An optimal local theory for reaction-diffusion equations driven by non-trace-class noise
arXiv:2606.07417
The paper develops a general local existence and uniqueness theory for stochastic reaction‑diffusion equations driven by rough, possibly non‑trace‑class noise, handling low‑regularity solutions and identifying critical initial data spaces.
Abstract
We study local well-posedness for a class of stochastic reaction-diffusion equations driven by multiplicative, possibly colored, noise. The interaction between rough stochastic forcing and polynomial nonlinearities naturally leads to solutions with low spatial regularity, making the treatment of the nonlinear terms delicate. Our main contribution is a general local existence and uniqueness theory for SPDEs with rough noise and highly irregular initial data. The framework also yields new results in standard noise regimes, including trace-class noise and space-time white noise. We identify the critical initial-data spaces for a wide range of nonlinearities, and we establish instantaneous parabolic regularization, general blow-up criteria, and sufficient conditions for positivity preservation. We apply the abstract theory to several prototypical models, including the stochastic Allen-Cahn, Burgers, Fisher-KPP, and coupled Gray-Scott equations. Finally, in the one-dimensional space-time white-noise setting, we combine our local theory with existing global a priori results in a highly singular regime.
46 pages