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Conservative stochastic differential equations: Mathematical and numerical analysis

Authors: Erwan Faou and Tony Lelièvre
Journal: Math. Comp. 78 (2009), 2047-2074
MSC (2000): Primary 60H10, 60H30, 58J65, 65C20
Published electronically: January 30, 2009
MathSciNet review: 2521278
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Abstract: We consider stochastic differential equations on the whole Euclidean space possessing a scalar invariant along their solutions. The stochastic dynamics therefore evolves on a hypersurface of the ambient space. Using orthogonal coordinate systems, we show the existence and uniqueness of smooth solutions of the Kolmogorov equation under some ellipticity conditions over the invariant hypersurfaces. If we assume, moreover, the existence of an invariant measure, we show the exponential convergence of the solution towards its average. In the second part, we consider numerical approximation of the stochastic differential equation, and show the convergence and numerical ergodicity of a class of projected schemes. In the context of molecular dynamics, this yields numerical schemes that are ergodic with respect to the microcanonical measure over isoenergy surfaces.

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Additional Information

Erwan Faou
Affiliation: INRIA & Ecole Normale Supérieure de Cachan Bretagne, Avenue Robert Schumann, 35170 Bruz, France

Tony Lelièvre
Affiliation: INRIA Rocquencourt, MICMAC project-team, B.P. 105, 78153 Le Chesnay Cedex, France
Address at time of publication: CERMICS, Ecole Nationale des Ponts (ParisTech), 6 & 8 Av. B. Pascal, 77455 Marne-la-Vallée, France

Keywords: Stochastic differential equations, invariant preservation, numerical approximation of invariant measure, microcanonical sampling
Received by editor(s): February 22, 2008
Received by editor(s) in revised form: September 17, 2008
Published electronically: January 30, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.