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January  2019, 18(1): 323-340. doi: 10.3934/cpaa.2019017

On the positive semigroups generated by Fleming-Viot type differential operators

1. 

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Campus Universitario, Via Edoardo Orabona n. 4, 70125 Bari, Italy

2. 

Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Campus di Macchia Romana, Viale Dell' Ateneo Lucano n. 10, 85100 Potenza, Italy

* Corresponding author

Received  January 2018 Revised  April 2018 Published  August 2018

Fund Project: Work partially supported by the Italian INDAM-GNAMPA.

In this paper we study a class of degenerate second-order elliptic differential operators, often referred to as Fleming-Viot type operators, in the framework of function spaces defined on the $d$-dimensional hypercube $Q_d$ of $\mathbf{R}^d$, $d ≥1$.

By making mainly use of techniques arising from approximation theory, we show that their closures generate positive semigroups both in the space of all continuous functions and in weighted $L^{p}$-spaces.

In addition, we show that the semigroups are approximated by iterates of certain polynomial type positive linear operators, which we introduce and study in this paper and which generalize the Bernstein-Durrmeyer operators with Jacobi weights on $[0, 1]$.

As a consequence, after determining the unique invariant measure for the approximating operators and for the semigroups, we establish some of their regularity properties along with their asymptotic behaviours.

Citation: Francesco Altomare, Mirella Cappelletti Montano, Vita Leonessa. On the positive semigroups generated by Fleming-Viot type differential operators. Communications on Pure and Applied Analysis, 2019, 18 (1) : 323-340. doi: 10.3934/cpaa.2019017
References:
[1]

A. A. AlbaneseM. Campiti and E. M. Mangino, Regularity properties of semigroups generated by some Fleming-Viot type operators, J. Math. Anal. Appl., 335 (2007), 1259-1273.  doi: 10.1016/j.jmaa.2007.02.042.

[2]

A. A. Albanese and E. M. Mangino, On the sectoriality of a class of degenerate elliptic operators arising in population genetics, J. Evol. Equ., 15 (2015), 131-147.  doi: 10.1007/s00028-014-0253-3.

[3]

A. A. Albanese and E. M. Mangino, Analytic semigroups and some degenerate evolution equations defined on domains with corners, Discrete Contin. Dynam. Systems, 35 (2015), 595-615.  doi: 10.3934/dcds.2015.35.595.

[4]

F. Altomare, Korovkin-type theorems and approximation by positive linear operators, Surveys in Approximation Theory, 5 (2010), 92–164. Available from http://www.math.technion.ac.il/sat/papers/13/, ISSN 1555-578X.

[5]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, de Gruyter Studies in Mathematics, 17, Walter de Gruyter, Berlin-New York, 1994. doi: 10.1515/9783110884586.

[6]

F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Raşa, Markov Operators, Positive Semigroups and Approximation Processes, de Gruyter Studies in Mathematics, 61, Walter de Gruyter GmbH, Berlin/Boston, 2014.

[7]

F. AltomareM. Cappelletti MontanoV. Leonessa and I. Raşa, Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators, J. Math. Anal. Appl., 458 (2018), 153-173.  doi: 10.1016/j.jmaa.2017.08.034.

[8]

F. Altomare and I. Raşa, On some classes of diffusion equations and relates approximation problems, in Trends and Applications in Constructive Approximation (eds. M. G. de Bruin, H. D. Mache and J. Szabados), Internat. Ser. Numer. Math., 151, Birkhäuser Velag, Basel, (2005), 13–26. doi: 10.1007/3-7643-7356-3_2.

[9]

F. Altomare and I. Raşa, Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups, Boll. Unione Mat. Ital.(9), V (2012), 1-17. 

[10]

A. Attalienti and M. Campiti, Degenerate evolution problems and Beta-type operators, Studia Math., 140 (2000), 117-139. 

[11]

H. Bauer, Measure and Integration Theory, de Gruyter Studies in Mathematics, 26, Walter de Gruyter GmbH, Berlin/Boston, 2011. doi: 10.1515/9783110866209.

[12]

E. E. Berdysheva and K. Jetter, Multivariate Bernstein Durrmeyer operators with arbitrary weight functions, J. Approx. Theory, 162 (2010), 576-598.  doi: 10.1016/j.jat.2009.11.005.

[13]

H. Berens and Y. Xu, On Bernstein-Durrmeyer polynomials with Jacobi weights, in Approximation Theory and Functional Analysis (ed. C. K. Chui), Academic Press, Boston, (1991), 25–46.

[14]

S. Cerrai and Ph. Clément, Schauder estimates for a degenerate second order elliptic operator on a cube, J. Diff. Eq., 242 (2007), 287-321.  doi: 10.1016/j.jde.2007.08.002.

[15]

M. Campiti and I. Raşa, Qualitative properties of a class of Fleming-Viot operators, Acta Math. Hungar., 103 (2004), 55-69.  doi: 10.1023/B:AMHU.0000028236.59446.da.

[16]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, 6, Walter de Guyter, Berlin/New York, 1985. doi: 10.1515/9783110844641.

[17]

D. Mugnolo and A. Rhandi, On the domain of a Fleming-Viot-type operator on an Lp-space with invariant measure, Note Mat., 31 (2011), 139-148.  doi: 10.1285/i15900932v31n1p139.

