Large deviations for the occupation time functional of a Poisson system of independent Brownian particles. (English) Zbl 0813.60029

The authors study the large deviations and the central limit theorem for the occupation time functional of a Poisson system of independent Brownian particles in \(\mathbb{R}^ d\). They extend the results of J. T. Cox and D. Griffeath for random walks [Z. Wahrscheinlichkeitstheorie Verw. Geb. 66, 543-558 (1984; Zbl 0551.60028)] and partially the results of T.-Y. Lee [Ann. Probab. 16, No. 4, 1537-1558 (1988; Zbl 0661.60046) and ibid. 17, No. 1, 46-57 (1989; Zbl 0664.60032)] to functional spaces. The order of the large deviations remains the same, namely \(T^{1/2}\) and \(T/ \log T\) in the recurrent dimensions one and two, and \(T\) for the higher transient dimensions. Explicit expressions for the corresponding rate functions and covariance functionals are given and a specific microcanonical principle is considered. In one dimension the function-space rate function is not the standard one; thus an untypical end point is reached via a nonlinear profile.


60F10 Large deviations
60F05 Central limit and other weak theorems
60J65 Brownian motion
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
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[1] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[2] Borodin, A. N., Brownian local time, Russian Math. Surveys, 44, 2, 1-51 (1989) · Zbl 0697.60080
[3] Cox, J. T.; Durrett, R., Large deviations for independent random walks, Probab. Theory Rel. Fields, 84, 67-82 (1990) · Zbl 0692.60028
[4] Cox, J. T.; Griffeath, D., Large deviations for Poisson systems of independent random walks, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 66, 543-558 (1984) · Zbl 0551.60028
[5] Deuschel, J.-D.; Stroock, D. W., Large Deviations (1989), Academic Press: Academic Press New York · Zbl 0682.60018
[6] Deuschel, J.-D.; Stroock, D. W., A function space large deviation principle for certain stochastic integrals, Probab. Theory Rel. Fields, 83, 279-307 (1989) · Zbl 0682.60018
[7] Deuschel, J.-D.; Stroock, D. W.; Zessin, H., Microcanonical distributions for lattice gases, Comm. Math. Phys., 139, 83-101 (1991) · Zbl 0727.60025
[8] Donsker, M. D.; Varadhan, S. R.S., Large deviations for noninteracting infinite-particle systems, J. Statist. Phys., 46, 1195-1232 (1987) · Zbl 0682.60020
[9] Ellis, R. S., Entropy, Large Deviations, and Statistical Mechanics (1985), Springer: Springer Berlin · Zbl 0566.60097
[10] Karlin, S.; Taylor, H. M., (A First Course in Stochastic Process (1975), Academic Press: Academic Press New York) · Zbl 0315.60016
[11] Lee, T. Y., Large deviations for noninteracting infinite particle systems, The Ann. Probab., 16, 1537-1558 (1988) · Zbl 0661.60046
[12] Lee, T. Y., Large deviations for systems of noninteracting recurrent particles, The Ann. Probab., 17, 1, 46-57 (1989) · Zbl 0664.60032
[13] Lee, T. Y.; Remillard, B., Occupation times in systems of null recurrent Markov processes, Probab. Theory Rel. Fields, 98, 245-260 (1994) · Zbl 0792.60027
[14] Revuz, D.; Yor, M., Continuous Martingales and Brownian Motion (1991), Springer: Springer Berlin · Zbl 0731.60002
[15] Zabreyko, P. P.; Koshelev, A. I.; Krasnosel’skii, M. A.; Mikhlin, S. G.; Rakovshchik, L. S.; Stet’senko, V. Y., Integral Equations — A Reference Text (1975), Noordhoff: Noordhoff Leyden · Zbl 0293.45001
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