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February  2021, 41(2): 849-898. doi: 10.3934/dcds.2020302

Well-posedness of some non-linear stable driven SDEs

Noufel Frikha 1, , Valentin Konakov 2, and  Stéphane Menozzi 3,4,,

1. 

Université de Paris, Laboratoire de Probabilités, Statistiques et Modélisation (LPSM), UMR 8001, F-75013 Paris, France
2. 

Laboratory of Stochastic Analysis, Higher School of Economics, Pokrovsky boulevard 11, Moscow, Russian federation
3. 

Laboratoire de Modélisation Mathématique d'Evry (LaMME), UMR CNRS 8070, Université d'Evry Val d'Essonne, Université Paris-Saclay, 23 Boulevard de France 91037 Evry, France
4. 

Laboratory of Stochastic Analysis, Higher School of Economics, Pokrovsky boulevard 11, Moscow, Russian federation

* Corresponding author: Stéphane Menozzi

Received  October 2019 Revised  June 2020 Published  August 2020

We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $ \alpha $-stable Lévy processes with values in $ {{{\mathbb R}}}^d $ under some mild Hölder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $ \alpha \in (0,1) $. New strong well-posedness results are also derived from the previous analysis.

Keywords: McKean-Vlasov stable SDEs, non-linear martingale problem, weak uniqueness, strong uniqueness, perturbative techniques of parametrix type.
Mathematics Subject Classification: Primary 60H10, 60G46; Secondary 60H30, 35R09.
Citation: Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
References:
[1]

R. F. Bass, Stochastic differential equations driven by symmetric stable processes, Séminaire de Probabilités XXXVI, 1801 (2004), 302–313. doi: 10.1007/978-3-540-36107-7_11.  Google Scholar

[2]

R. F. Bass, K. Burdzy and Z.-Q. Chen, Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stochastic Processes and their Applications, 111 (2004), 1-15.  doi: 10.1016/j.spa.2004.01.010.  Google Scholar

[3]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I, volume 83 of Probability Theory and Stochastic Modelling, Springer, Cham, 2018. Mean field FBSDEs, control, and games.  Google Scholar

[4]

P.-E. Chaudru de Raynal, Strong well posedness of McKean-Vlasov stochastic differential equations with Hölder drift, Stochastic Processes and their Applications, 130 (2020), 79-107.  doi: 10.1016/j.spa.2019.01.006.  Google Scholar

[5]

P.-E. Chaudru de Raynal and N. Frikha, Well-posedness for some non-linear diffusion processes and related PDE on the Wasserstein space, arXiv: 1811.06904, under revision for Journal de Mathématiques Pures et Appliquées, 2018. Google Scholar

[6]

P.-E. Chaudru de Raynal and N. Frikha, From the Backward Kolmogorov PDE on the Wasserstein space to propagation of chaos for McKean-Vlasov SDEs, accepted publication for Journal de Mathématiques Pures et Appliquées, 2019. Google Scholar

[7]

P.-E. Chaudru de Raynal, I. Honoré and S. Menozzi, Sharp Schauder Estimates for some Degenerate Kolmogorov Equations, To appear in Ann. Scie. Scuola Norm. Superiore, 2020. https://arXiv.org/abs/1810.12227. Google Scholar

[8]

P.-E. Chaudru de Raynal, S. Menozzi and E. Priola, Schauder estimates for drifted fractional operators in the supercritical case, Journal of Functional Analysis, 278 (2020), 108425, 57 pp. doi: 10.1016/j.jfa.2019.108425.  Google Scholar

[9]

Z. Q. Chen, X. Zhang and G. Zhao, Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts, https://arXiv.org/pdf/1709.04632.pdf, 2017. Google Scholar

[10]

T. Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations, Zeitschrift Für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 67 (1984), 331–348. doi: 10.1007/BF00535008.  Google Scholar

[11]

J. Gärtner, On the Mckean-Vlasov limit for interacting diffusions, Mathematische Nachrichten, 137 (1988), 197-248.  doi: 10.1002/mana.19881370116.  Google Scholar

[12]

C. Graham, Nonlinear diffusion with jumps, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 393-402.   Google Scholar

[13]

L. Huang and S. Menozzi, A parametrix approach for some degenerate stable driven SDEs, Annales Instit. H. Poincaré, 52 (2016), 1925–1975. doi: 10.1214/15-AIHP704.  Google Scholar

[14]

L. Huang, S. Menozzi and E. Priola, $L^p$ estimates for degenerate non-local Kolmogorov operators, Journal de Mathématiques Pures et Appliquées, 121 (2019), 162–215. doi: 10.1016/j.matpur.2017.12.008.  Google Scholar

[15]

