  A well-known result with respect to the one dimensional nearest-neighbor
symmetric simple exclusion process is the convergence to fractional Brownian
motion with Hurst parameter 1/4, in the sense of finite-dimensional
distributions, of the subdiffusively rescaled current across the origin, and
the subdiffusively rescaled tagged particle position.
  The purpose of this note is to improve this convergence to a functional
central limit theorem, with respect to the uniform topology, and so complete
the solution to a conjecture in the literature with respect to simple exclusion
processes.
