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Quasi-linear PDEs and forward-backward stochastic differential equations: weak solutions

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posted on 22.09.2017 by Chunrong Feng, Xince Wang, Huaizhong Zhao
In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space H1ρ(Rd). For this, we study first the solutions of forward-backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space L2ρ(Rd; Rd) ⊗ L2ρ(Rd; Rk) ⊗ L2ρ(Rd; Rk×d). This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.

Funding

We would like to acknowledge the financial support of Royal Society Newton Advanced Fellowship NA150344.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Journal of Differential Equations

Volume

264

Issue

2

Pages

959 - 1018

Citation

FENG, C., WANG, X. and ZHAO, H., 2017. Quasi-linear PDEs and forward-backward stochastic differential equations: weak solutions. Journal of Differential Equations, 264 (2), pp. 959-1018.

Publisher

© Elsevier

Version

AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Acceptance date

16/09/2017

Publication date

2017-09-29

Notes

This paper was published in the journal Journal of Differential Equations and the definitive published version is available at https://doi.org/10.1016/j.jde.2017.09.030.

ISSN

0022-0396

Language

en

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