On recurrence and transience of some Lévy-type processes in $\mathbb {R}$
Author:
Victoria Knopova
Journal:
Theor. Probability and Math. Statist. 108 (2023), 59-75
MSC (2020):
Primary 60G17; Secondary 60J25, 60G53
DOI:
https://doi.org/10.1090/tpms/1187
Published electronically:
May 2, 2023
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in $\mathbb {R}$, whose generator defined on the test functions is of the form \begin{equation*} Lf(x) =\int _{\mathbb {R}} \left ( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb {1}_{|u|\leq 1} \right ) \nu (x,du), \quad f\in C_\infty ^2(\mathbb {R}). \end{equation*} Here $\nu (x,du)$ is a Lévy-type kernel, whose tails are either extended regularly varying or decaying fast enough. For the proof the Foster–Lyapunov approach is used.
References
- Dominique Bakry, Patrick Cattiaux, and Arnaud Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (2008), no. 3, 727–759. MR 2381160, DOI 10.1016/j.jfa.2007.11.002
- Krzysztof Bogdan, PawełSztonyk, and Victoria Knopova, Heat kernel of anisotropic nonlocal operators, Doc. Math. 25 (2020), 1–54. MR 4077549, DOI 10.4171/dm/736
- Björn Böttcher, An overshoot approach to recurrence and transience of Markov processes, Stochastic Process. Appl. 121 (2011), no. 9, 1962–1981. MR 2819236, DOI 10.1016/j.spa.2011.05.010
- Björn Böttcher, René Schilling, and Jian Wang, Lévy matters. III, Lecture Notes in Mathematics, vol. 2099, Springer, Cham, 2013. Lévy-type processes: construction, approximation and sample path properties; With a short biography of Paul Lévy by Jean Jacod; Lévy Matters. MR 3156646, DOI 10.1007/978-3-319-02684-8
- Krzysztof Bogdan, Krzysztof Burdzy, and Zhen-Qing Chen, Censored stable processes, Probab. Theory Related Fields 127 (2003), no. 1, 89–152. MR 2006232, DOI 10.1007/s00440-003-0275-1
- Krzysztof Bogdan and PawełSztonyk, Harnack’s inequality for stable Lévy processes, Potential Anal. 22 (2005), no. 2, 133–150. MR 2137058, DOI 10.1007/s11118-004-0590-x
- Krzysztof Bogdan and PawełSztonyk, Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian, Studia Math. 181 (2007), no. 2, 101–123. MR 2320691, DOI 10.4064/sm181-2-1
- R. K. Getoor, Transience and recurrence of Markov processes, Seminar on Probability, XIV (Paris, 1978/1979) Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 397–409. MR 580144
- B. Grigelionis, The Markov property of random processes, Litovsk. Mat. Sb. 8 (1968), 489–502 (Russian). MR 0251810
- Randal Douc, Eric Moulines, Pierre Priouret, and Philippe Soulier, Markov chains, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2018. MR 3889011, DOI 10.1007/978-3-319-97704-1
- Randal Douc, Gersende Fort, Eric Moulines, and Philippe Soulier, Practical drift conditions for subgeometric rates of convergence, Ann. Appl. Probab. 14 (2004), no. 3, 1353–1377. MR 2071426, DOI 10.1214/105051604000000323
- Randal Douc, Gersende Fort, and Arnaud Guillin, Subgeometric rates of convergence of $f$-ergodic strong Markov processes, Stochastic Process. Appl. 119 (2009), no. 3, 897–923. MR 2499863, DOI 10.1016/j.spa.2008.03.007
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085, DOI 10.1002/9780470316658
- G. Fort and G. O. Roberts, Subgeometric ergodicity of strong Markov processes, Ann. Appl. Probab. 15 (2005), no. 2, 1565–1589. MR 2134115, DOI 10.1214/105051605000000115
