$\overline{\alpha }$	Rotation angle around reference axis $\mathbf{a}={\mathbf{e}}_z=(0,0,1)$	Remark 7
${\sigma }^{p,s}$	UIR of $G=SE\left(3\right)$	Equation (32)
${\overline{\sigma }}^{p,s}$	the action on the quotient corresponding to ${\sigma }^{p,s}$	Definition 7
${\tilde{K}}_t^{\alpha }$	The probability kernel on G	Equation (26)
$K_t^{\alpha }$	The probability kernel on $G/H$	Equation (27)
${\tilde{W}}_{\alpha }$	Solution of the PDE on G	Equation (10)
$W_{\alpha }$	Solution of the PDE on $G/H$	Equation (6)
${\tilde{Q}}_{\alpha }$	Evolution generator of the PDE on G	Equation (11)
$Q_{\alpha }$	Evolution generator of the PDE on $G/H$	Equation (7)
${\mathbf{R}}_{\mathbf{n}}$	Any rotation that maps $\mathbf{a}$ onto $\mathbf{n}$	Remark 2
${\mathbf{R}}_{\mathbf{v},\varphi }$	A counter-clockwise rotation about axis $\mathbf{v}$ with angle $\varphi$	Remark 2
${\mathbf{P}}_t$	Lévy Processes on $G/H$	Definition A1
${\overline{\mathbf{P}}}_t$	Lévy Processes on ${\mathbb{R}}^3\times {\mathbb{R}}^3$	Equation (A4)
$q_{t,\alpha }$	The kernel relating $K_t^{\alpha }$ and $K_t^1$	Equation (77)
$Y^{l,m}$	The ordinary spherical harmonics	Proposition 2
$Y_s^{l,m}$	The modified spherical harmonics according to [4]	Proposition 2
$Y_{\omega }^{l,m}$	The generalized spherical harmonics according to [40]	Definition 10
${\mathsf{\Phi }}_{\omega }^{l,m}$	The spheroidal wave basis function for ${\mathbb{L}}_2\left(S^2\right)$	Definition 11
$(\overline{\alpha },\beta ,\gamma )$	ZYZ Euler angles.	Equation (A12)