May  2020, 40(5): 2641-2669. doi: 10.3934/dcds.2020144

$\sigma$-finite invariant densities for eventually conservative Markov operators

Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, 060-0810, Japan

Received  March 2019 Revised  December 2019 Published  March 2020

We establish equivalent conditions for the existence of an integrable or locally integrable fixed point for a Markov operator with the maximal support. Maximal support means that almost all initial points will concentrate on the support of the invariant density under the iteration of the process. One of the equivalent conditions for the existence of a locally integrable fixed point is weak almost periodicity of the jump operator with respect to some sweep-out set. This result includes the case of the existence of an absolutely continuous $ \sigma $-finite invariant measure when we consider a nonsingular transformation on a probability space. Weak almost periodicity implies the Jacobs-Deleeuw-Glicksberg splitting and we show that constrictive Markov operators which guarantee the spectral decomposition are typical weakly almost periodic operators.

Citation: Hisayoshi Toyokawa. $\sigma$-finite invariant densities for eventually conservative Markov operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2641-2669. doi: 10.3934/dcds.2020144
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

J. AaronsonM. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548.  doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar

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D. W. Dean and L. Sucheston, On invariant measures for operators, Z. Wahrscheinlichkeitstheorie Verw. Geb., 6 (1966), 1-9.  doi: 10.1007/BF00531807.  Google Scholar

[4] J. Ding and A. H. Zhou, Statistical Properties of Deterministic Systems, Tsinghua University Texts, Springer-Verlag, Berlin, Tsinghua University Press, Beijing, 2009.  doi: 10.1007/978-3-540-85367-1.  Google Scholar
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N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York, Interscience Publishers, Ltd., London, 1958.  Google Scholar

[6]

E. Y. Emel'yanov, Invariant densities and mean ergodicity of Markov operators, Israel Journal of Math., 136 (2003), 373-379.  doi: 10.1007/BF02807206.  Google Scholar

[7]

E. Y. Emel'yanov, Non-Spectral Asymptotic Analysis of One-Parameter Operator Semigroups, Operator Theory: Advances and Applications, 173. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[8]

S. Eigen, A. Hajian, Y. Ito and V. Prasad, Weakly Wandering Sequences in Ergodic Theory, Springer Monographs in Mathematics, Springer, Tokyo, 2014. doi: 10.1007/978-4-431-55108-9.  Google Scholar

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S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand Mathematical Studies, No. 21. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969.  Google Scholar

[10]

A. B. Hajian and S. Kakutani, Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc., 110 (1964), 136-151.  doi: 10.1090/S0002-9947-1964-0154961-1.  Google Scholar

[11]

P. R. Halmos, Invariant measures, Ann. of Math., 48 (1947), 735-754.  doi: 10.2307/1969138.  Google Scholar

[12]

H. Y. Hu and L.-S. Young, Nonexistence of SBR measures for some diffeomorphisms that are "almost Anosov", Ergodic Theory Dynam. Systems, 15 (1995), 67-76.  doi: 10.1017/S0143385700008245.  Google Scholar

[13]

T. Inoue, Invariant measures for position dependent random maps with continuous random parameters, Studia Math., 208 (2012), 11-29.  doi: 10.4064/sm208-1-2.  Google Scholar

[14]

T. Inoue, First return maps of random map and invariant measures, Nonlinearity, 33 (2019). doi: 10.1088/1361-6544/ab4c83.  Google Scholar

[15]

T. Inoue and H. Ishitani, Asymptotic periodicity of densities and ergodic properties for nonsingular systems, Hiroshima Math. J., 21 (1991), 597-620.  doi: 10.32917/hmj/1206128723.  Google Scholar

[16]

Y. Ito, Invariant measures for Markov processes, Trans. Amer. Math. Soc., 110 (1964), 152-184.  doi: 10.1090/S0002-9947-1964-0158049-5.  Google Scholar

[17]

Y. Iwata and T. Ogihara, Random perturbations of non-singular transformation on [0, 1], Hokkaido Math. J., 42 (2013), 269-291.  doi: 10.14492/hokmj/1372859588.  Google Scholar

[18]

J. Komorník, Asymptotic periodicity of iterates of weakly constrictive Markov operators, Tohoku Math. J., 38 (1986), 15-27.  doi: 10.2748/tmj/1178228533.  Google Scholar

