  The following stochastic Cauchy initial-value problem is studied for the
parabolic heat equation on a domain $ \mathbf{Q}\subset{\mathbf{R}}^{n}$ with
random field initial data. \begin{align} &{\square}\widehat{u(x,t)} \equiv
\bigg(\frac{\partial}{\partial
t}-{\Delta}_{x}\bigg)\widehat{u(x,t)}=0,~x\in\mathbf{Q},t> 0 \end{align}
\begin{align} \widehat{u(x,0)}=\phi(x)+\mathscr{J}(x),~x\in\mathbf{Q},t=0
\end{align} where $\phi(x)\in C^{\infty}({\mathbf{Q}})$, and $\mathscr{J}(x)$
is a classical Gaussian random scalar field with expectation
$\mathbb{E}[\![\mathscr{J}(x)]\!]=0 $ and with a regulated covariance
$\mathbb{E}[\![ \mathscr{J}(x)\otimes\mathscr{J}(y)]\!]=\zeta J(x,y;\ell)$,
correlation length $\ell$ and $\mathbb{E}[\![
\mathscr{J}(x)\otimes\mathscr{J}(x)]\!]=\zeta<\infty$. The randomly perturbed
solution $\widehat{u(x,t)}$ is a stochastic convolution integral. This leads to
stochastic extensions and versions of some classical results for the heat
equation; in particular, a Li-Yau differential Harnack inequality \begin{align}
\mathbb{E}\left[\!\!\left[\frac{|\nabla\widehat{u(x,t)}|^{2}}{|\widehat{u(x,t)}|^{2}}
\right]\!\!\right]-\mathbb{E}\left[\!\!\left[\frac{\tfrac{\partial}{\partial
t}\widehat{u(x,t)}}{\widehat{u(x,t)}} \right]\!\!\right]\le
\frac{1}{2}n\frac{1}{t} \end{align} and a parabolic Harnack inequality. Decay
estimates and bounds for the volatility
$\mathbb{E}[\![|\widehat{u(x,t)}|^{2}]\!]$ and p-moments
$\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!] $ are derived. Since $\lim_{t\uparrow
\infty}\mathbb{E}[\![|\widehat{u(x,t)}|^{p}]\!]=0 $, the Cauchy evolution of
the randomly perturbed solution is stable since the heat equation smooths out
or dissipates volatility induced by initial data randomness as
$t\rightarrow\infty$.
