Electron. J. Diff. Eqns., Vol. 2003(2003), No. 111, pp. 1-21.

SDDEs limits solutions to sublinear reaction-diffusion SPDEs

Hassan Allouba

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.

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Hassan Allouba
Department of Mathematical Sciences
Kent State University, Kent, Ohio 44242, USA
email: allouba@mcs.kent.edu
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