Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

Nicolas Fournier and Liping Xu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study some jumping SDE and the corresponding Fokker–Planck (or Kolmogorov forward) equation, which is a non-local PDE. We assume only some measurability and growth conditions on the coefficients. We prove that for any weak solution $(f_{t})_{t\in[0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time marginals are given by $(f_{t})_{t\in[0,T]}$. As a corollary, we deduce that for any given initial condition, existence for the PDE is equivalent to weak existence for the SDE and uniqueness in law for the SDE implies uniqueness for the PDE. This extends some ideas of Figalli (J. Funct. Anal. 254 (2008) 109–153) concerning continuous SDEs and local PDEs.


On étudie certaines EDS à sauts et les équations de Fokker–Planck (ou Kolmogorov progressives) correspondantes, qui sont des EDP non-locales. On suppose seulement que les coefficients sont mesurables et à croissance au plus linéaire. On montre que pour toute solution faible $(f_{t})_{t\in[0,T]}$ de l’EDP, il existe une solution faible à l’EDS, dont les lois marginales sont données par $(f_{t})_{t\in[0,T]}$. On en déduit que pour toute donnée initiale, l’existence pour l’EDP est équivalente à l’existence faible pour l’EDS, et que l’unicité en loi pour l’EDS implique l’unicité pour l’EDP. Nous étendons ainsi des idées de Figalli (J. Funct. Anal. 254 (2008) 109–153) concernant des EDS continues et des EDP locales.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1163-1178.

Received: 26 December 2016
Revised: 1 May 2018
Accepted: 15 May 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes 40K05

Existence and uniqueness Weak solution Jumping SDEs Non-local PDEs


Fournier, Nicolas; Xu, Liping. On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1163--1178. doi:10.1214/18-AIHP914.

Export citation


  • [1] D. Aldous. Stopping times and tightness. Ann. Probab. 6 (2) (1978) 335–340.
  • [2] R. F. Bass. Stochastic differential equations with jumps. Probab. Surv. 1 (2004) 1–19.
  • [3] A. G. Bhatt and R. L. Karandikar. Invariant measures and evolution equations for Markov processes characterized via martingale problems. Ann. Probab. 21 (4) (1993) 2246–2268.
  • [4] S. N. Ethier and T. G. Kurtz. Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986.
  • [5] A. Figalli. Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254 (1) (2008) 109–153.
  • [6] N. Fournier and M. Hauray. Propagation of chaos for the Landau equation with moderately soft potentials. Ann. Probab. 44 (6) (2016) 3581–3660.
  • [7] J. Horowitz and R. L. Karandikar. Martingale problems associated with the Boltzmann equation. In Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989) 75–122. Progr. Probab. 18. Birkhäuser Boston, Boston, MA, 1990.
  • [8] J. Jacod. Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics 714. Springer, Berlin, 1979.
  • [9] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288, 2nd edition. Springer-Verlag, Berlin, 2003.
  • [10] W. Rudin. Real and Complex Analysis. McGraw-Hill Book Co., New York–Toronto, Ont.–London, 1966.
  • [11] R. Situ. Theory of Stochastic Differential Equations with Jumps and Applications. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York, 2005.
  • [12] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften 233. Springer-Verlag, Berlin–New York, 1979.
  • [13] H. Tanaka. Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 (1) (1978/79) 67–105.
  • [14] L. Xu. Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials. Ann. Appl. Probab. 28 (2) (2018) 1136–1189.