A Priori Estimates for Singular Fractional Stochastic Burgers Equations

We study the periodic fractional stochastic Burgers equation $({\mathrm{\partial }}_t+{\mathrm{\Lambda }}^{\gamma })u={\mathrm{\partial }}_x(u^2)+\mathrm{\mid }{\mathrm{\partial }}_x{\mathrm{\mid }}^{1-\alpha }\xi$, where $1<\gamma \le 2$ and $\xi$ is space-time white noise. Under the condition $\alpha >\max (7-4\gamma )\mathrm{/}2,(15-8\gamma )\mathrm{/}6$, 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.