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The Novikov and entropy conditions of multidimensional diffusion processes with singular drift
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  • Published: December 1993

The Novikov and entropy conditions of multidimensional diffusion processes with singular drift

  • Wolfgang Stummer1 nAff2 

Probability Theory and Related Fields volume 97, pages 515–542 (1993)Cite this article

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  • 21 Citations

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Summary

We consider multidimensional stochastic differential equations of the form

$$dX_t = b(t, X_t ) dt + dW_t 0 \leqq t \leqq R, R< \infty $$
((1))

with arbitrary initial (probability) distribution μ onR d,d≧1. The first aim of this paper is to give handy-to-verify analytic (i.e. non-stochastic) conditions for the existence of a weak solution of (1), where the driftb will be allowed to have singularities. These investigations are illustrated by various examples. We first concentrate on (a uniform form of) the Novikov condition

$$E_\mu \left[ {exp\left( {\frac{1}{2}\int\limits_0^R {|b(s, X_s )|^2 } ds} \right)} \right]< \infty .$$
((2))

and then investigate further sufficient conditions for the applicability of the Girsanov-Maruyama Theorem which are not covered by (2). The outcoming results improve some of those of Engelbert and Schmidt [8] (for time-independent driftsb(x)) and Portenko [28] (for time-dependent driftsb(t, x)). One of the examples involves a drift which is singular on a dense set inR d but nevertheless satisfies (2).

The second aim of this paper is to discuss some general properties and applications of (2). For instance, we investigate whether the factor 1/2 in the Novikov condition (2) “can be replaced” by 1/2±ε (ε>0). Furthermore, we give several equivalence characterizations of (2) (being connected to the well-known Khas'minskii-Lemma [17]). Finally, it is shown that under the Novikov condition (2), the diffusion process with driftb has finite relative entropy with respect to Wiener measure (and thus finite “energy”).

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Author notes
  1. Wolfgang Stummer

    Present address: School of Mathematical Sciences, Queen Mary and Westfield College, University of London, Mile End Road, E1 4NS, London, UK

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Zürich, Rämistrasse 74, CH-8001, Zürich, Switzerland

    Wolfgang Stummer

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  1. Wolfgang Stummer
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Stummer, W. The Novikov and entropy conditions of multidimensional diffusion processes with singular drift. Probab. Th. Rel. Fields 97, 515–542 (1993). https://doi.org/10.1007/BF01192962

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  • Received: 26 March 1991

  • Revised: 08 June 1993

  • Issue Date: December 1993

  • DOI: https://doi.org/10.1007/BF01192962

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Mathematics Subject Classification

  • 60H10
  • 60G46
  • 60J45
  • 35J10
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