The Novikov and entropy conditions of multidimensional diffusion processes with singular drift
type: Item
instance of: scholarly article
title: The Novikov and entropy conditions of multidimensional diffusion processes with singular drift
zbMATH Open document ID: 0794.60055
DOI: 10.1007/BF01192962
author: W. Stummer
publication date: 1994-08-15T00:00:00Z
review text: We consider multidimensional stochastic differential equations of the form \[ dX_ t=b(t,X_ t) dt+dW_ t,\quad 0 \leq t \leq R,\;R<\infty, \tag{1} \] with arbitrary initial (probability) distribution \(\mu\) on \(\mathbb{R}^ d\), \(d \geq 1\). 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 \[ E_ \mu \left[ \exp \left( {1 \over 2} \int^ R_ 0 | b(s,X_ s) |^ 2ds \right) \right]<\infty \tag{2} \] 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 \textit{H. J. Engelbert} and \textit{W. Schmidt} [Math. Nachr. 119, 97-115 (1984; Zbl 0565.60063)] (for time-independent drifts \(b(x))\) and \textit{N. I. Portenko} [Theory Probab. Appl. 20, 27-37 (1975); translation from Teor. Veroyatn. Primen. 20, 29-39 (1975; Zbl 0335.60050)] (for time-dependent drifts \(b(t,x))\). One of the examples involves a drift which is singular on a dense set in \(\mathbb{R}^ d\) 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 \(1/2 \pm \varepsilon\) \((\varepsilon>0)\). Furthermore, we give several equivalence characterizations of (2) [being connected to the well-known \textit{R. Z. Khas'minskij} lemma, ibid. 4, 309-318 (1960) resp. ibid. 4, 332-341 (1959; Zbl 0089.345)]. 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'').
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 569001
zbMATH Keywords: existence of a weak solution
zbMATH Keywords: Novikov condition
zbMATH Keywords: Girsanov-Maruyama theorem
zbMATH Keywords: Wiener measure
MaRDI profile type: Publication
cites work: Large deviations and the propagation of chaos for Schrödinger processes
cites work: Brownian motion and harnack inequality for Schrödinger operators
cites work: General Gauge Theorem for Multiplicative Functionals
cites work: Conditional transformation of drift formula and potential theory for \(\Delta +b(\cdot)\cdot \nabla\)
cites work: Q3972685
cites work: On Exponential Local Martingales Connected with Diffusion Processes
cites work: Q3704700
cites work: On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures
cites work: Q3217386
cites work: Q3799121
cites work: Q4039796
cites work: Q3258067
cites work: On the transformation of some classes of martingales by a change of law
cites work: On Positive Solutions of the Equation $\mathfrak{A}U + Vu = 0$
cites work: Q4139359
cites work: Q5832412
cites work: Continuous Markov processes and stochastic equations
cites work: The Bessel motion and a singular integral equation
cites work: Transformations of diffusion and Schrödinger processes
cites work: Q3210689
cites work: A diffusion process in a singular mean-drift-field
cites work: Q3676873
cites work: On an Identity for Stochastic Integrals
cites work: On the Existence of Solutions of Stochastic Differential Equations with Integrable Drift Coefficient
cites work: Diffusion Processes with Unbounded Drift Coefficient
cites work: Q3763296
cites work: Q3874719
cites work: Schrödinger semigroups
cites work: Q3862204
cites work: Q4002873
cites work: Q3995571
cites work: Q3483673
cites work: Measures charging no polar sets and additive functionals of Brownian motion
cites work: Conditional expectations of Brownian functionals and their applications
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 569001