We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ t μ = L * μ for bounded Borel measures on ℝd × (0, T), where L is a second order elliptic operator, for example, \( Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) \), and the equation is understood as the identity
for all smooth functions u with compact support in ℝd × (0, T). Our study are motivated by equations of such a type, namely, the Fokker–Planck–Kolmogorov equations for transition probabilities of diffusion processes. Solutions are considered in the class of probability measures and in the class of signed measures with integrable densities. We present some recent positive results, give counterexamples, and formulate open problems. Bibliography: 34 titles.
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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).
R. Z. Hasminskii, “Ergodic properties of reccurent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations” [in Russian], Teor. Veroyatn. Primen. 5, 196–214 (1960); English transl.: Theory Probab. Appl. 5, 179–196 (1960).
V. I. Bogachev, G. Da Prato, and M. Röckner, “Existence of solutions to weak parabolic equations for measures,” Proc. London Math. Soc. 88, No. 3, 753–774 (2004).
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).
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 uniqueness of a probability solution to the Cauchy problem for the Fokker–Planck–Kolmogorov equation” [in Russian], Theory Probab. Appl. 56 (2011), No. 1, 77–99 (2011); English tranls.: Theor. Probab. Appl. 56, No. 1 (2012).
L. Wu and Y. Zhang, “A new topological approach to the L∞-uniqueness of operators and L1-uniqueness of Fokker–Planck equations,” J. Funct. Anal. 241, 557–610 (2006).
D. L. Lemle, “L 1(ℝd , dx)-Uniqueness of weak solutions for the Fokker–Planck equation associated with a class of Dirichlet operators,” Elect. Research Announc. Math. Sci. 15, 65–70 (2008).
S. V. Shaposhnikov, “On the uniqueness of integrable and probability solutions to the Cauchy problem for the Fokker–Planck–Kolmogorov equation” [in Russian], Dokl. Ross. Akad. Nauk 439, No. 3, 331–335 (2011); English tranls.: Dokl. Math. (2011).
A. N. Tychonoff, “A uniqueness theorem for the heat equation” [in Russian], Mat. Sb. 42, 199–216 (1935).
D. V. Widder, “Positive temperatures on the infinite rod,” Trans. Am. Math. Soc. 55, No. 1, 85–95 (1944).
D. G. Aronson and P. Besala, “Uniqueness of solutions of the Cauchy problem for parabolic equations,” J. Math. Anal. Appl. 13, 516–526 (1966).
D. G. Aronson, “Non-negative solutions of linear parabolic equations,” Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser. 22, 607–694 (1968).
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey (1964).
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).
A. Figalli, “Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients,” J. Funct. Anal. 254, No. 1, 109–153 (2008).
K. Ishige and M. Murata, “Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains,” Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser. 30, No. 1, 171–223 (2001).
Y. Pinchover, “On uniqueness and nonuniqueness of positive Cauchy problem for parabolic equations with unbounded coefficients,” Math. Z. 233, 569–586 (1996).
O. A. Oleinik and E. V. Radkevich, “The method of introducing a parameter in the study of evolutionary equations” [in Russian], Usp. Mat. Nauk 33, No. 5, 6–76 (1978); English transl.: Russ. Math. Surv. 33, No. 5, 7–84 (1978).
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).
A. N. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,” Math. Ann. 104, No. 1, 415–458 (1931).
I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes. II, Springer, New York etc. (1975).
D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes! Springer, Berlin etc. (1979).
G. Metafune, D. Pallara, and M. Wacker, “Feller semigroupe om ℝN,” Semigroup Forum 65, No. 2, 159–205 (2002).
G. Metafune, D. Pallara, and A. Rhandi, “Global properties of transition probabilities of singular diffusions,” Theory Probab. Appl. 54, No. 1, 116–148 (2009).
V. I. Bogachev, M. Röckner, and W. Stannat, “Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions” [in Russian], Mat. Sb. 193, No. 7, 3–36 (2002); English transl.: Sb. Math. 193, No. 7, 945–976 (2002).
S. V. Shaposhnikov, “The nonuniqueness of solutions to elliptic equations for probability measures” [in Russian], Dokl. Akad. Nauk, Ross. Akad. Nauk 420, No. 3, 320–323 (2008); English transl.: Dokl. Math. 77! No. 3, 401–403 (2008).
S. V. Shaposhnikov, “On nonuniqueness of solutions to elliptic equations for probability measures,” J. Funct. Anal. 254, No. 10, 2690–2705 (2008).
V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, “On uniqueness problems related to elliptic equations for measures,” J. Math. Sci. (New York) 176, No. 6, 759–773 (2011).
S. Albeverio, V. Bogachev, and M. Röckner, “On uniqueness of invariant measures for finite- and infinite-dimensional diffusions” Commun. Pure Appl. Math. 52, No. 3, 325–362 (1999).
N. V. Krylov, “Parabolic and elliptic equations with VMO coefficients,” Commun. Partial Diff. Equations 32, No. 3, 453–475 (2007).
O. A. Ladyz’enskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type! Am. Math. Soc., Providence RI (1968).
V. I. Bogachev, N. V. Krylov, and M. Röckner, “Elliptic and parabolic equations for measures” [in Russian], Usp. Mat. Nauk 64, No. 6, 5–116 (2009); English transl.: Russ. Math. Surv. 64, No. 6, 973–1078 (2009).
F. O. Porper and S. D. Eidel’man, “Two-sided estimates of fundamental solutions of second order parabolic equations, and some applications” [in Russian], Usp. Mat. Nauk 39, No. 3, 107–156 (1984): English transl.: Russ. Math. Surv. 39, No. 3, 119–178 (1984).
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Dedicated to Professor N. V. Krylov on the occasion of his 70th birthday
Translated from Problems in Mathematical Analysis 61, October 2011, pp. 9–42.
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Bogachev, V.I., Röckner, M. & Shaposhnikov, S.V. On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures. J Math Sci 179, 7–47 (2011). https://doi.org/10.1007/s10958-011-0581-6
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DOI: https://doi.org/10.1007/s10958-011-0581-6