Spatial asymptotic conjecture for the stochastic heat equation with compactly supported initial data

Let $u_0$ satisfy

$$p_t*|u_0|(x)<\infty\quad\forall t>0,\ x\in\mathbb{R}^{\ell}.$$

Let $u(t,x)$ be the corresponding solution of the stochastic heat equation, and let $p_t*u_0(x)$ denote the heat-kernel convolution. In the space-time white-noise case, the spatial dimension and noise covariance are specialized so that the claimed constant below is well-defined.

Spatial asymptotic conjecture. In the particular case of space-time white noise,

$$\lim_{R\to\infty}(\log R)^{-\frac23}\sup_{|x|\le R}\left(\log u(t,x)-\log p_t*u_0(x)\right)=\frac34\left(\frac{2t}{3}\right)^{\frac13}\quad\mathrm{a.s.}$$

The preceding asymptotic formulas are established under the bounded-covariance and Dalang covariance conditions, while this space-time white-noise asymptotic is presented as a conjecture. It predicts the almost-sure spatial growth after normalization by the deterministic heat evolution of the initial data.