  Let $\lnlap$ be the logarithmic Laplacian operator with Fourier symbol $2\ln
|\zeta|$, we study the expression of the diffusion kernel which is associated
to the equation $$\partial_tu+ \lnlap u=0 \ \ {\rm in}\ \, (0,\tfrac N2) \times
\R^N,\quad\quad u(0,\cdot)=0\ \ {\rm in}\ \, \R^N\setminus \{0\}.$$ We apply
our results to give a classification of the solutions of $$\left\{
\arraycolsep=1pt \begin{array}{lll} \displaystyle \partial_tu+ \lnlap u=0\quad
\ &{\rm in}\ \ (0,T)\times \R^N\\[2.5mm]
  \phantom{ \lnlap \ \, } \displaystyle u(0,\cdot)=f\quad \ &{\rm{in}}\ \ \R^N
\end{array} \right. $$ and obtain an expression of the fundamental solution of
the associated stationary equation in $\R^N$, and of the fundamental solution
in a bounded domain, i.e. $$\lnlap u=k\delta_0\quad {\rm in}\ \
\cD'(\Omega)\quad \text{such that }\,u=0\quad {\rm in}\ \ \R^N\setminus\Omega.
$$
