  A multi cone domain $\Omega \subseteq \mathbb{R}^n$ is an open, connected set
that resembles a finite collection of cones far away from the origin. We study
the rate of decay in time of the heat kernel $p(t,x,y)$ of a Brownian motion
killed upon exiting $\Omega$, using both probabilistic and analytical
techniques. We find that the decay is polynomial and we characterize
$\lim_{t\to\infty} t^{1+\alpha}p(t,x,y)$ in terms of the Martin boundary of
$\Omega$ at infinity, where $\alpha>0$ depends on the geometry of $\Omega$. We
next derive an analogous result for $t^{\kappa/2}\mathbb{P}_x(T >t)$, with
$\kappa = 1+\alpha - n/2$, where $T$ is the exit time form $\Omega$. Lastly, we
deduce the renormalized Yaglom limit for the process conditioned on survival.
