This research demonstrates the well-posedness of a stochastic reaction-diffusion equation using L2 initial data, indicating robustness under random forces.
We consider a parabolic stochastic partial differential equation (SPDE) on $[0\,,1]$ that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an Llog L growth condition. We prove that the SPDE is well posed when the initial data is in L²[0\,,1]. This solves a strong form of an open problem.
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Foondun et al. (2025) studied this question.
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