Markov Processes and Their Applications to Partial Differential Equations: Kuznetsov’s Contributions
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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”.
KeywordsMarkov processes Transition function Excessive functions Superdiffusions Semilinear PDE Fine trace Kuznetsov measures
- Dyn69a.E. B. Dynkin, The boundary theory for Markov processes (discrete case), Russian Math. Surveys 24, 2 (1969), 89–157.Google Scholar
- Dyn69b.E. B. Dynkin, Exit space of a Markov process, Russian Math. Surveys 24, 4 (1969), 89–152.Google Scholar
- Dyn72.E. B. Dynkin, Integral representation of excessive measures and excessive functions, Russian Math. Surveys 27,1 (1972), 43–84.Google Scholar
- Dyn75.E. B. Dynkin, Markov representation of stochastic systems, Russian Math. Surveys 30,1 (1975), 64–104.Google Scholar
- 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
- Dyn02.E. B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, 2002. American Math Society, Colloquium Publications, Vol. 50.Google Scholar
- Dyn05.E. B. Dynkin, Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations, 2004. American Math Society, University Lecture Series, Vol. 34.Google Scholar
- Hun68.G. A. Hunt, Markov Chains and Martin Boundaries, Illinois J. Math. 4 (1968), 233–278, 365–410.Google Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Ms04.B. Mselaty, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Mem. Amer. Math. Soc. 168 (2004).Google Scholar
- 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
- Per91.E. A. Perkins, On the continuity of measure-valued processes, (1991), 261–268. Seminar on Stochastic Processes, Boston/Basel/Berlin.Google Scholar
- 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