- W. Hoh. Pseudo differential operators generating Markov processes. Habilitationsschrift, Universität Bielefeld 1998. https://www.math.uni-bielefeld.de/~hoh/temp/pdo_mp.pdf
- Walter Hoh, Pseudo differential operators with negative definite symbols of variable order, Rev. Mat. Iberoamericana 16 (2000), no. 2, 219–241. MR 1809340, DOI 10.4171/RMI/274
- Niels Jacob, Feller semigroups, Dirichlet forms, and pseudodifferential operators, Forum Math. 4 (1992), no. 5, 433–446. MR 1176881, DOI 10.1515/form.1992.4.433
- Niels Jacob, A class of Feller semigroups generated by pseudo-differential operators, Math. Z. 215 (1994), no. 1, 151–166. MR 1254818, DOI 10.1007/BF02571704
- Niels Jacob and Hans-Gerd Leopold, Pseudo-differential operators with variable order of differentiation generating Feller semigroups, Integral Equations Operator Theory 17 (1993), no. 4, 544–553. MR 1243995, DOI 10.1007/BF01200393
- N. Jacob, Pseudo differential operators and Markov processes. Vol. I, Imperial College Press, London, 2001. Fourier analysis and semigroups. MR 1873235, DOI 10.1142/9781860949746
- N. Jacob, Pseudo differential operators & Markov processes. Vol. II, Imperial College Press, London, 2002. Generators and their potential theory. MR 1917230, DOI 10.1142/9781860949562
- Victoria Knopova, Alexei Kulik, and René L. Schilling, Construction and heat kernel estimates of general stable-like Markov processes, Dissertationes Math. 569 (2021), 86. MR 4361582, DOI 10.4064/dm824-8-2021
- Koji Kikuchi and Akira Negoro, On Markov process generated by pseudodifferential operator of variable order, Osaka J. Math. 34 (1997), no. 2, 319–335. MR 1483853
- Victoria Knopova and Alexei Kulik, Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 1, 100–140 (English, with English and French summaries). MR 3765882, DOI 10.1214/16-AIHP796
- Victoria Knopova and Aleksei Kulik, Intrinsic compound kernel estimates for the transition probability density of Lévy-type processes and their applications, Probab. Math. Statist. 37 (2017), no. 1, 53–100. MR 3652202, DOI 10.19195/0208-4147.37.1.3
- Victoria Knopova and René L. Schilling, On level and collision sets of some Feller processes, ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), no. 2, 1001–1029. MR 3457549, DOI 10.4171/owr/2015/19
- Takashi Komatsu, Markov processes associated with certain integro-differential operators, Osaka Math. J. 10 (1973), 271–303. MR 359017
- Takashi Komatsu, On the martingale problem for generators of stable processes with perturbations, Osaka J. Math. 21 (1984), no. 1, 113–132. MR 736974
- Alexei Kulik, Ergodic behavior of Markov processes, De Gruyter Studies in Mathematics, vol. 67, De Gruyter, Berlin, 2018. With applications to limit theorems. MR 3791835
- Sean P. Meyn and R. L. Tweedie, Stability of Markovian processes. II. Continuous-time processes and sampled chains, Adv. in Appl. Probab. 25 (1993), no. 3, 487–517. MR 1234294, DOI 10.2307/1427521
- Sean P. Meyn and R. L. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab. 25 (1993), no. 3, 518–548. MR 1234295, DOI 10.2307/1427522
- S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1993. MR 1287609, DOI 10.1007/978-1-4471-3267-7
- S.P. Meyn, R.L. Tweedie. A survey of Foster–Lyapunov conditions for general state-space Markov processes, In: Proc. Workshop Stoch. Stability Stoch. Stabilization (Metz, France, June 1993), Springer, Berlin, 1993.
