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 and Continuous Dynamical Systems, 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.

[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.

[3]

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

[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.
[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[18]

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

[19]

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

[20]

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

[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.

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

[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.

[25]

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

[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.

[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.

[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.

[29]

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

[30]

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

[31]

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

[32]

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

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

[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.

[36]

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

[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.

[38]

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

[39]

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

[40]

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

[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.

[42]

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

[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.

[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.

[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.

[46]

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

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.

[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.

[3]

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

[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.
[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[18]

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

[19]

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

[20]

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

[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.

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

[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.

[25]

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

[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.

[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.

[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.

[29]

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

[30]

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

[31]

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

[32]

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

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

[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.

[36]

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

[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.

[38]

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

[39]

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

[40]

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

[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.

[42]

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

[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.

[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.

[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.

[46]

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

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