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On the cohomology of flows of stochastic and random differential equations
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  • Published: June 2001

On the cohomology of flows of stochastic and random differential equations

  • Peter Imkeller1 &
  • Christian Lederer1 

Probability Theory and Related Fields volume 120, pages 209–235 (2001)Cite this article

  • 245 Accesses

  • 14 Citations

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Abstract.

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise.

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  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. e-mail: imkeller@mathematik.hu-berlin.de; lederer@mathematik.hu-berlin.de, , , , , , DE

    Peter Imkeller & Christian Lederer

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  1. Peter Imkeller
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  2. Christian Lederer
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Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001

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Imkeller, P., Lederer, C. On the cohomology of flows of stochastic and random differential equations. Probab Theory Relat Fields 120, 209–235 (2001). https://doi.org/10.1007/PL00008781

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  • Issue Date: June 2001

  • DOI: https://doi.org/10.1007/PL00008781

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  • Mathematics Subject Classification (2000): Primary 60 H 10, 58 F 25; Secondary 60 J 60, 58 F 12
  • Key words or phrases: Stochastic differential equations – Random dynamical systems – Stochastic flows – Solvable Lie algebra – Nilpotent Lie algebra – Random cohomology – Conjugation of flows – Random attractor – Duffing-van der Pol oscillator
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