- R. Mikulevičius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Liet. Mat. Rink. 32 (1992), no. 2, 299–331 (English, with Lithuanian summary); English transl., Lithuanian Math. J. 32 (1992), no. 2, 238–264 (1993). MR 1246036, DOI 10.1007/BF02450422
- R. Mikulevičius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Liet. Mat. Rink. 32 (1992), no. 2, 299–331 (English, with Lithuanian summary); English transl., Lithuanian Math. J. 32 (1992), no. 2, 238–264 (1993). MR 1246036, DOI 10.1007/BF02450422
- Akira Negoro, Stable-like processes: construction of the transition density and the behavior of sample paths near $t=0$, Osaka J. Math. 31 (1994), no. 1, 189–214. MR 1262797
- Y\B{o}ichi Ōshima, On conservativeness and recurrence criteria for Markov processes, Potential Anal. 1 (1992), no. 2, 115–131. MR 1245880, DOI 10.1007/BF01789234
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- Nikola Sandrić, Long-time behavior of stable-like processes, Stochastic Process. Appl. 123 (2013), no. 4, 1276–1300. MR 3016223, DOI 10.1016/j.spa.2012.12.004
- Nikola Sandrić, Ergodicity of Lévy-type processes, ESAIM Probab. Stat. 20 (2016), 154–177. MR 3528622, DOI 10.1051/ps/2016009
- Nikola Sandrić, On transience of Lévy-type processes, Stochastics 88 (2016), no. 7, 1012–1040. MR 3529858, DOI 10.1080/17442508.2016.1178749
- Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 2013. Translated from the 1990 Japanese original; Revised edition of the 1999 English translation. MR 3185174
- René L. Schilling, Conservativeness and extensions of Feller semigroups, Positivity 2 (1998), no. 3, 239–256. MR 1653474, DOI 10.1023/A:1009748105208
- René L. Schilling and Jian Wang, Some theorems on Feller processes: transience, local times and ultracontractivity, Trans. Amer. Math. Soc. 365 (2013), no. 6, 3255–3286. MR 3034465, DOI 10.1090/S0002-9947-2012-05738-2
- O. Stramer and R. L. Tweedie, Stability and instability of continuous-time Markov processes, Probability, statistics and optimisation, Wiley Ser. Probab. Math. Statist. Probab. Math. Statist., Wiley, Chichester, 1994, pp. 173–184. MR 1320750
- Matsuyo Tomisaki, Comparison theorems on Dirichlet norms and their applications, Forum Math. 2 (1990), no. 3, 277–295. MR 1050410, DOI 10.1515/form.1990.2.277
- R. L. Tweedie, Topological conditions enabling use of Harris methods in discrete and continuous time, Acta Appl. Math. 34 (1994), no. 1-2, 175–188. MR 1273853, DOI 10.1007/BF00994264
- Jian Wang, Exponential ergodicity and strong ergodicity for SDEs driven by symmetric $\alpha$-stable processes, Appl. Math. Lett. 26 (2013), no. 6, 654–658. MR 3028071, DOI 10.1016/j.aml.2013.01.004
- Jian Wang, Criteria for ergodicity of Lévy type operators in dimension one, Stochastic Process. Appl. 118 (2008), no. 10, 1909–1928. MR 2454470, DOI 10.1016/j.spa.2007.11.003
References
- D. Barky, P. Cattiaux, A. Guillin. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (2008), no. 3, 727–759. MR 2381160
- K. Bogdan, V. Knopova, P. Sztonyk. Heat kernel of anisotropic nonlocal operators, Doc. Math. 25 (2020), 1–54. MR 4077549
- B. Böttcher. An overshoot approach to recurrence and transience of Markov processes, Stochastic Process. Appl. 121 (2011), no. 9, 1962–1981. MR 2819236
- B. Böttcher, R. Schilling, J. Wang. Lévy-Type processes: Construction, approximation and sample path properties, Lecture Notes in Mathematics vol. 2099 (Lévy Matters III), Springer, Cham, 2013. MR 3156646
- K. Bogdan, K. Burdzy, and Z.Q. Chen. Censored stable processes, Probab. Theory Related Fields 127 (2003), no. 1, 89–152. MR 2006232
- K. Bogdan, P. Sztonyk. Harnack’s inequality for stable Lévy processes, Potential Anal. 22 (2005), no. 2, 133–150. MR 2137058
- K. Bogdan, P. Sztonyk. Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian, Studia Math. 181 (2007), no. 2, 101–123. MR 2320691
