# 1.2. Statement of the main result. We are now prepared to state our main result.

Theorem 1.1. Let 3 \( < p < \) 5 and \( \begin{array} { r } { \frac { p - 1 } { p + 1 } < s < 1 } \end{array} \). For real-valued \( f = ( f _ { 1 } , f _ { 2 } ) \in H _ { x } ^ { s } ( \mathbb { R } ^ { 3 } ) \ \times \)\( H _ { x } ^ { s - 1 } ( \mathbb { R } ^ { 3 } ) \), let \( f ^ { \omega } = ( f _ { 1 } ^ { \omega } , f _ { 2 } ^ { \omega } ) \) be the randomized initial data defined in (1.1) and let \( u _ { f } ^ { \omega } \) be the free wave evolution (1.1) of \( f ^ { \omega } \). Then for almost every \( \omega \in { \Omega } \), there exists a unique global solution

\[ ( 8 ) \qquad \qquad \qquad \qquad ( u , \partial _ { t } u ) \in ( u _ { f } ^ { \omega } , \partial _ { t } u _ { f } ^ { \omega } ) + C \bigl ( \mathbb { R } ; H _ { x } ^ { 1 } ( \mathbb { R } ^ { 3 } ) \times L _ { x } ^ { 2 } ( \mathbb { R } ^ { 3 } ) \bigr ) \]

to the nonlinear wave equation

\[ \begin{array} { r l } { ( 9 ) \qquad \qquad \qquad \qquad \qquad } & { { } \left\{ - \partial _ { t } ^ { 2 } u + \Delta u = | u | ^ { p - 1 } u ~ o n ~ \mathbb { R } \times \mathbb { R } ^ { 3 } , \right. } \\ { \left. ( u , \partial _ { t } u ) | _ { t = 0 } = ( f _ { 1 } ^ { \omega } , f _ { 2 } ^ { \omega } ) . \right. } \end{array} \]

Here, uniqueness holds in the sense that upon writing

\[ ( u , \partial _ { t } u ) = ( u _ { f } ^ { \omega } , \partial _ { t } u _ { f } ^ { \omega } ) + ( v , \partial _ { t } v ) , \]

there exists a unique global solution

\[ ( v , \partial _ { t } v ) \in C \big ( \mathbb { R } ; H _ { x } ^ { 1 } ( \mathbb { R } ^ { 3 } ) \big ) \cap L _ { t , l o c } ^ { \frac { 2 p } { p - 3 } } L _ { x } ^ { 2 p } \big ( \mathbb { R } \times \mathbb { R } ^ { 3 } \big ) \times C \big ( \mathbb { R } ; L _ { x } ^ { 2 } ( \mathbb { R } ^ { 3 } ) \big ) \]

to the forced nonlinear wave equation

\[ \quad ( 1 0 ) \qquad \qquad \qquad \qquad \left\{ - \partial _ { t } ^ { 2 } v + \Delta v = | u _ { f } ^ { \omega } + v | ^ { p - 1 } ( u _ { f } ^ { \omega } + v ) ~ o n ~ \mathbb { R } \times \mathbb { R } ^ { 3 } , \right. \]

Remark 1.2. In contrast to the mild uniqueness of the authors’ previous work [25, The-orem 1.1], Theorem 1.1 yields the more standard notion of uniqueness for solutions to semilinear wave equations, compare with [25, Remark 4.3]. Moreover, the threshold for the allowable regularity in Theorem 1.1 has been significantly lowered as compared to that in[25, Theorem 1.1]. In particular, we prove the existence of global solutions for initial data at super-critical regularities for all 3 \( < p < \) 5, see Figure 1.1.

While the randomization (1.1) does not regularize at the level of Sobolev spaces, the free evolution of the randomized initial data (1.1) almost surely satisfies better space-time integrability properties. For this reason one can show that the nonlinear component of the solution lies in a better space, in this case \( H _ { x } ^ { 1 } ( \mathbb { R } ^ { 3 } ) \times L _ { x } ^ { 2 } ( \mathbb { R } ^ { 3 } ) \), by constructing local solutions via a fixed point argument centered at the free evolution \( u _ { f } ^ { \omega } \). We will see that to conclude global existence, it suffices to control the growth of the \( H _ { x } ^ { 1 } ( \mathbb { R } ^ { 3 } ) \times L _ { x } ^ { 2 } ( \mathbb { R } ^ { 3 } ) \) norm of the nonlinear component of the solution. The main novelty of this paper is the derivation of probabilistic growth estimates for the modified energy functional

\[ \quad ( 1 1 ) \qquad \qquad E ( v ) = \int _ { \mathbb { R } ^ { 3 } } { \frac { 1 } { 2 } } | \nabla _ { x } v | ^ { 2 } + { \frac { 1 } { 2 } } | \partial _ { t } v | ^ { 2 } + { \frac { 1 } { 2 } } | v | ^ { 2 } + { \frac { 1 } { p + 1 } } | u _ { f } ^ { \omega } + v | ^ { p + 1 } \, d x \]

for 3 \( < p < \) 5, where v is the nonlinear component of the solution to (1). Consequently, we will be able to conclude that almost surely, we have the necessary control to extend the local solutions that we construct to global ones.

We consider this modified energy functional for two reasons. The first is that the appear-ance of the free evolution of the randomized initial data in the potential term creates an