Electronic Journal of Probability

Invariance principle for non-homogeneous random walks

Nicholas Georgiou, Aleksandar Mijatović, and Andrew R. Wade

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Abstract

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in ${\mathbb R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X} $ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq 2$. To characterize $\mathcal{X} $, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in ${\mathbb R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X} $ and thus develop the excursion theory of $\mathcal{X} $ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X} $ in ${\mathbb R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X} $ is time-reversible. If so, the excursions of $\mathcal{X} $ in ${\mathbb R}^d$ generalize the classical Pitman–Yor splitting-at-the-maximum property of Bessel excursions.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 48, 38 pp.

Dates
Received: 23 April 2018
Accepted: 27 March 2019
First available in Project Euclid: 18 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1558145016

Digital Object Identifier
doi:10.1214/19-EJP302

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces 60J60: Diffusion processes [See also 58J65]
Secondary: 60F17: Functional limit theorems; invariance principles 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J55: Local time and additive functionals

Keywords
non-homogeneous random walk invariance principle diffusion limits excursions skew product rapid spinning recurrence transience

Rights
Creative Commons Attribution 4.0 International License.

Citation

Georgiou, Nicholas; Mijatović, Aleksandar; Wade, Andrew R. Invariance principle for non-homogeneous random walks. Electron. J. Probab. 24 (2019), paper no. 48, 38 pp. doi:10.1214/19-EJP302. https://projecteuclid.org/euclid.ejp/1558145016


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References

  • [1] Alili, L., Chaumont, L., Graczyk, P. and Żak, T.: Inversion, duality and Doob $h$-transforms for self-similar Markov processes. Electron. J. Probab. 22 (2017), paper no. 20, 18 pp.
  • [2] Barlow, M., Pitman, J. and Yor, M.: On Walsh’s Brownian motions. Séminaire de Probabilités, XXIII, pp. 275–293, Lecture Notes in Math. 1372, Springer-Verlag, Berlin, 1989.
  • [3] Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge, 1996.
  • [4] Billingsley, P.: Convergence of Probability Measures. 2nd ed., John Wiley & Sons, Inc., New York, 1999.
  • [5] Cherny, A.S.: Convergence of some integrals associated with Bessel processes. Theory Probab. Appl. 45 (2001) 195–209. Translated from Teor. Veroyatnost. i Primenen 45 (2000) 251–267 (Russian).
  • [6] Dugundji, J.: Topology. Allyn and Becon, Inc., Boston, 1966.
  • [7] Embrechts, P. and Hofert, M.: A note on generalized inverses. Math. Meth. Oper. Res. 77 (2013) 423–432.
  • [8] Ethier, S.N. and Kurtz, T.G.: Markov Processes. Characterization and Convergence. John Wiley & Sons, Inc., New York, 1986.
  • [9] Georgiou, N., Menshikov, M.V., Mijatović, A. and Wade, A.R.: Anomalous recurrence properties of many-dimensional zero-drift random walks. Adv. in Appl. Probab. 48A (2016) 99–118.
  • [10] Georgiou, N., Mijatović, A. and Wade, A.R.: A radial invariance principle for non-homogeneous random walks. Electron. Commun. Probab. 23 (2018) paper no. 56, 11 pp.
  • [11] Hsu, E.P.: Stochastic Analysis on Manifolds. American Mathematical Society, Providence, 2002.
  • [12] Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. 2nd ed., North-Holland Publishing Company, Amsterdam, 1989.
  • [13] Itô, K. and McKean Jr., H.P.: Diffusion Processes and Their Sample Paths. 2nd corr. printing, Springer-Verlag, Berlin, 1974.
  • [14] Jost, J.: Riemannian Geometry and Geometric Analysis. 4th ed., Springer-Verlag, Berlin, 2005.
  • [15] Kent, J.: Time-reversible diffusions. Adv. in Appl. Probab. 10 (1978) 819–835.
  • [16] Kent, J.: Eigenvalue expansion for diffusion hitting times. Z. Wahr. ver. Gebiete 52 (1980) 309–319.
  • [17] Kingman, J.F.C.: Poisson Processes. Oxford University Press, Oxford, 1993.
  • [18] Krylov, N.V.: Controlled Diffusion Processes. Reprint of the 1980 edition, Springer-Verlag, Berlin, 2009.
  • [19] Lamperti, J.: A new class of probability limit theorems. J. Math. Mech. 11 (1962) 749–772.
  • [20] Menshikov, M., Popov, S. and Wade, A.: Non-homogeneous Random Walks. Cambridge University Press, Cambridge, 2016.
  • [21] Mijatović, A. and Urusov, M.: Convergence of integral functionals of one-dimensional diffusions. Electron. Commun. Probab. 17 (2012) paper no. 61, 13 pp.
  • [22] Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge University Press, Cambridge, 1995.
  • [23] Pitman, J. and Yor, J.: A decomposition of Bessel bridges. Z. Wahr. ver. Gebiete 59 (1982) 425–457.
  • [24] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion. 3rd ed., Springer-Verlag, Berlin, 1999.
  • [25] Rogers, L.C.G. and Williams, D.: Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus. Reprint of the second (1994) edition. Cambridge University Press, Cambridge, 2000.
  • [26] Stroock, D.W. and Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer-Verlag, Berlin-New York, 1979.
  • [27] Stroock, D.W. and Yor, M.: Some remarkable martingales. Séminaire de Probabilités XV 1979/80. Lecture Notes in Mathematics 850 (1981) pp. 590–603.
  • [28] Vuolle-Apiala, J.: Excursion theory for rotation invariant Markov processes. Probab. Theory Relat. Fields 93 (1992) 153–158.