A Criterium for the Strict Positivity of the Density of the Law of a Poisson Process
© Rémi Léandre. 2011
Received: 26 August 2010
Accepted: 9 January 2011
Published: 24 January 2011
We translate in semigroup theory our result (Léandre, 1990) giving a necessary condition so that the law of a Markov process with jumps could have a strictly positive density. This result express, that we have to jump in a finite number of jumps in a "submersive" way from the starting point to the end point if the density of the jump process is strictly positive in . We use the Malliavin Calculus of Bismut type of (Léandre, (2008;2010)) translated in semi-group theory as a tool, and the interpretation in semi-group theory of some classical results of the stochastic analysis for Poisson process as, for instance, the formula giving the law of a compound Poisson process.
We are interested in this paper in the following problem.
This problem was solved by using the Malliavin Calculus. See the survey paper of Léandre  on that. For various applications of the Malliavin Calculus on heat kernels, we refer to the review of Kusuoka , Léandre , and Watanabe .
If is a submersion in some sense in , then we can apply in some sense the implicit function theorem in order to get a lower bound of the law of by a measure having a strictly positive density in the values of in with respect of the Lebesgue measure on . The problem is that the solution of the stochastic differential equation (1.3) is only almost surely defined. So the use of the implicit theorem leds to some difficulties which were overcome by Bismut in . The use of Bismut's procedure allows to  to solve Problem*. See  for a translation of the proof of  in semigroup theory.
Plenty of the standard tools of stochastic analysis were translated recently by Léandre in semigroup theory. See the review [9, 10] on that. Problem* was solved for a diffusion by using the Malliavin Calculus of Bismut type in semigroup theory in .
Tortrat  studied the support of the law of : if and , then belong to . If has a finite mass , then the process has the law of a compound Poisson process: is sum of his jumps. There is only a finite number of jumps. The jumps are all independents with law and the times where the jumps occur follow the law of a standard Poisson process with parameter . We will give a proof, uniquely based upon algebraic computations on semi groups, of this fact in the paper.
Problem* was solved in  by using the Malliavin Calculus for jump processes (see [15–17] for related works). is called the Lévy measure. For that we need some regularity on the Lévy measure . Under regularity assumption on ,  used another time the implicit function theorem, when we can jumps in a finite number of jumps in a "submersive" way between the starting point and the end point. Recently we have translated in semigroup theory plenty of tools of the stochastic analysis for Poisson processes [18–23]. Our goal is to translate in semigroup theory the result of .
For material on stochastic differential equations driven by jump processes, we refer to the books [13, 24, 25]. For the analytic side of the theory of Markov processes with jumps, we refer to the books [11–13].
This paper enters in a general program which would like that stochastic analysis tools become available for partial differential equation different of the parabolic equations whose generators satisfy the maximum principle [26, 27].
2. Statement of the Main Theorems
We do the following hypothesis.
if is bounded differentiable. If , the is classically related to fractional powers of the Laplacian .
We do the following Hypothesis.
3. Two Results on Jump Processes Translated in Semigroup Theory
Theorem 3.1 (compound Poisson process).
We suppose that the total mass of is finite and is equal to the constant quantity and that depends continuously of for the strong topology. generates a semigroup on the space of continuous functions endowed with the uniform norm.
It is a bounded operator on the set of uniformly bounded functions endowed with the uniform topology. Therefore it generates a semigroup on the set of bounded continuous functions. We get the following translation of (2.20) in semigroup theory.
4. Proof of Theorem 2.2
and the result goes. It remains to remark that under the previous condition has a polynomial behaviour whose component tends to zero and to apply Theorem 3 of . This comes from the fact that is a polynomial in and is differentiable bounded in because we keep only bounded values of due to the apparition of .
Proof of Theorem 2.2.
