Linear Gaussian-preserving dynamics

Statement

For $d\geq1$ , a real linear map $T_Q(x)=Qx$ preserves standard Gaussian probability $\gamma_d$ if and only if $Q\in O(d)$ . None of these transformations is ergodic on the full Gaussian probability space. An affine map $x\mapsto Qx+b$ preserves $\gamma_d$ exactly when $b=0$ and $Q\in O(d)$ .

Proof

Preserving mean and covariance forces $b=0$ and $QQ^{\mathsf T}=I_d$ . Conversely, an orthogonal map preserves the norm, the Gaussian density, and Lebesgue measure. Every such map fixes the ball $\{\|x\|<1\}$ , whose Gaussian probability lies strictly between zero and one. Hence ergodicity fails.

Koopman interpretation

For linear observables $f_v(x)=v^{\mathsf T}x$ , forward composition gives $U_Qf_v=f_{Q^{\mathsf T}v}$ . This finite-dimensional subrepresentation can have no fixed vectors while the full Koopman representation has one: $r(x)=\|x\|^2-d$ is a nonzero mean-zero fixed vector. The bounded function $e^{-\|x\|^2}$ also witnesses a nontrivial fixed-point algebra.

Thus studying only coordinate observables can miss an invariant nonlinear quantity.