Abstract
The asymptotic behavior of linear stochastic systems in Rd of the form ẋ = A(ξ(t))x, x(o) = xo ε Rd, ξ(t) stationary stochastic process, is investigated by means of geometric nonlinear control theory. Concerning the rotational behavior it is proved that the projection onto the unit sphere has a unique invariant probability. Concerning the stability it is proved that the solution x(t;xo) has an exponential growth rate
which is independent of chance and of xo and equal to the biggest Lyapunov exponent from the multiplicative ergodic theorem.
Keywords
- Lyapunov Exponent
- Stochastic System
- Principal Eigenvalue
- Integral Manifold
- Exponential Growth Rate
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Arnold, L., Kliemann, W., Oeljeklaus, E. (1986). Lyapunov exponents of linear stochastic systems. In: Arnold, L., Wihstutz, V. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1186. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076835
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DOI: https://doi.org/10.1007/BFb0076835
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