# On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

Article

First Online:

Received:

- 122 Downloads
- 10 Citations

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂for all smooth functions u with compact support in ℝ

_{ t }*μ*=*L*^{*}*μ*for bounded Borel measures on ℝ^{ d }× (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$$ \int \left( {{\partial_t}u + Lu} \right)d\mu = 0 $$

^{ d }× (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.## Keywords

Probability Measure Cauchy Problem Lebesgue Measure Parabolic Equation Lyapunov Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- 1.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).MATHCrossRefGoogle Scholar - 2.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).MathSciNetGoogle Scholar - 3.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).MathSciNetMATHCrossRefGoogle Scholar - 4.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).MATHCrossRefGoogle Scholar - 5.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).MATHCrossRefGoogle Scholar - 6.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).Google Scholar - 7.L. Wu and Y. Zhang, “A new topological approach to the
*L∞*-uniqueness of operators and*L*1-uniqueness of Fokker–Planck equations,”*J. Funct. Anal.***241**, 557–610 (2006).MathSciNetMATHCrossRefGoogle Scholar - 8.D. L. Lemle, “
*L*^{1}(ℝ^{d}*, 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).MathSciNetMATHGoogle Scholar - 9.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).Google Scholar - 10.A. N. Tychonoff, “A uniqueness theorem for the heat equation” [in Russian],
*Mat. Sb.***42**, 199–216 (1935).MATHGoogle Scholar - 11.D. V. Widder, “Positive temperatures on the infinite rod,”
*Trans. Am. Math. Soc.***55**, No. 1, 85–95 (1944).MathSciNetMATHCrossRefGoogle Scholar - 12.D. G. Aronson and P. Besala, “Uniqueness of solutions of the Cauchy problem for parabolic equations,”
*J. Math. Anal. Appl.***13**, 516–526 (1966).MathSciNetMATHCrossRefGoogle Scholar - 13.D. G. Aronson, “Non-negative solutions of linear parabolic equations,”
*Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV Ser.***22**, 607–694 (1968).MathSciNetMATHGoogle Scholar - 14.A. Friedman,
*Partial Differential Equations of Parabolic Type*, Prentice-Hall, Englewood Cliffs, New Jersey (1964).MATHGoogle Scholar - 15.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).MATHCrossRefGoogle Scholar - 16.A. Figalli, “Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients,”
*J. Funct. Anal.***254**, No. 1, 109–153 (2008).MathSciNetMATHCrossRefGoogle Scholar - 17.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).MathSciNetMATHGoogle Scholar - 18.Y. Pinchover, “On uniqueness and nonuniqueness of positive Cauchy problem for parabolic equations with unbounded coefficients,”
*Math. Z.***233**, 569–586 (1996).MathSciNetGoogle Scholar - 19.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).MathSciNetGoogle Scholar - 20.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).MathSciNetMATHGoogle Scholar - 21.A. N. Kolmogoroff, “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung,”
*Math. Ann.***104**, No. 1, 415–458 (1931).MathSciNetCrossRefGoogle Scholar - 22.I. I. Gihman and A. V. Skorohod,
*The Theory of Stochastic Processes. II*, Springer, New York etc. (1975).MATHGoogle Scholar - 23.D. W. Stroock and S. R. S. Varadhan,
*Multidimensional Diffusion Processes*! Springer, Berlin etc. (1979).Google Scholar - 24.G. Metafune, D. Pallara, and M. Wacker, “Feller semigroupe om ℝ
^{N},”*Semigroup Forum***65**, No. 2, 159–205 (2002).MathSciNetMATHCrossRefGoogle Scholar - 25.G. Metafune, D. Pallara, and A. Rhandi, “Global properties of transition probabilities of singular diffusions,”
*Theory Probab. Appl.***54**, No. 1, 116–148 (2009).MathSciNetGoogle Scholar - 26.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).Google Scholar - 27.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).MathSciNetGoogle Scholar - 28.S. V. Shaposhnikov, “On nonuniqueness of solutions to elliptic equations for probability measures,”
*J. Funct. Anal.***254**, No. 10, 2690–2705 (2008).MathSciNetMATHCrossRefGoogle Scholar - 29.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).CrossRefGoogle Scholar - 30.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).MATHCrossRefGoogle Scholar - 31.N. V. Krylov, “Parabolic and elliptic equations with VMO coefficients,”
*Commun. Partial Diff. Equations***32**, No. 3, 453–475 (2007).MATHCrossRefGoogle Scholar - 32.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).Google Scholar - 33.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).Google Scholar - 34.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).MathSciNetGoogle Scholar

## Copyright information

© Springer Science+Business Media, Inc. 2011