[19]

J. Komorník, Asymptotic decomposition of smoothing positive operators, Acta Univ. Carolina. Math. Physica, 30 (1989), 77-81.   Google Scholar

[20]

U. Krengel, Ergodic Theorems, De Gruyter Studies in Mathematics, 6. Walter de Gruyter & Co., Berlin, 1985. doi: 10.1515/9783110844641.  Google Scholar

[21]

A. LambertS. Siboni and S. Vaienti, Statistical properties of a nonuniformly hyperbolic map of the interval, J. Statist. Phys., 72 (1993), 1305-1330.  doi: 10.1007/BF01048188.  Google Scholar

[22] A. Lasota and M. C. Mackey, Probabilistic Properties of Deterministic Systems, Cambridge University Press, Cambridge, 1985.  doi: 10.1017/CBO9780511897474.  Google Scholar
[23]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Second edition, Applied Mathematical Sciences, 97. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[24]

A. LasotaT.-Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. of the Amer. Math. Soc., 286 (1984), 751-764.  doi: 10.1090/S0002-9947-1984-0760984-4.  Google Scholar

[25]

C. LiveraniB. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.  Google Scholar

[26]

M. Mori, On the intermittency of a piecewise linear map (Takahashi model), Tokyo J. Math., 16 (1993), 411-428.  doi: 10.3836/tjm/1270128495.  Google Scholar

[27]

J. Myjak and T. Szarek, On the existence of an invariant measure for Markov-Feller operators, J. Math. Anal. Appl., 294 (2004), 215-222.  doi: 10.1016/j.jmaa.2004.02.011.  Google Scholar

[28]

N. Provatas and M. C. Mackey, Asymptotic periodicity and banded chaos, Phys. D, 53 (1991), 295-318.  doi: 10.1016/0167-2789(91)90067-J.  Google Scholar

[29]

H. L. Royden and P. M. Fitzpatrick, Real Analysis, Fourth Edit, Pearson, 2010. Google Scholar

[30]

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math., 43 (1995), 245-262.   Google Scholar

[31]

R. Rudnicki, Asymptotic stability of Markov operators: A counter-example, Bull. Polish Acad. Sci. Math., 45 (1997), 1-5.   Google Scholar

[32]

F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.  Google Scholar

[33] F. Schweiger, Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, Oxford, 2000.   Google Scholar
[34]

F. H. Simons and D. A. Overdijk, Recurrent and sweep-out sets for Markov processes, Monatsh. Math., 86 (1978/79), 305-326.  doi: 10.1007/BF01300246.  Google Scholar

[35]

J. Socała, On the existence of invariant densities for Markov operators, Ann. Polon. Math., 48 (1988), 51-56.  doi: 10.4064/ap-48-1-51-56.  Google Scholar

[36]

E. Straube, On the existence of invariant, absolutely continuous measures, Comm. Math. Phys., 81 (1981), 27-30.  doi: 10.1007/BF01941798.  Google Scholar

[37]

T. Szarek, Invariant measures for Markov operators with application to function systems, Studia Math., 154 (2003), 207-222.  doi: 10.4064/sm154-3-2.  Google Scholar

[38]

T. Szarek, The uniqueness of invariant measures for Markov operators, Studia Math., 189 (2008), 225-233.  doi: 10.4064/sm189-3-2.  Google Scholar

[39]

L. Sucheston, Banach limits, Amer. Math. Monthly, 74 (1967), 308-311.  doi: 10.2307/2316038.  Google Scholar

[40]

L. Sucheston, On existence of finite invariant measures, Math. Zeitschrift, 86 (1964), 327-336.  doi: 10.1007/BF01110407.  Google Scholar

[41]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.  doi: 10.1007/BF02788928.  Google Scholar

[42]

M. Thaler, Transformations on [0, 1] with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar

[43]

T. YoshidaH. Mori and H. Shigematsu, Analytic study of chaos of the tent map: Band structures, power spectra, and critical behaviors, J. Statist. Phys., 31 (1983), 279-308.  doi: 10.1007/BF01011583.  Google Scholar

[44]

K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff's process and mean ergodic theorem, Annal. of Math., 42 (1941), 188-228.  doi: 10.2307/1968993.  Google Scholar

