We consider multidimensional stochastic differential equations of the form

dX t =b(t,X t )dt+dW t ,0≤t≤R,R<∞,(1)

with arbitrary initial (probability) distribution μ on ℝ d , d≥1. The first aim of this paper is to give handy-to- verify analytic (i.e. nonstochastic) conditions for the existence of a weak solution of (1), where the drift b 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 μ exp1 2∫ 0 R |b(s,X s )| 2 ds<∞(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 H. J. Engelbert and W. Schmidt [Math. Nachr. 119, 97-115] (for time-independent drifts b(x)) and N. I. Portenko [Theory Probab. Appl. 20, 27-37 (1975); translation from Teor. Veroyatn. Primen. 20, 29-39] (for time-dependent drifts b(t,x)). One of the examples involves a drift which is singular on a dense set in ℝ 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 R. Z. Khas'minskij lemma, ibid. 4, 309-318 (1960) resp. ibid. 4, 332-341]. Finally, it is shown that under the Novikov condition (2), the diffusion process with drift b has finite relative entropy with respect to Wiener measure (and thus finite“energy”).

Stummer, W (1993). *The Novikov and entropy conditions of multidimensional diffusion processes with singular drift.* Probability Theory and Related Fields, 97(4):515-542.

## Abstract

We consider multidimensional stochastic differential equations of the form

dX t =b(t,X t )dt+dW t ,0≤t≤R,R<∞,(1)

with arbitrary initial (probability) distribution μ on ℝ d , d≥1. The first aim of this paper is to give handy-to- verify analytic (i.e. nonstochastic) conditions for the existence of a weak solution of (1), where the drift b 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 μ exp1 2∫ 0 R |b(s,X s )| 2 ds<∞(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 H. J. Engelbert and W. Schmidt [Math. Nachr. 119, 97-115] (for time-independent drifts b(x)) and N. I. Portenko [Theory Probab. Appl. 20, 27-37 (1975); translation from Teor. Veroyatn. Primen. 20, 29-39] (for time-dependent drifts b(t,x)). One of the examples involves a drift which is singular on a dense set in ℝ 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 R. Z. Khas'minskij lemma, ibid. 4, 309-318 (1960) resp. ibid. 4, 332-341]. Finally, it is shown that under the Novikov condition (2), the diffusion process with drift b has finite relative entropy with respect to Wiener measure (and thus finite“energy”).

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 1993 |

Deposited On: | 29 Nov 2010 16:29 |

Last Modified: | 05 Apr 2016 13:28 |

Publisher: | Springer |

ISSN: | 0178-8051 |

Publisher DOI: | https://doi.org/10.1007/BF01192962 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1246978 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0794.60055 |

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