# American Institute of Mathematical Sciences

January  2020, 5: 7 doi: 10.1186/s41546-020-00049-8

## Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

 1. Univ Brest, UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Brest, France, and Shandong University, Jinan, China 2. Department of Mathematics, University of Central Florida, Orlando, Florida, United States 3. Department of Mathematics, University of Southern California, Los Angeles, California, United States

Received  March 11, 2020 Published  November 2020

We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. If the diffusion coefficient is semilinear (i.e, linear in the gradient of the solution and nonlinear in the solution; the drift can still be fully nonlinear), we establish a complete theory, including global existence and a comparison principle.
Citation: Rainer Buckdahn, Christian Keller, Jin Ma, Jianfeng Zhang. Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 7-. doi: 10.1186/s41546-020-00049-8
##### References:
 [1] Buckdahn, R., I. Bulla, and J. Ma. (2011). On Pathwise Stochastic Taylor Expansions, Math. Control Relat. Fields 1, no. 4, 437-468. [2] Buckdahn, R. and J. Li. (2008). Stochastic differential games and viscosity solutions of Hamilton-JacobiBellman-Isaacs equations, SIAM J. Control Optim. 47, no. 1, 444-475. [3] Buckdahn, R. and J. Ma. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I, Stoch. Process. Appl. 93, no. 2, 181-204. [4] Buckdahn, R. and J. Ma. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II, Stoch. Process. Appl. 93, no. 2, 205-228. [5] Buckdahn, R. and J. Ma. (2002). Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs, Ann. Probab. 30, no. 3, 1131-1171. [6] Buckdahn, R. and J. Ma. (2007). Pathwise stochastic control problems and stochastic HJB equations, SIAM J. Control Optim. 45, no. 6, 2224-2256. [7] Buckdahn, R., J. Ma, and J. Zhang. (2015). Pathwise Taylor expansions for random fields on multiple dimensional paths, Stoch. Process. Appl. 125, 2820-2855. [8] Caruana, M., P. Friz, and H. Oberhauser. (2011). A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 27-46. [9] Crandall, M.G., H. Ishii, and P.-L. Lions. (1992). User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27, no. 1, 1-67. [10] Da Prato, G. and L. Tubaro. (1996). Fully nonlinear stochastic partial differential equations, SIAM J. Math. Anal. 27, no. 1, 40-55. [11] Davis, M. and G. Burstein. (1992). A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control, Stochast. Stoch. Rep. 40, no. 3-4, 203-256. [12] Diehl, J. and P. Friz. (2012). Backward stochastic differential equations with rough drivers, Ann. Prob. 40, 1715-1758. [13] Diehl, J., P. Friz, and P. Gassiat. (2017). Stochastic control with rough paths, Appl. Math. Optim. 75, no. 2, 285-315. [14] Diehl, J., P. Friz, and H. Oberhauser. (2014). Regularity theory for rough partial differential equations and parabolic comparison revisited, Springer, Cham. [15] Diehl, J., H. Oberhauser, and S. Riedel. (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations, Stoch. Process. Appl. 125, no. 1, 161-181. [16] Dupire, B. (2019). Functional Itô calculus, Quant. Finan. 19, no. 5, 721-729. [17] Ekren, I., C. Keller, N. Touzi, and J. Zhang. (2014). On viscosity solutions of path dependent PDEs, Ann. Probab. 42, 204-236. [18] Ekren, I., N. Touzi, and J. Zhang. (2016). Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part I, Ann. Probab. 44, 1212-1253. [19] Ekren, I., N. Touzi, and J. Zhang. (2016). Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part II, Ann. Probab. 44, 2507-2553. [20] Friz, P., P. Gassiat, P.L. Lions, and P.E. Souganidis. (2017). Eikonal equations and pathwise solutions to fully non-linear SPDEs, Stochast. Partial Differ. Equ. Anal. Comput. 