[45]

M. Yuri, On a Bernoulli property for multidimensional mappings with finite range structure, Tokyo J. Math., 9 (1986), 457-485.  doi: 10.3836/tjm/1270150732.  Google Scholar

[46]

M. Yuri, Invariant measures for certain multi-dimensional maps, Nonlinearity, 7 (1994), 1093-1124.  doi: 10.1088/0951-7715/7/3/018.  Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

J. AaronsonM. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337 (1993), 495-548.  doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar

[3]

D. W. Dean and L. Sucheston, On invariant measures for operators, Z. Wahrscheinlichkeitstheorie Verw. Geb., 6 (1966), 1-9.  doi: 10.1007/BF00531807.  Google Scholar

[4] J. Ding and A. H. Zhou, Statistical Properties of Deterministic Systems, Tsinghua University Texts, Springer-Verlag, Berlin, Tsinghua University Press, Beijing, 2009.  doi: 10.1007/978-3-540-85367-1.  Google Scholar
[5]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York, Interscience Publishers, Ltd., London, 1958.  Google Scholar

[6]

E. Y. Emel'yanov, Invariant densities and mean ergodicity of Markov operators, Israel Journal of Math., 136 (2003), 373-379.  doi: 10.1007/BF02807206.  Google Scholar

[7]

E. Y. Emel'yanov, Non-Spectral Asymptotic Analysis of One-Parameter Operator Semigroups, Operator Theory: Advances and Applications, 173. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[8]

S. Eigen, A. Hajian, Y. Ito and V. Prasad, Weakly Wandering Sequences in Ergodic Theory, Springer Monographs in Mathematics, Springer, Tokyo, 2014. doi: 10.1007/978-4-431-55108-9.  Google Scholar

[9]

S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand Mathematical Studies, No. 21. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969.  Google Scholar

[10]

A. B. Hajian and S. Kakutani, Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc., 110 (1964), 136-151.  doi: 10.1090/S0002-9947-1964-0154961-1.  Google Scholar

[11]

P. R. Halmos, Invariant measures, Ann. of Math., 48 (1947), 735-754.  doi: 10.2307/1969138.  Google Scholar

[12]

H. Y. Hu and L.-S. Young, Nonexistence of SBR measures for some diffeomorphisms that are "almost Anosov", Ergodic Theory Dynam. Systems, 15 (1995), 67-76.  doi: 10.1017/S0143385700008245.  Google Scholar

[13]

T. Inoue, Invariant measures for position dependent random maps with continuous random parameters, Studia Math., 208 (2012), 11-29.  doi: 10.4064/sm208-1-2.  Google Scholar

[14]

T. Inoue, First return maps of random map and invariant measures, Nonlinearity, 33 (2019). doi: 10.1088/1361-6544/ab4c83.  Google Scholar

[15]

T. Inoue and H. Ishitani, Asymptotic periodicity of densities and ergodic properties for nonsingular systems, Hiroshima Math. J., 21 (1991), 597-620.  doi: 10.32917/hmj/1206128723.  Google Scholar

[16]

Y. Ito, Invariant measures for Markov processes, Trans. Amer. Math. Soc., 110 (1964), 152-184.  doi: 10.1090/S0002-9947-1964-0158049-5.  Google Scholar

[17]

Y. Iwata and T. Ogihara, Random perturbations of non-singular transformation on [0, 1], Hokkaido Math. J., 42 (2013), 269-291.  doi: 10.14492/hokmj/1372859588.  Google Scholar

[18]

J. Komorník, Asymptotic periodicity of iterates of weakly constrictive Markov operators, Tohoku Math. J., 38 (1986), 15-27.  doi: 10.2748/tmj/1178228533.  Google Scholar

[19]

J. Komorník, Asymptotic decomposition of smoothing positive operators, Acta Univ. Carolina. Math. Physica, 30 (1989), 77-81.   Google Scholar

[20]

U. Krengel, Ergodic Theorems, De Gruyter Studies in Mathematics, 6. Walter de Gruyter & Co., Berlin, 1985. doi: 10.1515/9783110844641.  Google Scholar

[21]