- R. Getoor. Transience and recurrence of Markov processes, Seminar on Probability, XIV (Paris, 1978/1979) (French), 397–409, Lecture Notes in Math., 784, Springer, Berlin. MR 580144
- B. Grigelionis. The Markov property of random processes (Russian), Litovsk. Mat. Sb. 8 (1968), 489–502. MR 0251810
- R. Douc, E. Moulines, P. Priouret, P. Soulier. Markov chains, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2018. MR 3889011
- R. Douc, G. Fort, E. Moulines, P. Soulier. Practical drift conditions for subgeometric rates of convergence, Ann. Appl. Probab. 14 (2004), no. 3, 1353–1377. MR 2071426
- R. Douc, G. Fort, A. Guillin. Subgeometric rates of convergence of $f$-ergodic strong Markov processes, Stochastic Process. Appl. 119 (2009), no. 3, 897–923. MR 2499863
- S.N. Ethier, T.G. Kurtz. Markov Processes, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 838085
- G. Fort and G.O. Roberts. Subgeometric ergodicity of strong Markov processes, Ann. Appl. Probab. 15 (2005), no. 2, 1565–1589. MR 2134115
- W. Hoh. Pseudo differential operators generating Markov processes. Habilitationsschrift, Universität Bielefeld 1998. https://www.math.uni-bielefeld.de/~hoh/temp/pdo_mp.pdf
- W. Hoh. Pseudo-differential operators with negative defnite symbols of variable order, Rev. Mat. Iberoamericana 16 (2000), no. 2, 219–241. MR 1809340
- N. Jacob. Feller semigroups, Dirichlet forms, and pseudo differential operators, Forum Math. 4 (1992), no. 5, 433–446. MR 1176881
- N. Jacob. A class of Feller semigroups generated by pseudo-differential operators, Math. Z. 215 (1994), no. 1, 151–166. MR 1254818
- N. Jacob, H.-G. Leopold. Pseudo-differential operators with variable order of differentiation generating Feller semigroups, Integr. Equ. Oper. Theory 17 (1993), no. 4, 544–553. MR 1243995
- N. Jacob. Pseudo differential operators and Markov processes, Vol. I: Fourier Analysis and Semigroups, Imperial College Press, London, 2001. MR 1873235
- N. Jacob. Pseudo differential operators and Markov processes, II: Generators and their potential theory, Imperial College Press, London, 2002. MR 1917230
- V. Knopova, A. Kulik. R. Schilling. Construction and heat kernel estimates of general stable-like Markov processes, Dissertationes Math. 569 (2021), 86 pp. MR 4361582
- K. Kikuchi, A. Negoro. On Markov process generated by pseudo-differential operator of variable order, Osaka J. Math. 34 (1997), no. 2, 319–335. MR 1483853
- V. Knopova, A. Kulik. Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 1, 100–140. MR 3765882
- V. Knopova, A. Kulik. Intrinsic compound kernel estimates for the transition probability density of Lévy-type processes and their applications, Probab. Math. Statist. 37 (2017), no. 1, 53–100. MR 3652202
- V. Knopova, R. L. Schilling. On level and collision sets of some Feller processes, ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), no. 2, 1001–1029. MR 3457549
- T. Komatsu. Markov processes associated with certain integro-differentrial operators, Osaka Math. J. 10 (1973), 271–303. MR 359017
- T. Komatsu. On the martingale problem for generators of stable processes with perturbations, Osaka J. Math. 21 (1984), no. 1, 113–132. MR 736974
- A. Kulik. Ergodic behaviour of Markov processes, De Gruyter Studies in Mathematics, 67, De Gruyter, Berlin, 2018. MR 3791835
- S.P. Meyn, R.L. Tweedie. Stability of Markovian processes II: Continuous-time processes and sampled chains, Adv. in Appl. Probab. 25 (1993), no. 3, 487–517. MR 1234294
- S.P. Meyn, R.L. Tweedie. Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab. 25 (1993), no. 3, 518–548. MR 1234295
- S.P. Meyn, R.L. Tweedie. Markov chains and stochastic stability, Communications and Control Engineering Series, Springer-Verlag London, Ltd., London, 1993. MR 1287609
- S.P. Meyn, R.L. Tweedie. A survey of Foster–Lyapunov conditions for general state-space Markov processes, In: Proc. Workshop Stoch. Stability Stoch. Stabilization (Metz, France, June 1993), Springer, Berlin, 1993.