Z. Hao, Z. Wang and M. Wu, Schauder's estimates for nonlocal equations with singular Lévy measures, arXiv: 2002.09887, 2020. Google Scholar

[16]

Z. Hao, M. Wu and X. Zhang, Schauder's estimate for nonlocal kinetic equations and its applications, J. Math. Pures Appl. (9), 140 (2020), 139–184, arXiv: 1903.09967. doi: 10.1016/j.matpur.2020.06.003.  Google Scholar

[17]

X. Huang and F. F. Yang, Distribution dependent SDEs with Hölder continuous drift and $\alpha$-stable noise, arXiv: 1910.03299, 2019. Google Scholar

[18] N. Jacob, Pseudo Differential Operators and Markov Processes, volume 1, Imperial College Press, 2005.  doi: 10.1142/9781860947155.  Google Scholar
[19]

B. Jourdain, S. Méléard and W. A. Woyczynski, A probabilistic approach for nonlinear equations involving the fractional Laplacian and a singular operator, Potential Anal., 23 (2005), 55-81.  doi: 10.1007/s11118-004-3264-9.  Google Scholar

[20]

B. Jourdain, S. Méléard and W. A. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli, 11 (2005), 689-714.  doi: 10.3150/bj/1126126765.  Google Scholar

[21]

B. Jourdain, S. Méléard and W. A. Woyczynski, Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA Lat. Am. J. Probab. Math. Stat., 4 (2008), 1-29.   Google Scholar

[22]

B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations, ESAIM: PS, 1 (1997), 339-355.  doi: 10.1051/ps:1997113.  Google Scholar

[23]

M. Kac, Foundations of kinetic theory, In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics, Berkeley, Calif., 1956,171–197. University of California Press.  Google Scholar

[24]

V. Konakov and S. Menozzi, Weak error for stable driven stochastic differential equations: expansion of the densities, J. Theoret. Probab., 24 (2011), 454-478.  doi: 10.1007/s10959-010-0291-x.  Google Scholar

[25]

V. Kolokoltsov, Symmetric stable laws and stable-like jump diffusions, Proc. London Math. Soc., 80 (2000), 725–768. doi: 10.1112/S0024611500012314.  Google Scholar

[26] V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, volume 182 of Cambridge Tracts in Mathematics., Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511760303.  Google Scholar
[27]

N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35 (2010), 1-22.  doi: 10.1080/03605300903424700.  Google Scholar

[28]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12. AMS, 1996. doi: 10.1090/gsm/012.  Google Scholar

[29]

D. Lacker, On a strong form of propagation of chaos for McKean-Vlasov equations, Electron. Commun. Probab., 23 (2018), Paper No. 45, 11 pp. doi: 10.1214/18-ECP150.  Google Scholar

[30]

J. Li and H. Min, Weak solutions of mean-field stochastic differential equations and application to zero-sum stochastic differential games, SIAM Journal on Control and Optimization, 54 (2016), 1826-1858.  doi: 10.1137/15M1015583.  Google Scholar

[31]

H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences of the United States of America, 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.  Google Scholar

[32]

H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), 1967, 41–57.  Google Scholar

[33]

R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., 40 (2014), 539-563.  doi: 10.1007/s11118-013-9359-4.  Google Scholar

[34]

Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, Preprint, arXiv: 1603.02212, 2018. Google Scholar

[35]

K. Oelschlager, A martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Probab., 12 (1984), 458-479.  doi: 10.1214/aop/1176993301.  Google Scholar

[36]

E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka J. Math., 49 (2012), 421-447.   Google Scholar

[37]

E. Priola, Davie's type uniqueness for a class of SDEs with jumps, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 694–725. doi: 10.1214/16-AIHP818.  Google Scholar

[38]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, arXiv e-prints, arXiv: 1809.02216, Sep 2018. Google Scholar

[39]

M. Scheutzow, Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 43 (1987), 246-256.  doi: 10.1017/S1446788700029384.  Google Scholar

[40]

T. Shiga and H. Tanaka, Central limit theorem for a system of Markovian particles with mean field interactions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 69 (1985), 439–459. doi: 10.1007/BF00532743.  Google Scholar

[41]

A.-S. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour XIX –- 1989, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. doi: 10.1007/BFb0085169.  Google Scholar

[42]

P. Sztonyk, Estimates of tempered stable densities, J. Theoret. Probab., 23 (2010), 127-147.  doi: 10.1007/s10959-009-0208-8.  Google Scholar

[43]

H. Tanaka, M. Tsuchiya and S. Watanabe, Perturbation of drift-type for Lévy processes, J. Math. Kyoto Univ., 14 (1974), 73-92.  doi: 10.1215/kjm/1250523280.  Google Scholar