[18]

R. Nagel (Ed.), One-parameter Semigroups of Positive Operators, Lecture Notes in Math., 1184, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[19]

R. Păltănea, Sur un opérateur polynomial défini sur l'ensemble des fonctions intégrables, Univ. Babeş-Bolyai, Cluj-Napoca, 83 (1983), 101-106. 

[20]

T. Vladislav and I. Raşa, Analiza Numerica, Aproximare, problema lui Cauchy abstracta, proiectori Altomare, Ed. Tehnica, Bucuresti, 1999.

[21]

S. Waldron, A generalised beta integral and the limit of Bernstein-Durrmeyer operator with Jacobi weights, J. Approx. Theory, 122 (2003), 141-150.  doi: 10.1016/S0021-9045(03)00041-8.

show all references

References:
[1]

A. A. AlbaneseM. Campiti and E. M. Mangino, Regularity properties of semigroups generated by some Fleming-Viot type operators, J. Math. Anal. Appl., 335 (2007), 1259-1273.  doi: 10.1016/j.jmaa.2007.02.042.

[2]

A. A. Albanese and E. M. Mangino, On the sectoriality of a class of degenerate elliptic operators arising in population genetics, J. Evol. Equ., 15 (2015), 131-147.  doi: 10.1007/s00028-014-0253-3.

[3]

A. A. Albanese and E. M. Mangino, Analytic semigroups and some degenerate evolution equations defined on domains with corners, Discrete Contin. Dynam. Systems, 35 (2015), 595-615.  doi: 10.3934/dcds.2015.35.595.

[4]

F. Altomare, Korovkin-type theorems and approximation by positive linear operators, Surveys in Approximation Theory, 5 (2010), 92–164. Available from http://www.math.technion.ac.il/sat/papers/13/, ISSN 1555-578X.

[5]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, de Gruyter Studies in Mathematics, 17, Walter de Gruyter, Berlin-New York, 1994. doi: 10.1515/9783110884586.

[6]

F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Raşa, Markov Operators, Positive Semigroups and Approximation Processes, de Gruyter Studies in Mathematics, 61, Walter de Gruyter GmbH, Berlin/Boston, 2014.

[7]

F. AltomareM. Cappelletti MontanoV. Leonessa and I. Raşa, Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators, J. Math. Anal. Appl., 458 (2018), 153-173.  doi: 10.1016/j.jmaa.2017.08.034.

[8]

F. Altomare and I. Raşa, On some classes of diffusion equations and relates approximation problems, in Trends and Applications in Constructive Approximation (eds. M. G. de Bruin, H. D. Mache and J. Szabados), Internat. Ser. Numer. Math., 151, Birkhäuser Velag, Basel, (2005), 13–26. doi: 10.1007/3-7643-7356-3_2.

[9]

F. Altomare and I. Raşa, Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups, Boll. Unione Mat. Ital.(9), V (2012), 1-17. 

[10]

A. Attalienti and M. Campiti, Degenerate evolution problems and Beta-type operators, Studia Math., 140 (2000), 117-139. 

[11]

H. Bauer, Measure and Integration Theory, de Gruyter Studies in Mathematics, 26, Walter de Gruyter GmbH, Berlin/Boston, 2011. doi: 10.1515/9783110866209.

[12]

E. E. Berdysheva and K. Jetter, Multivariate Bernstein Durrmeyer operators with arbitrary weight functions, J. Approx. Theory, 162 (2010), 576-598.  doi: 10.1016/j.jat.2009.11.005.

[13]

H. Berens and Y. Xu, On Bernstein-Durrmeyer polynomials with Jacobi weights, in Approximation Theory and Functional Analysis (ed. C. K. Chui), Academic Press, Boston, (1991), 25–46.

[14]

S. Cerrai and Ph. Clément, Schauder estimates for a degenerate second order elliptic operator on a cube, J. Diff. Eq., 242 (2007), 287-321.  doi: 10.1016/j.jde.2007.08.002.

[15]

M. Campiti and I. Raşa, Qualitative properties of a class of Fleming-Viot operators, Acta Math. Hungar., 103 (2004), 55-69.  doi: 10.1023/B:AMHU.0000028236.59446.da.

[16]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, 6, Walter de Guyter, Berlin/New York, 1985. doi: 10.1515/9783110844641.

[17]

D. Mugnolo and A. Rhandi, On the domain of a Fleming-Viot-type operator on an Lp-space with invariant measure, Note Mat., 31 (2011), 139-148.  doi: 10.1285/i15900932v31n1p139.

[18]

R. Nagel (Ed.), One-parameter Semigroups of Positive Operators, Lecture Notes in Math., 1184, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0074922.

[19]

R. Păltănea, Sur un opérateur polynomial défini sur l'ensemble des fonctions intégrables, Univ. Babeş-Bolyai, Cluj-Napoca, 83 (1983), 101-106. 

[20]

T. Vladislav and I. Raşa, Analiza Numerica, Aproximare, problema lui Cauchy abstracta, proiectori Altomare, Ed. Tehnica, Bucuresti, 1999.

[21]

S. Waldron, A generalised beta integral and the limit of Bernstein-Durrmeyer operator with Jacobi weights, J. Approx. Theory, 122 (2003), 141-150.  doi: 10.1016/S0021-9045(03)00041-8.

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