A Priori Estimates for Singular Fractional Stochastic Burgers Equations

Abstract

We study the periodic fractional stochastic Burgers equation $(\partial_t+\Lambda^\gamma)u=\partial_x(u^2)+|\partial_x|^{1-\alpha}\xi$, where $1<\gamma\leq 2$ and $\xi$ is space-time white noise. Under the condition $\alpha>\max{(7-4\gamma)/2,(15-8\gamma)/6}$, we establish pathwise $L^1$, energy, and Besov estimates for a transformed remainder. The argument combines a response decomposition with a conservative Zvonkin transformation. For $1<\gamma<2$, this produces a compensated nonlocal diffusion operator whose coercivity is coupled to a cubic-increment estimate adapted from the modified Karman--Howarth--Monin argument.