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Markov Processes and Their Applications to Partial Differential Equations: Kuznetsov’s Contributions

  • E. B. DynkinEmail author
Conference paper
  • 843 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 38)

Abstract

We describe some directions of research in probability theory and related problems of analysis to which S. E. Kuznetsov has made fundamental contributions.A Markov process (understood as a random path X t , 0 ≤ t < such that past before t and future after t are independent given X t ) is determined by a probability measure P on a path space. This measure can be constructed starting from a transition function and probability distribution of X 0. For a number of applications, it is also important to consider a path in both, forward and backward directions which leads to a concept of dual processes. In 1973, Kuznetsov constructed, as a substitute for such a pair of processes, a single random process (X t , ℙ) determined on a random time interval (α, β). The corresponding forward and backward transition functions define a dual pair of processes. A σ-finite measure ℙ became, under the name “Kuznetsov measure,” an important tool for research on Markov processes and their applications.

In 1980, Kuznetsov proved that every Markov process in a Borel state space has a transition function (a problem that was open for many years). In 1992, he used this result to obtain simple necessary and sufficient conditions for existence of a unique decomposition of excessive functions into extreme elements—a significant extension of a classical result on positive superharmonic functions.

Intimate relations between the Brownian motion and differential equations involving the Laplacian Δ were known for a long time. Applications of probabilistic tools to classical potential theory and to study of linear PDEs are more recent. Even more recent is application of such tools to nonlinear PDEs. In a series of publications, starting from 1994, Dynkin and Kuznetsov investigated a class of semilinear elliptic equations by using super-Brownian motion and more general measure-valued Markov processes called superdiffusions. The main directions of this work were (a) description of removable singularities of solutions and (b) characterization of all positive solutions. One of the principal tools for solving the second problem was the fine trace of a solution on the boundary invented by Kuznetsov.

The same class of semilinear equations was the subject of research by Le Gall who applied a path-valued process Brownian snake instead of the super-Brownian motion. A slightly more general class of equations was studied by analysts including H. Brezis, M. Marcus and L. Veron. In the opinion of Brezis: “it is amazing how useful for PDEs are the new ideas coming from probability. This is an area where the interaction of probability and PDEs is most fruitful and exiting”.

Keywords

Markov processes Transition function Excessive functions Superdiffusions Semilinear PDE Fine trace Kuznetsov measures 

