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

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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.

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*Censored stable processes*, Probab. Theory Related Fields **127** (2003), no. 1, 89–152. MR **2006232**
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*Harnack’s inequality for stable Lévy processes*, Potential Anal. **22** (2005), no. 2, 133–150. MR **2137058**
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*Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian*, Studia Math. **181** (2007), no. 2, 101–123. MR **2320691**
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*Subgeometric rates of convergence of $f$-ergodic strong Markov processes*, Stochastic Process. Appl. **119** (2009), no. 3, 897–923. MR **2499863**
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*Subgeometric ergodicity of strong Markov processes*, Ann. Appl. Probab. **15** (2005), no. 2, 1565–1589. MR **2134115**
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*Pseudo-differential operators with negative defnite symbols of variable order*, Rev. Mat. Iberoamericana **16** (2000), no. 2, 219–241. MR **1809340**
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*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**
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*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**

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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