No Access

Typical long-time behavior of ground state-transformed jump processes

Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

E-mail Address: kamil.kaleta@pwr.edu.pl

Search for more papers by this author

and

József Lőrinczi

Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK

E-mail Address: J.Lorinczi@lboro.ac.uk

Search for more papers by this author

https://doi.org/10.1142/S0219199719500020Cited by:2 (Source: Crossref)

Abstract

We consider a class of Lévy-type processes derived via a Doob transform from Lévy processes conditioned by a control function called potential. These ground state transformed processes (also called $P {(ϕ)}_{1}$ -processes) have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise upper envelopes for the almost sure long-time behavior of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.

Keywords:

AMSC: 47D08, 60G51, 47D03, 47G20

References

1. J. A. D. Appleby, X. Mao and H. Z. Wu , On the almost sure running maxima of solutions of affine stochastic functional differential equations, SIAM J. Math. Anal. 42 (2010) 646–678. Crossref, Web of Science, Google Scholar
2. J. A. D. Appleby and H. Wu , Solutions of stochastic differential equations obeying the law of the iterated logarithm, with applications to financial markets, Electron. J. Probab. 14 (2009) 912–959. Crossref, Web of Science, Google Scholar
3. F. Avram, N. N. Leonenko and N. Šuvak , On spectral analysis of heavy-tailed Kolmogorov–Pearson diffusions, Markov Proc. Relat. Fields 19 (2013) 249–298. Web of Science, Google Scholar
4. J. Bertoin , Lévy Processes (Cambridge University Press, 1996). Google Scholar
5. V. Betz and J. Lőrinczi , Uniqueness of Gibbs measures relative to Brownian motion, Ann. Inst. H. Poincaré 39 (2003) 877–889. Crossref, Web of Science, Google Scholar
6. L. Breiman , A delicate law of the iterated logarithm for non-decreasing stable processes, Ann. Math. Stat. 39 (1968) 1814–1824; Erratum 41 (1970) 1126. Crossref, Google Scholar
7. R. Carmona, W. C. Masters and B. Simon , Relativistic Schrödinger operators: Asymptotic behaviour of the eigenfunctions, J. Funct. Anal. 91 (1990) 117–142. Crossref, Web of Science, Google Scholar
8. M. Csörgő and P. Révész , Strong Approximations in Probability and Statistics (Academic Press, 1981). Google Scholar
9. E. B. Davies , Heat Kernels and Spectral Theory (Cambridge University Press, 1989). Crossref, Google Scholar
10. S. O. Durugo and J. Lőrinczi , Spectral properties of the massless relativistic quartic oscillator, J. Differential Equations 264 (2018) 3775–3809. Crossref, Web of Science, Google Scholar
11. A. Dvoretzky and P. Erdős , Some problems on random walk in space, in Proc. 2nd Berkeley Symp. on Mathematical Statistics and Probability (University of California Press, 1951), pp. 353–367. Google Scholar
12. D. Freedman , Brownian Motion and Diffusion, 2nd edition (Springer, 1982). Google Scholar
13. B. Fristedt , Sample functions of stochastic processes with stationary independent increments, in Advances in Probability, Vol. 3, eds. P. Ney and S. Port (Marcel Dekker, 1974), pp. 241–396. Google Scholar
14. S. Gheryani, F. Hiroshima, J. Lorinczi, A. Majid and H. Ouerdiane, Functional central limit theorems and $P {(ϕ)}_{1}$ -processes for the classical and relativistic Nelson models, preprint (2016); arXiv:1605.01791 (under review). Google Scholar
15. S. Gheryani, F. Hiroshima, J. Lőrinczi, A. Majid and H. Ouerdiane , $P {(ϕ)}_{1}$ -process for the spin-boson model and a functional central limit theorem for associated additive functionals, Stochastics 89 (2017) 1104–1115. Crossref, Google Scholar
16. B. V. Gnedenko , O roste odnorodnykh sluchainykh protsessov s nezavisimymi prirashcheniyami (Sur la croissance des processus stochastiques homogènes à accroissements indépendants; in Russian with a summary in French), Izv. Akad. Nauk SSSR Ser. Mat. 7 (1943) 89–110. Google Scholar
17. K. Kaleta and J. Lőrinczi , Fractional $P {(ϕ)}_{1}$ -processes and Gibbs measures, Stoch. Proc. Appl. 122 (2012) 3580–3617. Crossref, Web of Science, Google Scholar
18. K. Kaleta and J. Lőrinczi , Pointwise estimates of the eigenfunctions and intrinsic ultracontractivity-type properties of Feynman–Kac semigroups for a class of Lévy processes, Ann. Probab. 43 (2015) 1350–1398. Crossref, Web of Science, Google Scholar
