Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator

\begin{equation*} \begin{array}{rccc} R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) & \to & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) \\ & f & \mapsto & \int f(x)\, N(\cdot,\sigma^2)(\mathrm{d}x). \end{array} \end{equation*}

I am interested in the eigenspace of $E_1 := \mathrm{kernel(I-R)},$ in particular in the dimension of $E_1.$

Obviously, the indicator function $\mathbb{1}_{\mathbb{R}}: x \mapsto 1$ and the identity $\mathrm{id}_{\mathbb{R}}: x \mapsto x$ are both eigenfunctions to the eigenvalue $1,$ that is, $\mathbb{1}_{\mathbb{R}},\ \mathrm{id}_{\mathbb{R}} \in E_1.$

Are there more linearly independent eigenfunctions?

Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator

\begin{equation*} \begin{array}{rccc} R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) & \to & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) \\ & f & \mapsto & \int f(x)\, N(\cdot,\sigma^2)(\mathrm{d}x). \end{array} \end{equation*}

I am interested in the eigenspace $E_1 := \mathrm{kernel(I-R)},$ in particular in the dimension of $E_1.$

Obviously, the indicator function $\mathbb{1}_{\mathbb{R}}: x \mapsto 1$ and the identity $\mathrm{id}_{\mathbb{R}}: x \mapsto x$ are both eigenfunctions to the eigenvalue $1,$ that is, $\mathbb{1}_{\mathbb{R}},\ \mathrm{id}_{\mathbb{R}} \in E_1.$

Are there more linearly independent eigenfunctions?