# On the existence of diffusions with singular drift coefficient

On the existence of diffusions with singular drift coefficient Let $$L = \frac{1}{2}\sum\limits_{i,j = 1}^d {a^{ij} } (x)\frac{{\partial ^2 }}{{\partial x^i \partial x^j }} + \sum\limits_{i = 1}^d {b^i (x)\frac{\partial }{{\partial x^i }}}$$ be an operator inR d, where the matrix (a ij ) is bounded, Hölder continuous and uniformly positive definite, and (b i (x)) is Borel measurable. In this paper we prove the existence ofL-diffusion under the hypothesis that $$\mathop {\sup }\limits_x \int_{|y - x| \leqslant \frac{1}{2}} {g_d (x - y)} |b(y)|^2 dy< \infty ,$$ whereg 1(z)=1,g 2(z)=−ln|z| andg d (z)=|z|2−d ford≥3. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# On the existence of diffusions with singular drift coefficient

, Volume 4 (1) – Jul 13, 2005
7 pages

/lp/springer-journals/on-the-existence-of-diffusions-with-singular-drift-coefficient-bRHUDVsUW9
Publisher
Springer Journals
Copyright © 1988 by Science Press, Beijing, China and Allerton Press, Inc. New York, U.S.A.
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/BF02018710
Publisher site
See Article on Publisher Site

### Abstract

Let $$L = \frac{1}{2}\sum\limits_{i,j = 1}^d {a^{ij} } (x)\frac{{\partial ^2 }}{{\partial x^i \partial x^j }} + \sum\limits_{i = 1}^d {b^i (x)\frac{\partial }{{\partial x^i }}}$$ be an operator inR d, where the matrix (a ij ) is bounded, Hölder continuous and uniformly positive definite, and (b i (x)) is Borel measurable. In this paper we prove the existence ofL-diffusion under the hypothesis that $$\mathop {\sup }\limits_x \int_{|y - x| \leqslant \frac{1}{2}} {g_d (x - y)} |b(y)|^2 dy< \infty ,$$ whereg 1(z)=1,g 2(z)=−ln|z| andg d (z)=|z|2−d ford≥3.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jul 13, 2005

### References

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