Abstract
The asymptotic behavior of the linear stochastic differential equation in Rd
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
exists a.s. and is essentially independent of chance and of xo. Here we prove that
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<∞ and γ=∞ are characterized. This answers the question of when sample stability (λ<O) implies moment stability (g(p)<O) for all p>O.
In case trace A = trace Bi=O for all i we have g(−d)=O, this enabling us to characterize the cases λ=O and λ>O by a simple criterion.
Keywords
- Lyapunov Exponent
- Stochastic Differential Equation
- Fundamental Matrix
- Lyapunov Spectrum
- Linear Stochastic System
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References
Arnold, L.: A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math. 44(1984), 793–802.
Arnold, L., Kliemann, W., and Oeljeklaus, E.: Lyapunov exponents of linear stochastic systems. These Proceedings.
Bhattacharya, R.N.: On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 60(1982), 185–201.
Bony, J.-M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19(1969), 277–304.
Borel, A.: Les bouts des éspaces homogènes de groupes de Lie. Ann. of Math. 58(1953), 443–457.
Carverhill, A.: Flows of stochastic dynamical systems: ergodic theory. Stochastics 14(1985), 273–317.
Chappell, M.: Bounds for average Lyapunov exponents of gradient stochastic systems. These Proceedings.
Greiner, G.: Spektrum und Asymptotic stark stetiger Halbgruppen positiver Operatoren. Sitzungsber. Heidelberger Akad. d. Wiss. (math.-naturw. Klasse), 3. Abhandlung (1982).
Ichihara, K., and Kunita, H.: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitstheorie verw. Gebiete 30(1974), 235–254, and 39(1977), 81–84 (Supplements).
Ikeda, N., and Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North Holland 1981.
Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1980.
Kliemann, W.: Recurrence and invariant measures for degenerate diffusions. Annals of Prob. (to appear).
Kunita, H.: Diffusion processes and control systems. Lecture Notes, University of Paris VI, 1974.
Kunita, H.: Supports of diffusion processes and controllability problems. Proceed. Intern. Symp. Stochastic Diff. Equs., Kyoto 1976, 163–185. New York: Wiley 1978.
Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. École d′ Été de Probabilités de Saint-Flour XII — 1982. Lecture Notes in Mathematics 1097, 143–303. Berlin-Heidelberg-New York-Tokyo: Springer 1984.
Ledrappier, F.: Quelques proprietés des éxposants caracteristiques. École d' Été de Probabilités de Saint-Flour XII-1982. Lecture Notes in Mathematics 1097, 305–396. Berlin-Heidelberg-New York-Tokyo: Springer 1984.
Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19(1968), 197–231.
Pardoux, E., and Protter, P.: Two-sided stochastic integral and calculus (to appear).
Pignol, M.: Stabilité stochastique des pales d' helicoptère. Thèse de troisieme cycle, Université de Provence. Marseille 1985.
Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. math. de l'IHES 50(1979), 27–58.
Sussmann, H., and Jurdjevic, V.: Controllability of nonlinear systems. J. Diff. Equs. 12(1972), 95–116.
Willems, J.L., and Willems, J.C.: Robust stabilization of uncertain systems. SIAM J. Control and Optim. 21(1983), 352–374.
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© 1986 Springer-Verlag
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Arnold, L., Oeljeklaus, E., Pardoux, E. (1986). Almost sure and moment stability for linear ito equations. In: Arnold, L., Wihstutz, V. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1186. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076837
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DOI: https://doi.org/10.1007/BFb0076837
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