Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 111, pp. 1-21.

Title: SDDEs limits solutions to sublinear reaction-diffusion SPDEs

Author: Hassan Allouba  (Kent State Univ., Ohio, USA}

Abstract: 
  We start by introducing a new definition of solutions to heat-based
  SPDEs driven by space-time white noise: SDDEs
  (stochastic differential-difference equations)  limits solutions.
  In contrast to the standard direct definition of   SPDEs solutions;
  this new notion, which builds on and refines our SDDEs approach to
  SPDEs from earlier work, is entirely based on the approximating SDDEs.
  It is applicable to, and gives a multiscale view of, a variety of SPDEs.
  We extend this approach in related work to other heat-based SPDEs
  (Burgers, Allen-Cahn, and others) and to the difficult case of SPDEs with
  multi-dimensional spacial variable.  We focus here on
  one-spacial-dimensional reaction-diffusion SPDEs; and we prove the
  existence of a SDDEs limit solution to these equations under
  less-than-Lipschitz conditions on the drift and the diffusion coefficients,
  thus extending our earlier SDDEs work to the nonzero drift case.
  The regularity of this solution is obtained  as a by-product
  of the existence estimates.  The uniqueness in law of our SPDEs follows,
  for a large class of such drifts/diffusions, as a simple extension of our
  recent Allen-Cahn uniqueness result.  We also examine briefly,
  through order parameters $\epsilon_1$ and $\epsilon_2$ multiplied by the
  Laplacian and the noise, the effect of letting
  $\epsilon_1,\epsilon_2\to 0$ at different speeds.  More precisely,
  it is shown that the ratio $\epsilon_2/\epsilon_1^{1/4}$ determines
  the behavior as $\epsilon_1,\epsilon_2\to 0$.
 
Submitted October 3, 2002. Published November 5, 2003.
Math Subject Classifications: 60H15, 35R60.
Key Words: Reaction-diffusion SPDE; SDDE; SDDE limits solutions; multiscale.