Random data Cauchy theory for supercritical Fourth-order Schrödinger equation

This work investigates a cubic fourth-order Schrödinger equation on $ \mathbb{R}^d, d = 5, 6, $ with randomized initial data. We prove the probabilistic local well-posedness of solutions for supercritical equations, assuming the initial data $ f $ lies in the Sobolev space $ H^s(\mathbb{R}^d) $ with regularity exponent $ s $ satisfying $ \frac{(d-1)(d-4)}{2d+16} < s < s_c = \frac{d-4}{2} $. This result is achieved via an iterative approach grounded in a partial power series expansion with respect to random initial data. We improve the results presented in Dinh [10]. Finally, we improve the lower bound for the regularity exponent $ s $ from $ \frac{(d-1)(d-4)}{2d+16} $ to $\frac{(d-4)^2}{2(d+2)}$, as presented in our main result.