5, 256-277. [21] Friz, P. and M. Hairer. (2014). A course on rough paths:With an introduction to regularity structures, Universitext, Springer, Cham. [22] Friz, P. and H. Oberhauser. (2011). On the splitting-up method for rough (partial) differential equations, J. Differ. Equ. 251, no. 2, 316-338. [23] Friz, P. and H. Oberhauser. (2014). Rough path stability of (semi-)linear SPDEs, Probab. Theory Relat. Fields 158, 401-434. [24] Gilbarg, D. and N. Trudinger. (1983). Elliptic Partial Differential Equations of second order, second edition, Springer-Verlag, Germany. [25] Gubinelli, M. (2004). Controlling rough paths, J. Funct. Anal. 216, no. 1, 86-140. [26] Gubinelli, M., S. Tindel, and I. Torrecilla. (2014). Controlled viscosity solutions of fully nonlinear rough PDEs. arXiv preprint, arXiv:1403.2832. [27] Keller, C. and J. Zhang. (2016). Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients, Stoch. Process. Appl. 126, 735-766. [28] Krylov, N.V. (1999). An analytic approach to SPDEs, Stoch. Partial Differ. Equ. Six Perspect. Math. Surv. Monogr. Amer. Math. Soc. Providence RI 64, 185-242. [29] Kunita, H. (1997). Stochastic flows and stochastic differential equations, Cambridge University Press, Cambridge. [30] Lieberman, G. (1996). Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge. [31] Lions, P.-L. and P. E. Souganidis. (1998). Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 326, no. 9, 1085-1092. [32] Lions, P.-L. and P. E. Souganidis. (1998). Fully nonlinear stochastic partial differential equations:nonsmooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math. 327, no. 8, 735-741. [33] Lions, P.-L. and P. E. Souganidis. (2000). Fully nonlinear stochastic PDE with semilinear stochastic dependence, C. R. Acad. Sci. Paris Sér. I Math. 331, no. 8, 617-624. [34] Lions, P.-L. and P.E. Souganidis. (2000). Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. l'Acad. Sci.-Ser. I-Math. 331, no. 10, 783-790. [35] Lunardi, A. (1995). Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16, Birkhäuser Verlag, Basel. [36] Lyons, T. (1998). Differential equations driven by rough signals, Rev. Mat. Iberoam. 14, no. 2, 215-310. [37] Matoussi, A., D. Possamai, and W. Sabbagh. (2018). Probabilistic interpretation for solutions of Fully Nonlinear Stochastic PDEs, Probab. Theory Relat. Fields. https://doi.org/10.1007/s00440-018-0859-4. [38] Mikulevicius, R. and G. Pragarauskas. (1994). Classical solutions of boundary value problems for some nonlinear integro-differential equations, Lithuanian Math. J. 34, no. 3, 275-287. [39] Musiela, M. and T. Zariphopoulou. (2010). Stochastic partial differential equations and portfolio choice, Contemporary Quantitative Finance, Springer, Berlin. [40] Nadirashvili, N. and S. Vladut. (2007). Nonclassical solutions of fully nonlinear elliptic equations, Geom. Funct. Anal. 17, no. 4, 1283-1296. [41] Pardoux, E. and S. Peng. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98, 209-227. [42] Peng, S. (1992). Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim 30, no. 2, 284- 304. [43] Pham, T. and J. Zhang. (2014). Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation, SIAM J. Control. Optim. 52, 2090-2121. [44] Rozovskii, B.L. (1990). Stochastic Evolution Systems:Linear Theory and Applications to Non-linear Filtering, Kluwer Academic Publishers, Boston. [45] Safonov, M.V. (1988). Boundary value problems for second-order nonlinear parabolic equations, (Russian), Funct. Numer. Methods Math. Phys. "Naukova Dumka" Kiev. 274, 99-203. [46] Safonov, M.V. (1989). Classical solution of second-order nonlinear elliptic equations, Math. USSR-Izv 33, no. 3, 597-612. [47] Seeger, B. (2018). Perron's method for pathwise viscosity solutions, Commun. Partial Differ. Equ. 43, no. 6, 998-1018. [48] Seeger, B. (2018). Homogenization of pathwise Hamilton-Jacobi equations, J. Math. Pures Appl. 110, 1-31. [49] Seeger, B. (2020). Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations, Ann. Appl. Probab. 30, no. 4, 1784-1823. [50] Souganidis, P.E. (2019). Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence, Singular random dynamics, Lecture Notes in Math. vol. 2253, Springer, Cham. [51] Zhang, J. (2017). Backward Stochastic Differential Equations-from linear to fully nonlinear theory, Springer, New York.

