Summary
LetX t ∈Rd be the solution of the stochastic equationdX t =b(X t )dt+δ(X t )dW t , whereW t denotes a standard Wiener process. The aim of the paper is to clarify under which conditions the drift term or the diffusion term is of negligible significance for the long term behaviour ofX t .
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Arnold, L., Oeljeklaus, E., Paradoux, E.: Almost sure and moment stability for linear Ito equations. Lect. Notes Math. vol. 1186, Berlin Heidelberg New York: Springer 1986
Bhattacharya, R.N.: Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Probab.6, 541–553 (1978)
Clark, C.R.: Asymptotic properties of some multidimensional diffusion. Ann. Probab.15, 985–1008 (1987)
Cranston, M.: Invariant δ-Fields for a class of diffusions. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 161–180 (1983)
Durrett, R.: Brownian motion and martingales in analysis. Belmont: Wadsworth 1984
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin Heidelberg New York: Springer 1977
Has'minskii, R.Z.: Stochastic stability of differential equations. Rockville: Alphen 1980
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North Holland 1981
Keller, G., Kersting, G., Rösler, U.: On the asymptotic behaviour of solutions of stochastic differential equations. Z. Wahrscheinlichkeitstheor. Verw. Geb.68, 163–189 (1984)
Pinsky, R.: Recurrence, transience and bounded harmonic functions for diffusions in the plane. Ann. Probab.15, 954–984 (1987)
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Kersting, G. Asymptotic properties of solutions of multidimensional stochastic differential equations. Probab. Th. Rel. Fields 82, 187–211 (1989). https://doi.org/10.1007/BF00354759
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DOI: https://doi.org/10.1007/BF00354759