Abstract: The Central Limit Theorem in the PDE setting can be stated as follows: the solution to the heat equation with $L^1$ initial data $u_0$ behaves asymptotically as the mass $M=\int u_0$ times the fundamental solution $h_t$ (the heat kernel). More precisely,
\begin{align*}
\|h_t\|_{L^{p}(\mathbb{R}^n)}^{-1}\,\|u_0\ast h_t-\, M \, h_{t}\|_{L^{p}(\mathbb{R}^n)}\,
\longrightarrow\,0, \quad \text{as } t \rightarrow +\infty, \quad 1\leq p \leq \infty.
\end{align*}Since the heat kernel is rather sensitive to the underlying geometry, it is a natural task to examine whether such a large-time behavior remains true when changing the setting. We show that on (real) hyperbolic space not only the quantity corresponding to $M$ should be varying with $p$, but it should also be a \textit{function} rather than a constant; in fact, this function has a completely different expression for $1\leq p<2$ and for $2< p\leq \infty$, the case $p=2$ bridging both worlds. The results extend to symmetric spaces of non-compact type, and recover and extend results by V\'azquez, Anker et al, Naik et al.
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