- R. Mikulevicius, H. Pragarauskas. On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder Spaces, Lithuanian Math. J. 32 (1992), no. 2, 238–264. MR 1246036
- R. Mikulevicius, H. Pragarauskas. On the martingale problem associated with nondegenerate Lévy operators, Lithuanian Math. J. 32 (1992), 297–311. MR 1246036
- A. Negoro. Stable-like processes: construction of the transition density and the behavior of sample paths near $t=0$, Osaka J. Math. 31 (1994), no. 1, 189–214. MR 1262797
- Y. Oshima. On conservativeness and recurrence criteria for Markov processes, Potential Anal. 1 (1992), no. 2, 115–131. MR 1245880
- D. Revuz, M. L. Yor. Continuous martingales and Brownian motion, third edition, Grundlehren der mathematischen Wissenschaften, vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357
- N. Sandrić. Long-time behavior of stable-like processes, Stochastic Process. Appl. 123 (2013), no. 4, 1276–1300. MR 3016223
- N. Sandrić. Ergodicity of Lévy-type processes, ESAIM Probab. Stat. 20 (2016), 154–177. MR 3528622
- N. Sandrić. On transience of Lévy-type processes, Stochastics 88 (2016), no. 7, 1012–1040. MR 3529858
- K. Sato. Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 2013. MR 3185174
- R. L. Schilling. Conservativeness and extensions of Feller semigroups, Positivity 2 (1998), no. 3, 239–256. MR 1653474
- R. L. Schilling, J. Wang. Some theorems on Feller processes: transience, local times and ultracontractivity, Trans. Amer. Math. Soc. 365 (2013), no. 6, 3255–3286. MR 3034465
- O. Stramer, R.L. Tweedie. Stability and instability of continuous-time Markov processes. In: Probability, statistics and optimisation: a Tribute to Peter Whittle, Wiley Ser. Probab. Math. Statist., Wiley, Chichester, 173–184,. MR 1320750
- M. Tomisaki. Comparison theorems on Dirichlet norms and their applications, Forum Math. 2 (1990), no. 3, 277–295. MR 1050410
- R. L. Tweedie. Topological conditions enabling use of Harris methods in discrete and continuous time, Acta Appl. Math. 34 (1994), no. 1–2, 175–188. MR 1273853
- J. Wang. Exponential ergodicity and strong ergodicity for SDEs driven by symmetric $\alpha$-stable processes, Appl. Math. Lett. 26 (2013), no. 6, 654–658. MR 3028071
- J. Wang. Criteria for ergodicity of Lévy type operators in dimension one, Stochastic Process. Appl. 118 (2008), no. 10, 1909–1928. MR 2454470
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2020):
60G17,
60J25,
60G53
Retrieve articles in all journals
with MSC (2020):
60G17,
60J25,
60G53
Additional Information
Victoria Knopova
Affiliation:
Taras Shevchenko National University of Kyiv, Faculty of Mechanics and Mathematics, Hlushkova Avenue, 4e, 02127, Kyiv, Ukraine
Email:
vicknopova@knu.ua
Keywords:
Recurrence,
transience,
Lévy-type process,
Foster–Lyapunov criteria,
Lyapunov function
Received by editor(s):
August 4, 2021
Accepted for publication:
March 13, 2022
Published electronically:
May 2, 2023
Additional Notes:
The Grant “PK-0122U001843 Time-inhomogeneous or time-nondeterministic stochastic dynamic systems: asymptotic behaviour and statistic analysis” is gratefully acknowledged.
Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv
