Bernoulli

  • Bernoulli
  • Volume 25, Number 4A (2019), 2439-2478.

Stationary distributions and convergence for Walsh diffusions

Tomoyuki Ichiba and Andrey Sarantsev

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a stochastic differential equation with certain drift and diffusion coefficients, as long as it stays away from the origin. As it hits the origin, it instantaneously chooses a new direction according to a given probability law, called the spinning measure. A special example is a real-valued diffusion with skew reflections at the origin. This process continuously (in the weak sense) depends on the spinning measure. We determine a stationary measure for such process, explore long-term convergence to this distribution and establish an explicit rate of exponential convergence.

Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 2439-2478.

Dates
Received: October 2017
Revised: June 2018
First available in Project Euclid: 13 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1568362032

Digital Object Identifier
doi:10.3150/18-BEJ1059

Mathematical Reviews number (MathSciNet)
MR4003554

Zentralblatt MATH identifier
07110101

Keywords
ergodic process invariant measure Lyapunov function reflected diffusion stationary distribution stochastic differential equation Walsh Brownian motion Walsh diffusion

Citation

Ichiba, Tomoyuki; Sarantsev, Andrey. Stationary distributions and convergence for Walsh diffusions. Bernoulli 25 (2019), no. 4A, 2439--2478. doi:10.3150/18-BEJ1059. https://projecteuclid.org/euclid.bj/1568362032


Export citation

References

  • [1] Bakry, D., Cattiaux, P. and Guillin, A. (2008). Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 727–759.
  • [2] Barlow, M., Pitman, J. and Yor, M. (1989). On Walsh’s Brownian motions. In Séminaire de Probabilités, XXIII. Lecture Notes in Math. 1372 275–293. Berlin: Springer.
  • [3] Chen, Z.-Q. and Fukushima, M. (2012). Symmetric Markov Processes, Time Change, and Boundary Theory. London Mathematical Society Monographs Series 35. Princeton, NJ: Princeton Univ. Press.
  • [4] Chen, Z.-Q. and Fukushima, M. (2015). One-point reflection. Stochastic Process. Appl. 125 1368–1393.
  • [5] Cloez, B. and Hairer, M. (2015). Exponential ergodicity for Markov processes with random switching. Bernoulli 21 505–536.
  • [6] Davies, P.L. (1986). Rates of convergence to the stationary distribution for $k$-dimensional diffusion processes. J. Appl. Probab. 23 370–384.
  • [7] Douc, R., Fort, G. and Guillin, A. (2009). Subgeometric rates of convergence of $f$-ergodic strong Markov processes. Stochastic Process. Appl. 119 897–923.
  • [8] Down, D., Meyn, S.P. and Tweedie, R.L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 1671–1691.
  • [9] Evans, S.N. and Sowers, R.B. (2003). Pinching and twisting Markov processes. Ann. Probab. 31 486–527.
  • [10] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I. New York–London–Sydney: Wiley.
  • [11] Fitzsimmons, P.J. and Kuter, K.E. (2014). Harmonic functions on Walsh’s Brownian motion. Stochastic Process. Appl. 124 2228–2248.
  • [12] Freidlin, M. and Sheu, S.-J. (2000). Diffusion processes on graphs: Stochastic differential equations, large deviation principle. Probab. Theory Related Fields 116 181–220.
  • [13] Freidlin, M.I. and Wentzell, A.D. (1993). Diffusion processes on graphs and the averaging principle. Ann. Probab. 21 2215–2245.
  • [14] Hajri, H. (2011). Stochastic flows related to Walsh Brownian motion. Electron. J. Probab. 16 1563–1599.
  • [15] Hajri, H. and Touhami, W. (2014). Itô’s formula for Walsh’s Brownian motion and applications. Statist. Probab. Lett. 87 48–53.
  • [16] Ichiba, T., Karatzas, I., Prokaj, V. and Yan, M. (2018). Stochastic integral equations for Walsh semimartingales. Ann. Inst. Henri Poincaré Probab. Stat. 54 726–756.
  • [17] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. New York: Springer.
  • [18] Karatzas, I. and Yan, M. (2016). Semimartingales on rays, Walsh diffusions, and related problems of control and stopping. Available at arXiv:1603.03124.
  • [19] Lund, R.B., Meyn, S.P. and Tweedie, R.L. (1996). Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Probab. 6 218–237.
  • [20] Meyn, S.P. and Tweedie, R.L. (1993). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. in Appl. Probab. 25 487–517.
  • [21] Meyn, S.P. and Tweedie, R.L. (1993). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 518–548.
  • [22] Meyn, S.P. and Tweedie, R.L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 981–1011.
  • [23] Picard, J. (2005). Stochastic calculus and martingales on trees. Ann. Inst. Henri Poincaré Probab. Stat. 41 631–683.
  • [24] Rachev, S.T. (1991). Probability Metrics and the Stability of Stochastic Models. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Chichester: Wiley.
  • [25] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Berlin: Springer.
  • [26] Roberts, G.O. and Rosenthal, J.S. (1996). Quantitative bounds for convergence rates of continuous time Markov processes. Electron. J. Probab. 1 no. 9, approx. 21 pp.
  • [27] Roberts, G.O. and Tweedie, R.L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stochastic Process. Appl. 80 211–229.
  • [28] Roberts, G.O. and Tweedie, R.L. (2000). Rates of convergence of stochastically monotone and continuous time Markov models. J. Appl. Probab. 37 359–373.
  • [29] Sarantsev, A. (2016). Explicit rates of exponential convergence for reflected jump-diffusions on the half-line. ALEA Lat. Am. J. Probab. Math. Stat. 13 1069–1093.
  • [30] Sarantsev, A. (2017). Penalty method for obliquely reflected diffusions. Available at arXiv:1509.01777.
  • [31] Sarantsev, A. (2017). Reflected Brownian motion in a convex polyhedral cone: Tail estimates for the stationary distribution. J. Theoret. Probab. 30 1200–1223.
  • [32] Tsirelson, B. (1997). Triple points: From non-Brownian filtrations to harmonic measures. Geom. Funct. Anal. 7 1096–1142.
  • [33] Villani, C. (2009). Optimal Transport. Old and New. A Series of Comprehensive Studies in Mathematics 338. Springer.
  • [34] Walsh, J. (1978). A diffusion with a discontinuous local time. Astérisque 52–53 (53) 37–45.
  • [35] Watanabe, S. (1999). The existence of a multiple spider martingale in the natural filtration of a certain diffusion in the plane. In Séminaire de Probabilités, XXXIII. Lecture Notes in Math. 1709 277–290. Berlin: Springer.