The Novikov and entropy conditions of multidimensional diffusion processes with singular drift
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'').
scientific article; zbMATH DE number 569001
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