Heat kernel on affine buildings
Consider finite support probability distributions on an affinebuiding which are spherical : depends only on distance.
Plays the role of heat kernel on symmetric spaces. For such a kernel, one has rather sharp asymptotic estimates (Anker-Ji), used to dtermined the Martin boundary. There remains open questions about the asymptotic behaviour of Green function.
1. Case of
Theorem 1 (Lawler-Limic)
in some ball . Here , is the Fourier transform of probability distribution , , is the unique solution of .
is real analytic on the interior of the convex hull of the support of , but blows up at the boundary. Here is how.
2. Generalization to affine buildings
We have analogous estimates, with extra terms: distance to boundary of Weyl chamber.
Eralier results on on-diagonal behaviour (local limit theorem) by Sawyer, Gerl, Woess (trees), Lindbauer, Voit, Cartwright, Tolli and finally Parkinson. Off-diagonal behaviour by Lalley (trees), Anker, Schapira, Trojan for , general case here.
2.1. Spherical analysis
There is a spherical inversion formula reconstructing the probability distribution in terms of its Gelfand Fourier transform (goes back to Macdonald 1971 in special cases). Macdonald polynomial arise as multiplicative finctions on the algebra of finitely supported invariant operators.