[44]

C. Villani, Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]., Springer-Verlag, Berlin, 2009. Old and new. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[45]

T. Watanabe, Asymptotic estimates of multi-dimensional stable densities and their applications, Transactions of the American Mathematical Society, 359 (2007), 2851-2879.  doi: 10.1090/S0002-9947-07-04152-9.  Google Scholar

[46]

X. Zhang and G. Zhao, Dirichlet problem for supercritical non-local operators, arXiv: 1809.05712, 2018. Google Scholar

show all references

References:
[1]

R. F. Bass, Stochastic differential equations driven by symmetric stable processes, Séminaire de Probabilités XXXVI, 1801 (2004), 302–313. doi: 10.1007/978-3-540-36107-7_11.  Google Scholar

[2]

R. F. Bass, K. Burdzy and Z.-Q. Chen, Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stochastic Processes and their Applications, 111 (2004), 1-15.  doi: 10.1016/j.spa.2004.01.010.  Google Scholar

[3]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications. I, volume 83 of Probability Theory and Stochastic Modelling, Springer, Cham, 2018. Mean field FBSDEs, control, and games.  Google Scholar

[4]

P.-E. Chaudru de Raynal, Strong well posedness of McKean-Vlasov stochastic differential equations with Hölder drift, Stochastic Processes and their Applications, 130 (2020), 79-107.  doi: 10.1016/j.spa.2019.01.006.  Google Scholar

[5]

P.-E. Chaudru de Raynal and N. Frikha, Well-posedness for some non-linear diffusion processes and related PDE on the Wasserstein space, arXiv: 1811.06904, under revision for Journal de Mathématiques Pures et Appliquées, 2018. Google Scholar

[6]

P.-E. Chaudru de Raynal and N. Frikha, From the Backward Kolmogorov PDE on the Wasserstein space to propagation of chaos for McKean-Vlasov SDEs, accepted publication for Journal de Mathématiques Pures et Appliquées, 2019. Google Scholar

[7]

P.-E. Chaudru de Raynal, I. Honoré and S. Menozzi, Sharp Schauder Estimates for some Degenerate Kolmogorov Equations, To appear in Ann. Scie. Scuola Norm. Superiore, 2020. https://arXiv.org/abs/1810.12227. Google Scholar

[8]

P.-E. Chaudru de Raynal, S. Menozzi and E. Priola, Schauder estimates for drifted fractional operators in the supercritical case, Journal of Functional Analysis, 278 (2020), 108425, 57 pp. doi: 10.1016/j.jfa.2019.108425.  Google Scholar

[9]

Z. Q. Chen, X. Zhang and G. Zhao, Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts, https://arXiv.org/pdf/1709.04632.pdf, 2017. Google Scholar

[10]

T. Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations, Zeitschrift Für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 67 (1984), 331–348. doi: 10.1007/BF00535008.  Google Scholar

[11]

J. Gärtner, On the Mckean-Vlasov limit for interacting diffusions, Mathematische Nachrichten, 137 (1988), 197-248.  doi: 10.1002/mana.19881370116.  Google Scholar

[12]

C. Graham, Nonlinear diffusion with jumps, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 393-402.   Google Scholar

[13]

L. Huang and S. Menozzi, A parametrix approach for some degenerate stable driven SDEs, Annales Instit. H. Poincaré, 52 (2016), 1925–1975. doi: 10.1214/15-AIHP704.  Google Scholar

[14]

L. Huang, S. Menozzi and E. Priola, $L^p$ estimates for degenerate non-local Kolmogorov operators, Journal de Mathématiques Pures et Appliquées, 121 (2019), 162–215. doi: 10.1016/j.matpur.2017.12.008.  Google Scholar

[15]

Z. Hao, Z. Wang and M. Wu, Schauder's estimates for nonlocal equations with singular Lévy measures, arXiv: 2002.09887, 2020. Google Scholar

[16]

Z. Hao, M. Wu and X. Zhang, Schauder's estimate for nonlocal kinetic equations and its applications, J. Math. Pures Appl. (9), 140 (2020), 139–184, arXiv: 1903.09967. doi: 10.1016/j.matpur.2020.06.003.  Google Scholar

[17]

X. Huang and F. F. Yang, Distribution dependent SDEs with Hölder continuous drift and $\alpha$-stable noise, arXiv: 1910.03299, 2019. Google Scholar

[18] N. Jacob, Pseudo Differential Operators and Markov Processes, volume 1, Imperial College Press, 2005.  doi: 10.1142/9781860947155.  Google Scholar
[19]

