The Annals of Probability

Joint density for the local times of continuous-time Markov chains

David Brydges, Remco van der Hofstad, and Wolfgang König

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Abstract

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus.

We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan’s lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of ℤd tending to ℤd as time diverges. We finally discuss the relation of our density formula to the Ray–Knight theorem for continuous-time simple random walk on ℤ, which is analogous to the well-known Ray–Knight description of Brownian local times.

Article information

Source
Ann. Probab., Volume 35, Number 4 (2007), 1307-1332.

Dates
First available in Project Euclid: 8 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1181334246

Digital Object Identifier
doi:10.1214/009171906000001024

Mathematical Reviews number (MathSciNet)
MR2330973

Zentralblatt MATH identifier
1127.60076

Subjects
Primary: 60J55: Local time and additive functionals 60J27: Continuous-time Markov processes on discrete state spaces

Keywords
Local times density large deviations upper bound Ray–Knight theorem Varadhan’s lemma

Citation

Brydges, David; van der Hofstad, Remco; König, Wolfgang. Joint density for the local times of continuous-time Markov chains. Ann. Probab. 35 (2007), no. 4, 1307--1332. doi:10.1214/009171906000001024. https://projecteuclid.org/euclid.aop/1181334246


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References

  • Abdesselam, A. (2004). Grassmann–Berezin calculus and theorems of the matrix-tree type. Adv. in Appl. Math. 33 51–70.
  • Alicandro, R. and Cicalese, M. (2004). A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 1–37.
  • Berezin, F. A. (1987). Introduction to Superanalysis. Reidel, Dordrecht.
  • Biskup, M. and König, W. (2001). Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29 636–682.
  • Bolthausen, E., Deuschel, J.-D. and Tamura, I. (1995). Laplace approximations for large deviations of nonreversible Markov processes. The nondegenerate case. Ann. Probab. 23 236–267.
  • Brydges, D. C., Fröhlich, J. and Spencer, T. (1982). The random walk representation of classical spin systems and correlation inequalities. Comm. Math. Phys. 83 123–150.
  • Brydges, D. C., van der Hofstad, R. and König, W. (2005). Joint density for the local times of continuous-time Markov chains: Extended version. Available at.
  • Brydges, D. C. and Imbrie, J. Z. (2003). Branched polymers and dimensional reduction. Ann. Math. 158 1019–1039.
  • Brydges, D. C. and Imbrie, J. Z. (2003). Green's function for a hierarchical self-avoiding walk in four dimensions. Comm. Math. Phys. 239 549–584.
  • Brydges, D. C. and Muñoz-Maya, I. (1991). An application of Berezin integration to large deviations. J. Theor. Probab. 4 371–389.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
  • Donsker, M. D. and Varadhan, S. R. S. (1975). Asymptotic evaluation of certain Markov process expectations for large time. I–IV. Comm. Pure Appl. Math. 28 1–47, 279–301, 29 389–461 (1979) \bf36 183–212 (1983).
  • Dynkin, E. M. (1983). Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55 344–376.
  • Dynkin, E. M. (1984). Local times and quantum fields. In Seminar on Stochastic Processes 1983 (Gainesville, Fla., 1983) 69–83. Birkhäuser, Boston.
  • Dynkin, E. M. (1984). Polynomials of the occupation field and related random fields. J. Funct. Anal. 58 20–52.
  • Eisenbaum, N. (1994). Dynkin's isomorphism theorem and the Ray–Knight theorems. Probab. Theory Related Fields 99 321–335.
  • Eisenbaum, N. and Kaspi, H. (1993). A necessary and sufficient condition for the Markov properties of the local time process. Ann. Probab. 21 1591–1598.
  • Eisenbaum, N. and Kaspi, H. (1996). On the Markov property of local time for Markov processes on general graphs. Stoch. Proc. Appl. 64 153–172.
  • Eisenbaum, N., Kaspi, H., Marcus, M. B., Rosen, J. and Shi, Z. (2000). A Ray–Knight theorem for symmetric Markov processes. Ann. Probab. 28 1781–1796.
  • Gantert, N., König, W. and Shi, Z. (2007). Annealed deviations for random walk in random scenery. Ann. Inst. H. Poincaré Probab. Statist. 43 47–76.
  • Gärtner, J. (1977). On large deviations from the invariant measure. Theory Probab. Appl. 22 24–39.
  • Gärtner, J. and den Hollander, F. (1999). Correlation structure of intermittency in the parabolic Anderson model. Probab. Theory Related Fields 114 1–54.
  • Greven, A. and den Hollander, F. (1993). A variational characterization of the speed of a one-dimensional self-repellent random walk. Ann. Appl. Probab. 3 1067–1099.
  • van der Hofstad, R., König, W. and Mörters, P. (2006). The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 307–353.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in random environment. Compositio Math. 30 145–168.
  • Knight, F. B. (1963). Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 109 56–86.
  • Le Jan, Y. (1987). Temps local et superchamp. Séminaire de Probabilités XXI. Lecture Notes in Math. 1247 176–190. Springer, Berlin.
  • Luttinger, J. M. (1983). The asymptotic evaluation of a class of path integrals. II. J. Math. Phys. 24 2070–2073.
  • March, P. and Sznitman, A.-S. (1987). Some connections between excursion theory and the discrete Schrödinger equation with random potentials. Probab. Theory Related Fields 109 11–53.
  • Marcus, M. B. and Rosen, J. (1996). Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes. Ann. Probab. 24 1130–1177.
  • McKane, A. J. (1980). Reformulation of $n \rightarrow0$ models using anticommuting scalar fields. Physics Lett. A 76 22–24.
  • Parisi, G. and Sourlas, N. (1980). Self-avoiding walk and supersymmetry. J. Phys. Lett. 41 L403–L406.
  • Ray, D. (1963). Sojourn times of diffusion processes. Illinois J. Math. 7 615–630.
  • Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • Sheppard, P. (1985). On the Ray–Knight Markov property of local times. J. London Math. Soc. 31 377–384.
  • Symanzik, K. (1969). Euclidean quantum theory. In Local Quantum Theory (R. Jost, ed.) 152–226. Academic Press, New York.
  • Tóth, B. (1996). Generalized Ray–Knight theory and limit theorems for self-interacting random walks on $\mathbb{Z}^1$. Ann. Probab. 24 1324–1367.