The Dynkin Festschrift pp 387-397 | Cite as

# Regularity of Self-Diffusion Coefficient

## Abstract

We consider the infinite particle system in *Z* ^{ d } known commonly as symmetric random walk with simple exclusion. This consists of particles in *Z* ^{ d }, each trying to execute a more or less independent random walk with jump probabilities *p*(*x*). We assume that *p*(*x*) is symmetric, i.e., *p*(*x*) = *p*(−*x*), and that it has finite range. If *d* = 1, we assume, in addition, that *p*(1) +*p*(−1) < 1, so that only the one-dimensional nearest neighbor random walk is excluded. Particles wait for an exponential time, pick a random jump size *x* with probability *p*(*x*), and try to jump to a new site defined by a jump equal to *x*. If the site is free, the jump is completed and things start all over again. Otherwise the jump is disallowed and things start all over again with the particle remaining in the old site. All the particles are doing this simultaneously and independently of each other. See [3] for a discussion of the model.

## Keywords

Random Walk Dirichlet Form Variational Formula Infinitesimal Generator Neighbor Walk## Preview

Unable to display preview. Download preview PDF.

## References

- [1]C. Kipnis and S.R.S. Varadhan, Central limit theorem for additive functional of reversible Markov processes and application to simple exclusions,
*Comm. Math. Phys.*Vol. 104, (1986), 1–19.MathSciNetCrossRefMATHGoogle Scholar - [2]J. Quastel, Diffusion of color in simple exclusion process,
*Comm. Pure Appl. Math.*Vol. XLV (1992), 623–680.MathSciNetCrossRefGoogle Scholar - [3]H. Spohn,
*Large Scale Dynamics of Interacting Particles*, Springer-Verlag, Berlin, 1991.CrossRefMATHGoogle Scholar