Let $G$ be a compact matrix Lie group with Haar measure $\mu$. Then the heat kernel $\rho: G\times (0,\infty) \rightarrow \mathbb{R}$ satisfies
(1) $\rho(g_1g_2,t)=\rho(g_2g_1,t)$ for all positive $t$,
(2) $\int_{G} \rho(g_1g,t) \rho(g^{-1}g_2,t)d\mu(g)=\rho(g_1g_2,2t)$ for all $g_1,g_2 \in G$ and all $t$.
I'm interested whether there exist (non-trivial) one-parameter families of functions on $G$ satisfying (1) and a generalization of the convolution semigroup property (2) which involves more than two integrands.
More specifically, given a positive integer $k$, do there exist a construction of (non-constant) smooth functions $\rho_k: G\times (0,\infty)\rightarrow \mathbb{R}$ and $ \phi_k:(0,\infty) \rightarrow (0,\infty)$ such that $\rho_1=\rho$ (the heat kernel on $G$), $\phi_1(t)=2t$, and $\rho_k$ satisfies the condition (1) above as well as
$$\int_{G} \rho_k(g_1g,t) \rho_k(g^{-1}g_2,t)\cdots \rho_k(g_{2k-1}g,t) \rho_k(g^{-1}g_{2k},t)d\mu(g)=\rho_k(g_1g_2\cdots g_{2k-1} g_{2k},\phi_k(t))$$ for all $g_1,\ldots, g_{2k} \in G$ and all $t$ ?