5. Proof of Theorem 2.5
- Léandre R: Strange behaviour of the heat kernel on the diagonal. In Stochastic Processes, Physics and Geometry. Edited by: Albeverio Set al.. World Scientific Publishing; 1990:516-527.Google Scholar
- Kusuoka S: More recent theory of Malliavin calculus. Sugaku Expositions 1992,5(2):155-171.MathSciNetGoogle Scholar
- Léandre R: Quantitative and geometric applications of the Malliavin calculus. In French Japanese Seminar. Volume 73. Edited by: Métivier M, Watanabe S. American Mathematical Society, Providence, RI, USA; 1988:173-196. in Geometry of Random Motion, R. Durrett, M. Pinsky, Eds, vol. 73 of Contemporary Maths, American Mathematical Society, Providence, RI, USA, pp. 173–196, 1988Google Scholar
- Watanabe S: Stochastic analysis and its applications. Sugaku Expositions 1992,5(1):51-69.MathSciNetGoogle Scholar
- Ikeda N, Watanabe S: Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library. Volume 24. 2nd edition. North-Holland, Amsterdam, The Netherlands; 1989:xvi+555.MATHGoogle Scholar
- Bismut J-M: Large Deviations and the Malliavin Calculus, Progress in Mathematics. Volume 45. Birkhäuser, Boston, Mass, USA; 1984:viii+216.Google Scholar
- Ben Arous G, Léandre R: Décroissance exponentielle du noyau de la chaleur sur la diagonale. II. Probability Theory and Related Fields 1991,90(3):377-402. 10.1007/BF01193751MathSciNetView ArticleMATHGoogle Scholar
- Léandre R: Positivity theorem in semi-group theory. Mathematische Zeitschrift 2008,258(4):893-914. 10.1007/s00209-007-0204-6MathSciNetView ArticleMATHGoogle Scholar
- Léandre R: Applications of the Malliavin calculus of Bismut type without probability. In Simulation, Modelling and Optimization, WSEAS Transactions on Mathematics, pp. 1205–1211, 2006 Edited by: Madureira AM. 2006, 5: 559-564.Google Scholar
- Léandre R: Malliavin calculus of Bismut type in semi-group theory. Far East Journal of Mathematical Sciences 2008,30(1):1-26.MathSciNetMATHGoogle Scholar
- Jacob N: Pseudo Differential Operators and Markov Processes. Vol. I. Imperial College Press, London, UK; 2001:xxii+493.View ArticleMATHGoogle Scholar
- Jacob N: Pseudo Differential operators & Markov Processes. Vol. II. Imperial College Press, London, UK; 2002:xxii+453.View ArticleMATHGoogle Scholar
- Jacob N: Pseudo Differential Operators and Markov Processes. Vol. III. Imperial College Press, London, UK; 2005:xxviii+474.View ArticleMATHGoogle Scholar
- Tortrat A: Sur le support des lois indéfiniment divisibles dans les espaces vectoriels localement convexes. Annales de l'Institut Henri Poincaré 1977,13(1):27-43.MathSciNetMATHGoogle Scholar
- Fournier N: Strict positivity of the density for simple jump processes using the tools of support theorems. Application to the Kac equation without cutoff. The Annals of Probability 2002,30(1):135-170.MathSciNetView ArticleMATHGoogle Scholar
- Ishikawa Y: Support theorem for jump processes of canonical type. Proceedings of the Japan Academy. Series A 2001,77(6):79-83. 10.3792/pjaa.77.79MathSciNetView ArticleMATHGoogle Scholar
- Simon T: Support theorem for jump processes. Stochastic Processes and their Applications 2000,89(1):1-30. 10.1016/S0304-4149(00)00008-9MathSciNetView ArticleMATHGoogle Scholar
- Léandre R: Girsanov transformation for poisson processes in semi-group theory. In Proceedings of the International Conference on Numerical Analysis and Applied Mathematics, 2007 Edited by: Simos T. 936: 336-338.Google Scholar
- Léandre R: Malliavin calculus of Bismut type for Poisson processes without probability. In Fractional Order Systems. Volume 42. Edited by: Sabatier Jet al.. J.E.S.A.; 2008:715-733.Google Scholar
- Léandre R: Regularity of a degenerated convolution semi-group without to use the Poisson process. In Non Linear Science and Complexity. Edited by: Luo Aet al.. Springer; 2010:311-320.Google Scholar
- Léandre R: Wentzel-Freidlin estimates for jump process in semi-group theory: lower bound. In Proceedings of the International Conference of Differential Geometry and Dynamical Systems, 2010. Volume 17. Edited by: Balan Vet al.. B.S.G.; 107-113.Google Scholar
- Léandre R: Wentzel-Freidlin estimates for jump process in semi-group theory: upper bound. In Proceedings of the International Conference on Scientific Computing, 2010. Edited by: Arabnia Het al.. CSREA; 187-193.Google Scholar
- Léandre R: Varadhan estimates for a degeneratedconvolution semi-group: upper bound. Fractional Differentiation and Applications. In pressGoogle Scholar
- Applebaum D: Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics. Volume 93. Cambridge University Press, Cambridge, UK; 2004:xxiv+384.View ArticleMATHGoogle Scholar
- Jacod J: Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Mathematics. Volume 714. Springer, Berlin, Germany; 1979:x+539.Google Scholar
- Léandre R: Stochastic analysis without probability: study of some basic tools. Journal Pseudo Differential Operators and Application 2010,1(4):389-400. 10.1007/s11868-010-0020-3View ArticleMathSciNetMATHGoogle Scholar
- Léandre R: A path-integral approach to the Cameron-Martin-Maruyama-Girsanov formula associated to a bilaplacian. PreprintGoogle Scholar
- Bismut J-M: Calcul des variations stochastique et processus de sauts. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 1983,63(2):147-235. 10.1007/BF00538963MathSciNetView ArticleMATHGoogle Scholar
- Léandre R: Régularité de processus de sauts dégénérés. Comptes Rendus des Séances de l'Académie des Sciences. Série I 1983,297(11):595-598.MATHGoogle Scholar
- Léandre R: Régularité de processus de sauts dégénérés. Annales de l'Institut Henri Poincaré 1985,21(2):125-146.MATHGoogle Scholar
- Yosida K: Functional Analysis. 4th edition. Springer, New York, NY, USA; 1977:xii+496.Google Scholar
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