# 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

Figure 1.1. The dashed line is the critical regularity \( \textstyle s _ { c } = { \frac { 3 } { 2 } } - { \frac { 2 } { p - 1 } } \). The solid line is the threshold for the exponent s in Theorem 1.1. The dotted line is the threshold from the authors’ previous result in [25, Theorem 1.1].

important cancellation when computing the time derivative of the energy functional. Sec-ond, we need the appearance of the \( L _ { x } ^ { 2 } \) term in the energy in order to be able to estimate for 0 \( < \sigma < \) 1,

\[ \| | \nabla | ^ { \sigma } v | | _ { L _ { x } ^ { 2 } ( \mathbb { R } ^ { 3 } ) } ^ { 2 } \lesssim \| v \| _ { L _ { x } ^ { 2 } ( \mathbb { R } ^ { 3 } ) } ^ { 2 } + \| \nabla _ { x } v \| _ { L _ { x } ^ { 2 } ( \mathbb { R } ^ { 3 } ) } ^ { 2 } \lesssim E ( v ) . \]

Previously, energy methods for random data problems were used by Nahmod, Pavlovic´and Staffilani [26] in the context of the periodic Navier-Stokes equation in two and three dimensions and by Burq and Tzvetkov for the three-dimensional periodic defocusing cubic nonlinear wave equation [12]. Pocovnicu [29] and Oh and Pocovnicu [27] used probabilistic energy bounds in conjunction with a probabilistic perturbation theory for the energy critical nonlinear wave equation.

Remark 1.3. Our proof of the probabilistic energy estimates is inspired by the quintic case in [27], with some important differences. In [27], Oh and Pocovnicu only consider frequency truncated random initial data and show that almost surely the corresponding solutions satisfy energy bounds uniformly in the truncation parameter, which allows one to construct solutions using probabilistic perturbation theory. Instead, we study the Cauchy problem with super-critical random initial data directly. To do so we make use of the observation that although the term \( \partial _ { t } u _ { f } ^ { \omega } \in H _ { x } ^ { s - 1 } ( \mathbb { R } ^ { 3 } ) \) appears when taking the time derivative of our energy functional, this expression is always paired with a term at regularity \( H _ { x } ^ { 1 - s } ( \mathbb { R } ^ { 3 } ) \). We therefore have no problem achieving the necessary bounds to close our Gronwall argument, see the proof of Proposition 3.2 below for more details. Additionally, the presence of non-algebraic nonlinearities introduces some complications in our estimates. To overcome this difficulty, a more careful analysis using the fractional chain rule and interpolation in Sobolev spaces is necessary.

Remark 1.4. Our proof does not yield any improvement at \( p \) = 3. However, this case can be treated exactly as in the periodic case in [12] using the energy functional