A. LambertS. Siboni and S. Vaienti, Statistical properties of a nonuniformly hyperbolic map of the interval, J. Statist. Phys., 72 (1993), 1305-1330.  doi: 10.1007/BF01048188.  Google Scholar

[22] A. Lasota and M. C. Mackey, Probabilistic Properties of Deterministic Systems, Cambridge University Press, Cambridge, 1985.  doi: 10.1017/CBO9780511897474.  Google Scholar
[23]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Second edition, Applied Mathematical Sciences, 97. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[24]

A. LasotaT.-Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. of the Amer. Math. Soc., 286 (1984), 751-764.  doi: 10.1090/S0002-9947-1984-0760984-4.  Google Scholar

[25]

C. LiveraniB. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory Dynam. Systems, 19 (1999), 671-685.  doi: 10.1017/S0143385799133856.  Google Scholar

[26]

M. Mori, On the intermittency of a piecewise linear map (Takahashi model), Tokyo J. Math., 16 (1993), 411-428.  doi: 10.3836/tjm/1270128495.  Google Scholar

[27]

J. Myjak and T. Szarek, On the existence of an invariant measure for Markov-Feller operators, J. Math. Anal. Appl., 294 (2004), 215-222.  doi: 10.1016/j.jmaa.2004.02.011.  Google Scholar

[28]

N. Provatas and M. C. Mackey, Asymptotic periodicity and banded chaos, Phys. D, 53 (1991), 295-318.  doi: 10.1016/0167-2789(91)90067-J.  Google Scholar

[29]

H. L. Royden and P. M. Fitzpatrick, Real Analysis, Fourth Edit, Pearson, 2010. Google Scholar

[30]

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math., 43 (1995), 245-262.   Google Scholar

[31]

R. Rudnicki, Asymptotic stability of Markov operators: A counter-example, Bull. Polish Acad. Sci. Math., 45 (1997), 1-5.   Google Scholar

[32]

F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.  Google Scholar

[33] F. Schweiger, Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, Oxford, 2000.   Google Scholar
[34]

F. H. Simons and D. A. Overdijk, Recurrent and sweep-out sets for Markov processes, Monatsh. Math., 86 (1978/79), 305-326.  doi: 10.1007/BF01300246.  Google Scholar

[35]

J. Socała, On the existence of invariant densities for Markov operators, Ann. Polon. Math., 48 (1988), 51-56.  doi: 10.4064/ap-48-1-51-56.  Google Scholar

[36]

E. Straube, On the existence of invariant, absolutely continuous measures, Comm. Math. Phys., 81 (1981), 27-30.  doi: 10.1007/BF01941798.  Google Scholar

[37]

T. Szarek, Invariant measures for Markov operators with application to function systems, Studia Math., 154 (2003), 207-222.  doi: 10.4064/sm154-3-2.  Google Scholar

[38]

T. Szarek, The uniqueness of invariant measures for Markov operators, Studia Math., 189 (2008), 225-233.  doi: 10.4064/sm189-3-2.  Google Scholar

[39]

L. Sucheston, Banach limits, Amer. Math. Monthly, 74 (1967), 308-311.  doi: 10.2307/2316038.  Google Scholar

[40]

L. Sucheston, On existence of finite invariant measures, Math. Zeitschrift, 86 (1964), 327-336.  doi: 10.1007/BF01110407.  Google Scholar

[41]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., 37 (1980), 303-314.  doi: 10.1007/BF02788928.  Google Scholar

[42]

M. Thaler, Transformations on [0, 1] with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar

[43]

T. YoshidaH. Mori and H. Shigematsu, Analytic study of chaos of the tent map: Band structures, power spectra, and critical behaviors, J. Statist. Phys., 31 (1983), 279-308.  doi: 10.1007/BF01011583.  Google Scholar

[44]

K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff's process and mean ergodic theorem, Annal. of Math., 42 (1941), 188-228.  doi: 10.2307/1968993.  Google Scholar

[45]

M. Yuri, On a Bernoulli property for multidimensional mappings with finite range structure, Tokyo J. Math., 9 (1986), 457-485.  doi: 10.3836/tjm/1270150732.  Google Scholar

[46]

M. Yuri, Invariant measures for certain multi-dimensional maps, Nonlinearity, 7 (1994), 1093-1124.  doi: 10.1088/0951-7715/7/3/018.  Google Scholar

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