B. Jourdain, S. Méléard and W. A. Woyczynski, A probabilistic approach for nonlinear equations involving the fractional Laplacian and a singular operator, Potential Anal., 23 (2005), 55-81.  doi: 10.1007/s11118-004-3264-9.  Google Scholar

[20]

B. Jourdain, S. Méléard and W. A. Woyczynski, Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws, Bernoulli, 11 (2005), 689-714.  doi: 10.3150/bj/1126126765.  Google Scholar

[21]

B. Jourdain, S. Méléard and W. A. Woyczynski, Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA Lat. Am. J. Probab. Math. Stat., 4 (2008), 1-29.   Google Scholar

[22]

B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations, ESAIM: PS, 1 (1997), 339-355.  doi: 10.1051/ps:1997113.  Google Scholar

[23]

M. Kac, Foundations of kinetic theory, In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics, Berkeley, Calif., 1956,171–197. University of California Press.  Google Scholar

[24]

V. Konakov and S. Menozzi, Weak error for stable driven stochastic differential equations: expansion of the densities, J. Theoret. Probab., 24 (2011), 454-478.  doi: 10.1007/s10959-010-0291-x.  Google Scholar

[25]

V. Kolokoltsov, Symmetric stable laws and stable-like jump diffusions, Proc. London Math. Soc., 80 (2000), 725–768. doi: 10.1112/S0024611500012314.  Google Scholar

[26] V. N. Kolokoltsov, Nonlinear Markov Processes and Kinetic Equations, volume 182 of Cambridge Tracts in Mathematics., Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511760303.  Google Scholar
[27]

N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35 (2010), 1-22.  doi: 10.1080/03605300903424700.  Google Scholar

[28]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Mathematics 12. AMS, 1996. doi: 10.1090/gsm/012.  Google Scholar

[29]

D. Lacker, On a strong form of propagation of chaos for McKean-Vlasov equations, Electron. Commun. Probab., 23 (2018), Paper No. 45, 11 pp. doi: 10.1214/18-ECP150.  Google Scholar

[30]

J. Li and H. Min, Weak solutions of mean-field stochastic differential equations and application to zero-sum stochastic differential games, SIAM Journal on Control and Optimization, 54 (2016), 1826-1858.  doi: 10.1137/15M1015583.  Google Scholar

[31]

H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences of the United States of America, 56 (1966), 1907-1911.  doi: 10.1073/pnas.56.6.1907.  Google Scholar

[32]

H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), 1967, 41–57.  Google Scholar

[33]

R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem, Potential Anal., 40 (2014), 539-563.  doi: 10.1007/s11118-013-9359-4.  Google Scholar

[34]

Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, Preprint, arXiv: 1603.02212, 2018. Google Scholar

[35]

K. Oelschlager, A martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Probab., 12 (1984), 458-479.  doi: 10.1214/aop/1176993301.  Google Scholar

[36]

E. Priola, Pathwise uniqueness for singular SDEs driven by stable processes, Osaka J. Math., 49 (2012), 421-447.   Google Scholar

[37]

E. Priola, Davie's type uniqueness for a class of SDEs with jumps, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 694–725. doi: 10.1214/16-AIHP818.  Google Scholar

[38]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, arXiv e-prints, arXiv: 1809.02216, Sep 2018. Google Scholar

[39]

M. Scheutzow, Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 43 (1987), 246-256.  doi: 10.1017/S1446788700029384.  Google Scholar

[40]

T. Shiga and H. Tanaka, Central limit theorem for a system of Markovian particles with mean field interactions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 69 (1985), 439–459. doi: 10.1007/BF00532743.  Google Scholar

[41]

A.-S. Sznitman, Topics in propagation of chaos, Ecole d'Eté de Probabilités de Saint-Flour XIX –- 1989, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991. doi: 10.1007/BFb0085169.  Google Scholar

[42]

P. Sztonyk, Estimates of tempered stable densities, J. Theoret. Probab., 23 (2010), 127-147.  doi: 10.1007/s10959-009-0208-8.  Google Scholar

[43]

H. Tanaka, M. Tsuchiya and S. Watanabe, Perturbation of drift-type for Lévy processes, J. Math. Kyoto Univ., 14 (1974), 73-92.  doi: 10.1215/kjm/1250523280.  Google Scholar

[44]

C. Villani, Optimal Transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]., Springer-Verlag, Berlin, 2009. Old and new. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[45]

T. Watanabe, Asymptotic estimates of multi-dimensional stable densities and their applications, Transactions of the American Mathematical Society, 359 (2007), 2851-2879.  doi: 10.1090/S0002-9947-07-04152-9.  Google Scholar

[46]

X. Zhang and G. Zhao, Dirichlet problem for supercritical non-local operators, arXiv: 1809.05712, 2018. Google Scholar

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