19. K. Kaleta and J. Lőrinczi , Fall-off of eigenfunctions for non-local Schrödinger operators with decaying potentials, Potential Anal. 46 (2017) 647–688. Crossref, Web of Science, Google Scholar
20. K. Kaleta and R. L. Schilling, Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators, preprint (2018). Google Scholar
21. K. Kaleta and P. Sztonyk , Small time sharp bounds for kernels of convolution semigroups, J. Anal. Math. 132 (2017) 355–394. Crossref, Web of Science, Google Scholar
22. S. Keprta, Integral tests for Brownian motions and some related processes, PhD thesis, Carleton University, Ottawa (1997). Google Scholar
23. A. Ya. Khintchine , Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Kap. 5 (Springer, 1933). Crossref, Google Scholar
24. A. Ya. Khintchine , Dvo teoremy o stokhasticheskikh protssesakh s odnotipnymi prirashchenyami (Zwei Sätze über stochastische Prozesse mit stabilen Verteilungen; in Russian with a summary in German), Mat. Sb. 3 (1938) 577–584. Google Scholar
25. F. Kühn , Lévy-Type Processes: Moments, Construction and Heat Kernel Estimates, Lévy Matters, Vol. VI, Lecture Notes in Mathematics, Vol. 2187 (Springer, 2017). Crossref, Google Scholar
26. J. Lőrinczi, F. Hiroshima and V. Betz , Feynman–Kac-Type Theorems and Gibbs Measures on Path Space: With Applications to Rigorous Quantum Field Theory, 2nd edition: 2019, de Gruyter Studies in Mathematics, Vol. 34 (Walter de Gruyter, 2011). Crossref, Google Scholar
27. J. Lőrinczi and J. Małecki , Spectral properties of the massless relativistic harmonic oscillator, J. Differential Equations 253 (2012) 2846–2871. Crossref, Web of Science, Google Scholar
28. J. Lőrinczi and X. Yang , Multifractal properties of sample paths of ground state-transformed jump processes, Chaos Solitons Fractals 120 (2019) 83–94. Crossref, Web of Science, Google Scholar
29. X. Mao , Stochastic Differential Equations and Applications, 2nd edition (Horwood Publishing, 2008). Crossref, Google Scholar
30. M. Motoo , Proof of the law of iterated logarithm through diffusion equation, Ann. Inst. Statist. Math. 10 (1958) 21–28. Crossref, Web of Science, Google Scholar
31. I. Petrowsky , Zur ersten Randwertaufgabe der Wärmeleitungsgleichung, Comp. Math. 1 (1935) 383–419. Google Scholar
32. W. E. Pruitt and S. J. Taylor , Sample path properties of processes with stable components, Z. Wahrsch. verw. Gebiete 12 (1969) 267–289. Crossref, Google Scholar
33. M. Reed and B. Simon , Methods of Modern Mathematical Physics, 1. Functional Analysis (Academic Press, 1980). Google Scholar
34. P. Révész , Random Walk in Random and Non-Random Environments (World Scientific, 1990). Link, Google Scholar
35. J. Rosen and B. Simon , Fluctuations in $P {(ϕ)}_{1}$ -processes, Ann. Probab. 4 (1976) 155–174. Crossref, Web of Science, Google Scholar
36. N. Sandrić , Long-time behavior for a class of Feller processes, Trans. Amer. Math. Soc. 368 (2016) 1871–1910. Crossref, Web of Science, Google Scholar
37. K. Sato , Lévy Processes and Infinitely Divisible Distributions (Cambridge University Press, 1999). Google Scholar
38. R. Schilling , Growth and Hölder conditions for the sample paths of Feller processes, Probab. Theory Relat. Fields 112 (1998) 565–611. Crossref, Web of Science, Google Scholar
39. E. Seneta , Regularly Varying Functions, Lecture Notes in Mathematics, Vol. 508 (Springer, 1976). Crossref, Google Scholar
40. Y. Shiozawa and J. Wang , Rate functions for symmetric Markov processes via heat kernel, Potential Anal. 46 (2017) 23–53. Crossref, Web of Science, Google Scholar
41. B. Simon , Functional Integration and Quantum Physics, 2nd edition (AMS Chelsea Publishing, 2004). Crossref, Google Scholar
42. T. Sirao , On some asymptotic properties concerning homogenous differential processes, Nagoya Math. J. 6 (1953) 95–107. Crossref, Google Scholar
43. T. Sirao and T. Nisida , On some asymptotic properties concerning Brownian motion, Nagoya Math. J. 4 (1952) 97–101. Crossref, Google Scholar
44. V. M. Zolotarev , Analog of the iterated logarithm law for semi-continuous stable processes, Theory Probab. Appl. 9 (1964) 512–513. Google Scholar