Nahmod, Andrea R.Katsaros, Dean2024-12-062024-12-062024-0910.7275/55150https://hdl.handle.net/20.500.14394/55150This thesis studies semilinear wave equations with quadratric derivative non- linearity |∇u|2 (qDNLW) from the probabilistic perspective. We first adapt a method of Bjoern Bringmann in [Bri21] to the d = 2 setting. This method goes beyond the linear-nonlinear decomposition due to Bourgain ([Bou94] and [Bou96]). This is contained in Chapter III. We improve local-in-time well-posedness results in the probabilistic setting in spatial dimensions 2 and 3 by constructing the Random Averaging Operators for (qDNLW). Local well-posedness is proven for data in the spaces H^{3/2+}(\mathbb{T}^3) and H^{11/8+}(\mathbb{T}^2). The space H^{11/8+} (\mathbb{T}^2) is supercritical with respect to the deterministic scaling. Both these results improve over both the best probabilistic and best deterministic results. These thresholds however lie 1+ and 3/8+ above the respective probabilistic scalings for the problem (qDNLW). The argument is constructive in that it is shown that the solution has an explicit expression as the linear combination of a Gaussian sum with adapted random matrix coefficients and a smooth remainder term. This is contained in Chapter IV.Partial Differential Equations, Wave Equations, Derivative Wave Equations, Quadratic Derivative Wave Equations, Dispersive Equations, Random Data PDEs, Random Data Wave Equations, Periodic Wave Equations, Random Averaging Operators, Random Tensors, Periodic Wave Equations, Random Periodic Data, Propagation of Randomness, random flows.Dissertation (Open Access)https://orcid.org/0000-0002-8229-5185