Vấn đề Tính Độc Đáo cho Các Phương Trình Fokker–Planck–Kolmogorov Bị Định Hình

Journal of Mathematical Sciences - Tập 207 - Trang 147-165 - 2015
V. I. Bogachev1,2, M. Röckner3, S. V. Shaposhnikov1,2
1Moscow State University, Moscow, Russia
2St Tikhon’s Orthodox Humanitarian University, Moscow, Russia
3Universität Bielefeld, Bielefeld, Germany

Tóm tắt

Chúng tôi nghiên cứu tính độc đáo của các nghiệm đối với bài toán Cauchy cho phương trình Fokker–Planck–Kolmogorov với ma trận khuếch tán đặc biệt trong lớp các thước đo xác suất. Bài báo trình bày tổng quan về các kết quả và phương pháp đã biết. Ngoài ra, chúng tôi đưa ra các điều kiện đầy đủ mới cho tính độc đáo trong trường hợp các hệ số không bị chặn và ma trận khuếch tán một phần đặc biệt, cũng như trong trường hợp ma trận khuếch tán là ma trận Lipschitz bình phương.

Từ khóa

#Fokker–Planck–Kolmogorov equation #uniqueness of solutions #Cauchy problem #singular diffusion matrix #probability measures #Lipschitzian matrix.

Tài liệu tham khảo

A. N. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,” Math. Ann. 104, No. 1, 415–458 (1931). V. I. Bogachev, G. Da Prato, and M. Röckner, “On parabolic equations for measures,” Commun. Partial Diff. Equations 33, No. 1-3, 397–418 (2008). O. A. Manita and S. V. Shaposhnikov, “On the well-posedness of the Cauchy problem for Fokker–Planck–Kolmogorov equations with potential terms on arbitrary domains,” arXiv:1307.3662 (2013). V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, “On uniqueness problems related to the Fokker–Planck–Kolmogorov equations for measures” [in Russian], Probl. Mat. Anal. 61, 9–41 (2011); English transl.: J. Math. Sci. (New York) 179, No. 1, 1–41 (2011). V. I. Bogachev, G. Da Prato, M. Röckner, and W. Stannat, “Uniqueness of solutions to weak parabolic equations for measures,” Bull. London Math. Soc. 39, No. 4, 631–640 (2007). S. V. Shaposhnikov, “On the uniqueness of a probabilistic solution of the Cauchy problem for the Fokker–Planck–Kolmogorov equation” [in Russian], Teor. Veroyatnost. Primen. 56 No. 1, 77–99 (2011); English transl.: Theory Probab. Appl. 56, No. 1, 96–115 (2012). O. A. Ladyz’enskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967); English transl.: Am. Math. Soc., Providence, RI (1968). R. J. DiPerna and P. L. Lions, “Ordinary differential equations, transport theory and Sobolev spaces,” Invent. Math. 98, 511–547 (1989). L. Ambrosio, “Transport equation and Cauchy problem for non-smooth vector fields,” Lect. Notes Math. 1927, 2–41 (2008). L. Ambrosio and P. Bernard, “Uniqueness of signed measures solving the continuity equation for Osgood vector fields,” Rend. Lincei, Mat. Appl. 19, No. 3, 237–245 (2008). S. Maniglia, “Probabilistic representation and uniqueness results for measure-valued solutions of transport equation,” J. Math. Pures Appl. 87, 601–626 (2007). D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin etc. (1979). C. Le Bris and P. L. Lions, “Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients,” Commun. Partial Diff. Equations 33, 1272–1317 (2008). M. Röckner and X. Zhang, “Weak uniqueness of Fokker–Planck equations with degenerate and bounded coefficients,” C. R. Math. Acad. Sci. Paris 348, No. 7-8, 435–438 (2010). J. Wei and B. Liu, “L p-solutions of Fokker–Planck equations,” Nonlinear Anal. 85, 110–124 (2013). De Jun Luo, “Fokker–Planck type equations with Sobolev diffusion coefficients and BV drift coefficients,” Acta Math. Sin. (Engl. Ser.) 29, No. 2, 303–314 (2013). N. Belaribi and F. Russo, “Uniqueness for Fokker-Planck equations with measurable coefficients and applications to the fast diffusion equation,” Electron. J. Probab. 17, No. 84, 1–28 (2012). P. Blanchard, M. Röckner, and F. Russo, “Probabilistic representation for solutions of an irregular porous media type equation,” Ann. Probab. 38, No. 5, 1870–1900 (2010). V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, “On uniqueness of solutions to the Cauchy problem for degenerate Fokker–Planck–Kolmogorov equations,” J. Evol. Equ. 13, No. 3, 577–593 (2013). A. Figalli, “Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients,” J. Funct. Anal. 254, No. 1, 109–153 (2008). De Jun Luo, “Uniqueness of degenerate Fokker–Planck equations with weakly differentiable drift whose gradient is given by a singular integral,” Electron. Commun. Probab. 19, 1–14 (2014). A. A. Levakov and M. M. Vas’kovskii, “Existence of weak solutions of stochastic differential equations with discontinuous coefficients and a partially degenerate diffusion operator” [in Russian], Differ. Uravn. 43, No. 8, 1029–1042 (2007); English transl.: Differ. Equ. 43, No. 8, 1051–1066 (2007). A. Yu. Veretennikov, “On stochastic equations with degenerate diffusion according to a part of variables” [in Russian], Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 1, 188–196 (1983). A. Guillin and F.-Y.Wang, “Degenerate Fokker–Planck equations: Bismut formula, gradient estimate and Harnack inequality,” J. Differ. Equat. 253, No. 1, 20–40 (2012). O. A. Oleinik, “On smoothness of solutions to degenerate elliptic and parabolic equations” [in Russian], Dokl. Akad Nauk USSR 163, No. 3, 577–580 (1965); English transl.: Soviet Math. Dokl. 6, 972–976 (1965). X. Zhang, “Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients,” Rev. Mat. Iberoam. 29, No. 1, 25–52 (2013). S. V. Shaposhnikov, “Fokker–Planck–Kolmogorov equations with a potential and a nonuniformly elliptic diffusion matrix” [in Russian], Trans. Moscow Math. Soc. 74, No. 1, 17–34 (2013). V. I. Bogachev, G. Da Prato, M. R¨ockner, and S. V. Shaposhnikov, “An analytic approach to infinite-dimensional continuity and Fokker–Planck–Kolmogorov equations,” arXiv:1305.7348; Ann. Scuola Norm. Super. (2015). V. I. Bogachev, N. V. Krylov, and M. Röckner, “On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions,” Commun. Partial Diff. Equations 26, No. 11–12, 2037–2080 (2001).