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##### References:
 [1] Buckdahn, R., I. Bulla, and J. Ma. (2011). On Pathwise Stochastic Taylor Expansions, Math. Control Relat. Fields 1, no. 4, 437-468. [2] Buckdahn, R. and J. Li. (2008). Stochastic differential games and viscosity solutions of Hamilton-JacobiBellman-Isaacs equations, SIAM J. Control Optim. 47, no. 1, 444-475. [3] Buckdahn, R. and J. Ma. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I, Stoch. Process. Appl. 93, no. 2, 181-204. [4] Buckdahn, R. and J. Ma. (2001). Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II, Stoch. Process. Appl. 93, no. 2, 205-228. [5] Buckdahn, R. and J. Ma. (2002). Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs, Ann. Probab. 30, no. 3, 1131-1171. [6] Buckdahn, R. and J. Ma. (2007). Pathwise stochastic control problems and stochastic HJB equations, SIAM J. Control Optim. 45, no. 6, 2224-2256. [7] Buckdahn, R., J. Ma, and J. Zhang. (2015). Pathwise Taylor expansions for random fields on multiple dimensional paths, Stoch. Process. Appl. 125, 2820-2855. [8] Caruana, M., P. Friz, and H. Oberhauser. (2011). A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 27-46. [9] Crandall, M.G., H. Ishii, and P.-L. Lions. (1992). User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27, no. 1, 1-67. [10] Da Prato, G. and L. Tubaro. (1996). Fully nonlinear stochastic partial differential equations, SIAM J. Math. Anal. 27, no. 1, 40-55. [11] Davis, M. and G. Burstein. (1992). A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control, Stochast. Stoch. Rep. 40, no. 3-4, 203-256. [12] Diehl, J. and P. Friz. (2012). Backward stochastic differential equations with rough drivers, Ann. Prob. 40, 1715-1758. [13] Diehl, J., P. Friz, and P. Gassiat. (2017). Stochastic control with rough paths, Appl. Math. Optim. 75, no. 2, 285-315. [14] Diehl, J., P. Friz, and H. Oberhauser. (2014). Regularity theory for rough partial differential equations and parabolic comparison revisited, Springer, Cham. [15] Diehl, J., H. Oberhauser, and S. Riedel. (2015). A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations, Stoch. Process. Appl. 125, no. 1, 161-181. [16] Dupire, B. (2019). Functional Itô calculus, Quant. Finan. 19, no. 5, 721-729. [17] Ekren, I., C. Keller, N. Touzi, and J. Zhang. (2014). On viscosity solutions of path dependent PDEs, Ann. Probab. 42, 204-236. [18] Ekren, I., N. Touzi, and J. Zhang. (2016). Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part I, Ann. Probab. 44, 1212-1253. [19] Ekren, I., N. Touzi, and J. Zhang. (2016). Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs:Part II, Ann. Probab. 44, 2507-2553. [20] Friz, P., P. Gassiat, P.L. Lions, and P.E. Souganidis. (2017). Eikonal equations and pathwise solutions to fully non-linear SPDEs, Stochast. Partial Differ. Equ. Anal. Comput. 5, 256-277. [21] Friz, P. and M. Hairer. (2014). A course on rough paths:With an introduction to regularity structures, Universitext, Springer, Cham. [22] Friz, P. and H. Oberhauser. (2011). On the splitting-up method for rough (partial) differential equations, J. Differ. Equ. 251, no. 2, 316-338. [23] Friz, P. and H. Oberhauser. (2014). Rough path stability of (semi-)linear SPDEs, Probab. Theory Relat. Fields 158, 401-434. [24] Gilbarg, D. and N. Trudinger. (1983). Elliptic Partial Differential Equations of second order, second edition, Springer-Verlag, Germany. [25] Gubinelli, M. (2004). Controlling rough paths, J. Funct. Anal. 216, no. 1, 86-140. [26] Gubinelli, M., S. Tindel, and I. Torrecilla. (2014). Controlled viscosity solutions of fully nonlinear rough PDEs. arXiv preprint, arXiv:1403.2832. [27] Keller, C. and J. Zhang. (2016). Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients, Stoch. Process. Appl. 126, 735-766. [28] Krylov, N.V. (1999). An analytic approach to SPDEs, Stoch. Partial Differ. Equ. Six Perspect. Math. Surv. Monogr. Amer. Math. Soc. Providence RI 64, 185-242. [29] Kunita, H. (1997). Stochastic flows and stochastic differential equations, Cambridge University Press, Cambridge. [30] Lieberman, G. (1996). Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge. [31] Lions, P.-L. and P. E. Souganidis. (1998). Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 326, no. 9, 1085-1092. [32] Lions, P.-L. and P. E. Souganidis. (1998). Fully nonlinear stochastic partial differential equations:nonsmooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math. 327, no. 8, 735-741. [33] Lions, P.-L. and P. E. Souganidis. (2000). Fully nonlinear stochastic PDE with semilinear stochastic dependence, C. R. Acad. Sci. Paris Sér. I Math. 331, no. 8, 617-624. [34] Lions, P.-L. and P.E. Souganidis. (2000). Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. l'Acad. Sci.-Ser. I-Math. 331, no. 10, 783-790. [35] Lunardi, A. (1995). Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16, Birkhäuser Verlag, Basel. [36] Lyons, T. (1998). Differential equations driven by rough signals, Rev. Mat. Iberoam. 14, no. 2, 215-310. [37] Matoussi, A., D. Possamai, and W. Sabbagh. (2018). Probabilistic interpretation for solutions of Fully Nonlinear Stochastic PDEs, Probab. Theory Relat. Fields. https://doi.org/10.1007/s00440-018-0859-4. [38] Mikulevicius, R. and G. Pragarauskas. (1994). Classical solutions of boundary value problems for some nonlinear integro-differential equations, Lithuanian Math. J. 34, no. 3, 275-287. [39] Musiela, M. and T. Zariphopoulou. (2010). Stochastic partial differential equations and portfolio choice, Contemporary Quantitative Finance, Springer, Berlin. [40] Nadirashvili, N. and S. Vladut. (2007). Nonclassical solutions of fully nonlinear elliptic equations, Geom. Funct. Anal. 17, no. 4, 1283-1296. [41] Pardoux, E. and S. Peng. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98, 209-227. [42] Peng, S. (1992). Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim 30, no. 2, 284- 304. [43] Pham, T. and J. Zhang. (2014). Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation, SIAM J. Control. Optim. 52, 2090-2121. [44] Rozovskii, B.L. (1990). Stochastic Evolution Systems:Linear Theory and Applications to Non-linear Filtering, Kluwer Academic Publishers, Boston. [45] Safonov, M.V. (1988). Boundary value problems for second-order nonlinear parabolic equations, (Russian), Funct. Numer. Methods Math. Phys. "Naukova Dumka" Kiev. 274, 99-203. [46] Safonov, M.V. (1989). Classical solution of second-order nonlinear elliptic equations, Math. USSR-Izv 33, no. 3, 597-612. [47] Seeger, B. (2018). Perron's method for pathwise viscosity solutions, Commun. Partial Differ. Equ. 43, no. 6, 998-1018. [48] Seeger, B. (2018). Homogenization of pathwise Hamilton-Jacobi equations, J. Math. Pures Appl. 110, 1-31. [49] Seeger, B. (2020). Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations, Ann. Appl. Probab. 30, no. 4, 1784-1823. [50] Souganidis, P.E. (2019). Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence, Singular random dynamics, Lecture Notes in Math. vol. 2253, Springer, Cham. [51] Zhang, J. (2017). Backward Stochastic Differential Equations-from linear to fully nonlinear theory, Springer, New York.
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