## Annals of Probability

### Hitting times for special patterns in the symmetric exclusion process on ℤd

#### Abstract

We consider the symmetric exclusion process {ηt,t>0} on {0,1}d. We fix a pattern ${\mathcal{A}}:=\{\eta: \sum_{\Lambda}\eta(i)\ge k\}$, where Λ is a finite subset of ℤd and k is an integer, and we consider the problem of establishing sharp estimates for τ, the hitting time of ${\mathcal{A}}$. We present a novel argument based on monotonicity which helps in some cases to obtain sharp tail asymptotics for τ in a simple way. Also, we characterize the trajectories {ηs,st} conditioned on {τ>t}.

#### Article information

Source
Ann. Probab., Volume 32, Number 4 (2004), 3301-3323.

Dates
First available in Project Euclid: 8 February 2005

https://projecteuclid.org/euclid.aop/1107883354

Digital Object Identifier
doi:10.1214/009117904000000487

Mathematical Reviews number (MathSciNet)
MR2094446

Zentralblatt MATH identifier
1067.60096

#### Citation

Asselah, Amine; Pra, Paolo Dai. Hitting times for special patterns in the symmetric exclusion process on ℤ d. Ann. Probab. 32 (2004), no. 4, 3301--3323. doi:10.1214/009117904000000487. https://projecteuclid.org/euclid.aop/1107883354

#### References

• Asselah, A. and Castell, F. (2003). Existence of quasistationary measures for asymmetric attractive particle systems on $\ZZ^d$. Ann. Appl. Probab. 13 1569–1590.
• Asselah, A. and Dai Pra, P. (2001). Quasi-stationary measures for conservative dynamics in the infinite lattice. Ann. Probab. 29 1733–1754.
• Asselah, A. and Ferrari, P. (2002). Regularity of quasi-stationary measures for simple exclusion in dimension $d\ge 5$. Ann. Probab. 30 1913–1932.
• Bañuelos, R. and Davis, B. (1987). Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains. J. Funct. Anal. 84 188–200.
• Durrett, R. (1991). Probability: Theory and Examples. Wadsworth, Belmont, CA.
• Ferrari, P. A. and Goldstein, S. (1988). Microscopic stationary states for stochastic systems with particle flux. Probab. Theory Related Fields 78 455–471.
• Helmberg, G. (1969). Introduction to Spectral Theory in Hilbert Space. North-Holland, Amsterdam.
• Kondo, K. and Hara, T. (1987). Critical exponent of susceptibility for a class of general ferromagnets in $d>4$ dimensions. J. Math. Phys. 28 1206–1208.
• Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
• Martinez, S. and San Martin, J. (2001). Rates of decay and $h$-processes for one dimensional diffusions conditioned on non-absorption. J. Theoret. Probab. 14 199–212.
• Noble, B. and Daniel, J. W. (1988). Applied Linear Algebra. Prentice-Hall, Englewood Cliffs, NJ.
• Pinsky, R. G. (1990). The lifetimes of conditioned diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 26 87–99.
• Sethuraman, S. (2001). On extremal measures for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 37 139–154.