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.