On a Riemannian manifold, consider the Laplace-Beltrami operator $-\Delta$, and the associated extension problem

\begin{equation} \Delta v+\frac{(1-2\sigma)}{t}\frac{\partial v}{\partial t}+\frac{\partial^2 v}{\partial t^2}=0, \quad 0 <\sigma < 1,\quad t>0, \end{equation}

introduced by Caffarelli and Silvestre on Euclidean space to recover the fractional Laplacian $(-\Delta)^{\sigma}$, as $t$ approaches zero. On hyperbolic spaces this gives rise to a family of convolution operators, including the Poisson operator $e^{-t\sqrt{-\Delta}}$, $t>0$; moreover, the kernels of these operators are subordinated to the heat kernel.

Motivated by Euclidean results for the Poisson semigroup, but also by results on the heat semigroup on Riemannian manifolds and the influence of underlying geometry, we study the long-time asymptotic behavior of solutions to the extension problem for $L^1$ initial data. If the initial datum is radial, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence can break down in the non-radial case. The results extend to all noncompact symmetric spaces of arbitrary rank.

The video of this talk is available on the IISc Math Department channel.