounds for average Lyapunov exponents of gradient stochastic systems , 1986 .

[3]  A. Borel.  Les Bouts Des Espaces Homogenes de Groupes De Lie , 1953 .

[4]  G. Kallianpur.  Stochastic differential equations and diffusion processes , 1981 .

[5]  R. Bhattacharya.  On the functional central limit theorem and the law of the iterated logarithm for Markov processes , 1982 .

[6]  Hiroshi Kunita,et al.  A classification of the second order degenerate elliptic operators and its probabilistic characterization , 1974 .

[7]  W. Kliemann.  Recurrence and invariant measures for degenerate diffusions , 1987 .

[8]  École d'été de probabilités de Saint-Flour,et al.  École d'Été de Probabilités de Saint-Flour XII - 1982 , 1984 .

[9]  L. Arnold,et al.  A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems , 1984 .

[10]  J. Bony.  Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés , 1969 .

[11]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[12]  H. Sussmann,et al.  Controllability of nonlinear systems , 1972 .

[13]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[14]  D. Ruelle.  Ergodic theory of differentiable dynamical systems , 1979 .

[15]  F. Ledrappier.  Quelques proprietes des exposants caracteristiques , 1984 .

[16]  J. Willems,et al.  Robust Stabilization of Uncertain Systems , 1983 .

[17]  L. Arnold,et al.  Lyapunov exponents of linear stochastic systems , 1986 .

[18]  A. Carverhill.  Flows of stochastic dynamical systems: ergodic theory , 1985 .