Abstract
Consider a system of particles evolving as independent and identically distributed (i.i.d.) random walks. Initial fluctuations in the particle density get translated over time with velocity \(\vec{v}\), the common mean velocity of the random walks. Consider a box centered around an observer who starts at the origin and moves with constant velocity \(\vec{v}\). To observe interesting fluctuations beyond the translation of initial density fluctuations, we measure the net flux of particles over time into this moving box. We call this the “box-current” process.
We generalize this current process to a distribution-valued process. Scaling time by n and space by \(\sqrt{n}\) gives current fluctuations of order n d/4 where d is the space dimension. The scaling limit of the normalized current process is a distribution-valued Gaussian process with given covariance. The limiting current process is equal in distribution to the solution of a given stochastic partial differential equation which is related to the generalized Ornstein–Uhlenbeck process.
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References
Bojdecki, T., Gorostiza, L.G.: Langevin Equations for \(\mathcal{S}'(\mathbb{R}^{d})\)-Valued Gaussian Processes and Fluctuation Limits of Infinite Particle Systems. Probability Theory and Related Fields, vol. 73(2). Springer, Berlin (1986)
Dürr, D., Goldstein, S., Lebowitz, J.L.: Asymptotics of particle trajectories in infinite one-dimensional systems with collisions. Commun. Pure Appl. Math. 38, 573–597 (1985)
Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)
Holley, R., Stoock, D.: Generalized Ornstein–Uhlenbeck processes and infinite particle branching Brownian motions. Publ. RIMS Kyoto Univ. 14, 741–788 (1978)
Kallianpur, G., Xiong, J.: Stochastic Differential Equations in Infinite-Dimensional Spaces. Institute of Mathematical Statistics, Hayward (1995)
Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 320. Springer, Berlin (1999)
Kumar, R.: Space-time current process for independent random walks in one dimension. ALEA Lat. Am. J. Probab. Math. Stat. 4, 307–336 (2008)
Martin-Löf, A.: Limit theorems for the motion of a Poisson system of independent Markovian particles with high density. Probab. Theory Relat. Fields 34, 205–223 (1976)
Mitoma, I.: Tightness of probabilities on \(C([0,1];{\mathcal{S}}^{\prime})\) and \(D([0,1];{\mathcal{S}}^{\prime})\). Ann. Probab. 11, 989–999 (1983)
Seppäläinen, T.: Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks. Ann. Probab. 33, 759–797 (2005)
Sethuraman, S.: Diffusive variance for a tagged particle in d≤2 asymmetric simple exclusion. ALEA Lat. Am. J. Probab. Math. Stat. 1, 305–332 (2006) (electronic)
Sethuraman, S.: Superdiffusivity of occupation-time variance in 2-dimensional asymmetric exclusion processes with density ρ=1/2. J. Stat. Phys. 123, 787–802 (2006)
Sethuraman, S.: On diffusivity of a tagged particle in asymmetric zero-range dynamics. Ann. Inst. H. Poincaré Probab. Stat. 43, 215–232 (2007)
Sethuraman, S., Varadhan, S.R.S., Yau, H.-T.: Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Commun. Pure Appl. Math. 53, 972–1006 (2000)
Yau, H.-T.: (log t)2/3 law of the two dimensional asymmetric simple exclusion process. Ann. Math. 159, 377–405 (2004)
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Kumar, R. Current Fluctuations for Independent Random Walks in Multiple Dimensions. J Theor Probab 24, 1170–1195 (2011). https://doi.org/10.1007/s10959-010-0317-4
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DOI: https://doi.org/10.1007/s10959-010-0317-4
Keywords
- Independent random walks
- Hydrodynamic limit
- Current fluctuations
- Distribution-valued process
- Generalized Ornstein–Uhlenbeck process