References

  1. BP84.
    P. Baras and M. Pierre, Problems paraboliques semi-linéares avec donnees measures, Applicable Analysis 18 (1984), 111–149.MathSciNetzbMATHCrossRefGoogle Scholar
  2. BP84b.
    P. Baras and M. Pierre, Singularités éliminable pour des équations semi-linéares, Ann. Inst. Fourier Grenoble 34 (1984), 185–206.MathSciNetzbMATHCrossRefGoogle Scholar
  3. BS.
    H. Brezis and V. Strauss, Semilinear second-order Elliptic equations inL 1, J. Math. Soc. Japan 25 (1973), 565–590.MathSciNetzbMATHCrossRefGoogle Scholar
  4. BV.
    H. Brezis and L. Véron, Removable singularities of some removable equations, Arch. Rat. Mech. Anal. 75 (1980), 1–6.zbMATHCrossRefGoogle Scholar
  5. Daw75.
    D. A. Dawson, Stochastic evolution equations and related measure processes, J. Multivariate Anal. 3 (1975), 1–52.CrossRefGoogle Scholar
  6. Doo59.
    J. L. Doob, Discrete Potential Theory and Boundaries, J. Math. Mech 8 (1959), 433–458.MathSciNetzbMATHGoogle Scholar
  7. Dyn69a.
    E. B. Dynkin, The boundary theory for Markov processes (discrete case), Russian Math. Surveys 24, 2 (1969), 89–157.Google Scholar
  8. Dyn69b.
    E. B. Dynkin, Exit space of a Markov process, Russian Math. Surveys 24, 4 (1969), 89–152.Google Scholar
  9. Dyn72.
    E. B. Dynkin, Integral representation of excessive measures and excessive functions, Russian Math. Surveys 27,1 (1972), 43–84.Google Scholar
  10. Dyn75.
    E. B. Dynkin, Markov representation of stochastic systems, Russian Math. Surveys 30,1 (1975), 64–104.Google Scholar
  11. Dyn76.
    E. B. Dynkin, On a new approach to Markov processes, Proceedings of the Third Japan-USSR Symposium on Probability Theory (1976), 42–62. Lecture Notes in Mathematics, vol. 550.Google Scholar
  12. Dyn02.
    E. B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, 2002. American Math Society, Colloquium Publications, Vol. 50.Google Scholar
  13. Dyn05.
    E. B. Dynkin, Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations, 2004. American Math Society, University Lecture Series, Vol. 34.Google Scholar
  14. DK96.
    E. B. Dynkin and S.E. Kuznetsov, Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure Appl.Math. 49 (1996), 125–176.MathSciNetzbMATHCrossRefGoogle Scholar
  15. DK98a.
    E. B. Dynkin and S.E. Kuznetsov, Trace on the boundary for solutions of nonlinear partial differential equations, Trans. Amer. Math. Soc. 350 (1998), 4499–4519.MathSciNetzbMATHCrossRefGoogle Scholar
  16. DK98.
    E. B. Dynkin and S.E. Kuznetsov, Fine topology and fine trace on the boundary associated with a class of quasilinear partial differential equations, Comm. Pure Appl.Math. 51 (1998), 897–936.MathSciNetCrossRefGoogle Scholar
  17. GV91.
    A. Gmira and L. Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J. 64 (1991), 271–324.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Hun68.
    G. A. Hunt, Markov Chains and Martin Boundaries, Illinois J. Math. 4 (1968), 233–278, 365–410.Google Scholar
  19. Kel57.
    J. B. Keller, On the solutions of Δu = f(u), Comm. Pure Appl. Math. 10 (1957), 503–510.MathSciNetzbMATHCrossRefGoogle Scholar
  20. KW65.
    H. Kunita and T. Watanabe, Markov processes and Martin boundaries, Illinois J. Math. 9 (1965), 386–391.MathSciNetGoogle Scholar
  21. Kuz73.
    S. E. Kuznetsov, Construction of Markov processes with random birth and death times, Teoriya Veroyatn. i ee Primen. 18 (1973), 596–601. English Transl. in Theor. Prob. Appl., Vol 18 (1974).Google Scholar
  22. Kuz74.
    S. E. Kuznetsov, On decomposition of excessive functions, Dokl. AN SSSR 214 (1974), 276–278. English Transl. in Soviet Math. Dokl., Vol 15 (1974).Google Scholar
  23. Kuz80.
    S. E. Kuznetsov, Any Markov process in a Borel space has a transition function, Teoriya Veroyatn. i ee Primen. 25 (1980), 389–393. English Transl. in Theor. Prob. Appl., Vol 25 (1980).Google Scholar
  24. Kuz84.
    S. E. Kuznetsov, Specifications and stopping theorem for random fields, Teoriya Veroyatn. i ee Primen. 29 (1984), 65–77. English Transl. in Theor. Prob. Appl., Vol 29 (1985).Google Scholar
  25. Kuz86.
    S. E. Kuznetsov, On the existence of a homogeneous transition function, Teoriya Veroyatn. i ee Primen. 31 (1986), 290–300. English Transl. in Theor. Prob. Appl., Vol 31 (1987).Google Scholar
  26. Kuz92.
    S. E. Kuznetsov, More on existence and uniqueness of decomposition of excessive functions and measures to extremes, Seminaire de Probabilites XXVI. Lecture Notes in Mathematics 1526 (1992), 445–472.CrossRefGoogle Scholar
  27. Kuz92b.
    S. E. Kuznetsov, On existence of a dual semigroup, Seminaire de Probabilites XXVI. Lecture Notes in Mathematics 1526 (1992), 473–484.CrossRefGoogle Scholar
  28. Kuz98.
    S. E. Kuznetsov, σ-moderate solutions ofLu = u α and fine trace on the boundary, C.R. Acad. Sci. Paris, Série I 326 (1998), 1189–1194.Google Scholar
  29. Oss57.
    R. Osserman, On the inequality Δu ≥ f(u), Pacific J. Math. 7 (1957), 1641–1647.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Le94.
    J.-F. Le Gall, The Brownian snake and solutions of Δu = u2 in a domain, Probab. Theory Rel. Fields 102 (1995), 393–402.zbMATHCrossRefGoogle Scholar
  31. Le97.
    J.-F. Le Gall, A probabilistic Poisson representation for positive solutions of Δu = u2 in a planar domain, Comm. Pure Appl. Math. 50 (1997), 69–103.MathSciNetzbMATHCrossRefGoogle Scholar
  32. LN74.
    C. Loevner and L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, (1974), 245–272. Contributions to Analysis, Academic Press, Orlando, FL.Google Scholar
  33. Mar41.
    R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137–172.MathSciNetCrossRefGoogle Scholar
  34. Mey00.
    P.-A. Meyer, Dynkin and theory of Markov processes, Selected Papers of E. B. Dynkin with Commentary (2000), 763–772. Amer. Math. Soc., Providence, RI.Google Scholar
  35. Ms04.
    B. Mselaty, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Mem. Amer. Math. Soc. 168 (2004).Google Scholar
  36. MV.
    M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations, I. The subcritical case, Arch. Rat. Mech. Anal. 144 (1998), 201–231.zbMATHCrossRefGoogle Scholar
  37. MV98b.
    M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The supercritical case, J. Math. Pures Appl. 77 (1998), 481–524.MathSciNetzbMATHGoogle Scholar
  38. MV07.
    M. Marcus and L. Véron, The precise boundary trace of positive solutions of Δu = u q in the supercritical case, 446 (2007), 345–383. Perspectives in nonlinear partial differential equations, Contemp. Math., Amer. Math. Soc., Providence, RI.Google Scholar
  39. Per89.
    E. A. Perkins, The Hausdorff measure of the closed support of super-Brownian motion, Ann. Inst. H. Poincaré 25 (1989), 205–224.MathSciNetzbMATHGoogle Scholar
  40. Per91.
    E. A. Perkins, On the continuity of measure-valued processes, (1991), 261–268. Seminar on Stochastic Processes, Boston/Basel/Berlin.Google Scholar
  41. Wal72.
    J. B. Walsh, Transition functions of Markov processes, Sminaire de Probabilits VI (1972), 215–232. Lecture Notes in Math., Vol. 258, Springer, Berlin, 1972.Google Scholar
  42. Wat68.
    S. Watanabe, A limit theorem on branching processes and continuous state branching processes, J. Math. Kyoto Univ. 8 (1968), 141–167.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsMalott Hall Cornell UniversityIthacaUSA

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