The Annals of Probability

Markov processes on time-like graphs

Krzysztof Burdzy and Soumik Pal

Full-text: Open access

Abstract

We study Markov processes where the “time” parameter is replaced by paths in a directed graph from an initial vertex to a terminal one. Along each directed path the process is Markov and has the same distribution as the one along any other directed path. If two directed paths do not interact, in a suitable sense, then the distributions of the processes on the two paths are conditionally independent, given their values at the common endpoint of the two paths. Conditions on graphs that support such processes (e.g., hexagonal lattice) are established. Next we analyze a particularly suitable family of Markov processes, called harnesses, which includes Brownian motion and other Lévy processes, on such time-like graphs. Finally we investigate continuum limits of harnesses on a sequence of time-like graphs that admits a limit in a suitable sense.

Article information

Source
Ann. Probab., Volume 39, Number 4 (2011), 1332-1364.

Dates
First available in Project Euclid: 5 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1312555800

Digital Object Identifier
doi:10.1214/10-AOP583

Mathematical Reviews number (MathSciNet)
MR2857242

Zentralblatt MATH identifier
1234.60089

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60G60: Random fields 60J99: None of the above, but in this section

Keywords
Harness graphical Markov model time-like graphs

Citation

Burdzy, Krzysztof; Pal, Soumik. Markov processes on time-like graphs. Ann. Probab. 39 (2011), no. 4, 1332--1364. doi:10.1214/10-AOP583. https://projecteuclid.org/euclid.aop/1312555800


Export citation

References

  • [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [2] Dubins, L. E. (1968). On a theorem of Skorohod. Ann. Math. Statist. 39 2094–2097.
  • [3] Etheridge, A. M. (2000). An Introduction to Superprocesses. University Lecture Series 20. Amer. Math. Soc., Providence, RI.
  • [4] Fontes, L. R. G., Isopi, M., Newman, C. M. and Ravishankar, K. (2004). The Brownian web: Characterization and convergence. Ann. Probab. 32 2857–2883.
  • [5] Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston, MA.
  • [6] Geiger, D., Heckerman, D., King, H. and Meek, C. (2001). Stratified exponential families: Graphical models and model selection. Ann. Statist. 29 505–529.
  • [7] Hammersley, J. M. (1967). Harness. In Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, Calif., 1965/66), Vol. III: Physical Sciences 89–117. Univ. California Press, Berkeley, CA.
  • [8] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [9] Khoshnevisan, D. (2002). Multiparameter Processes: An Introduction to Random Fields. Springer, New York.
  • [10] Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Oxford Univ. Press, New York.
  • [11] Mansuy, R. and Yor, M. (2005). Harnesses, Lévy bridges and Monsieur Jourdain. Stochastic Process. Appl. 115 329–338.
  • [12] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390 (electronic).
  • [13] Soucaliuc, F., Tóth, B. and Werner, W. (2000). Reflection and coalescence between independent one-dimensional Brownian paths. Ann. Inst. H. Poincaré Probab. Statist. 36 509–545.
  • [14] Sun, R. and Swart, J. M. (2008). The Brownian net. Ann. Probab. 36 1153–1208.