The Central Limit Theorem of
probability represented in the PDE setting can be described as follows: consider the heat equation on \(\mathbb^n\),
\[
\partial_u(t,x)\,=\,\Delta_u(t,x),
\quad
u(0,x)\,=\,f(x),
\]
with initial data $f \in L^1(\mathbb^n)$.
Denote by $M = \int_f(x)\, dx$ the mass and by $h_t(x) = (4\pi t)^e^{-\frac{\|x\|^2}}$
the heat kernel. Then the following asymptotics are known to hold in $L^p(\mathbb^n)$, for all $1 \leq p \leq \infty$:
$$
\lim_ t^\|u(t,\cdot)-M\,h_t\|_=0.
$$


Analogous heat asymptotics may or may not hold on Riemannian manifolds. Our aim is to discuss noncompact symmetric spaces $G/K$ of arbitrary rank,
generalizing earlier results of J.L. Vzquez on real hyperbolic spaces.
More precisely, we discuss the heat equation related to the Laplace-Beltrami operator and to the distinguished Laplacian. In the first
case, if the data is bi-$K$-invariant, the convergence is true, but unlike the Euclidean case, if this symmetry on initial data is removed, the convergence may fail. In the case of the distinguished Laplacian, we observe phenomena
resembling to the Euclidean setting.

Joint work with J.-Ph. Anker (Université d’ Orléans, France) and H.-W. Zhang (Ghent University, Belgium).