Electronic Journal of Probability

Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients

Dario Trevisan

Full-text: Open access

Abstract

We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending the results first obtained by A. Figalli in [19]. Our main results are: a very general equivalence between different descriptions for multidimensional diffusion processes, such as Fokker-Planck equations and martingale problems, under minimal regularity and integrability assumptions; and new existence and uniqueness results for diffusions having weakly differentiable coefficients, by means of energy estimates and commutator inequalities. Our approach relies upon techniques recently developed jointly with L. Ambrosio in [6], to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations for commutators between smoothing operators and diffusion generators.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 22, 41 pp.

Dates
Received: 29 July 2015
Accepted: 3 March 2016
First available in Project Euclid: 18 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1458325000

Digital Object Identifier
doi:10.1214/16-EJP4453

Mathematical Reviews number (MathSciNet)
MR3485364

Zentralblatt MATH identifier
1336.60159

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 35Q84: Fokker-Planck equations

Keywords
Fokker-Planck equations martingale problem DiPerna-Lions flows

Rights
Creative Commons Attribution 4.0 International License.

Citation

Trevisan, Dario. Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab. 21 (2016), paper no. 22, 41 pp. doi:10.1214/16-EJP4453. https://projecteuclid.org/euclid.ejp/1458325000


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References

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