Infinitesimal generator of a Koopman flow

Let $\Phi^t$ be a probability-preserving flow such that $U_tf=f\circ\Phi^t$ is strongly continuous on $L^2$ . Its Koopman generator is

Lf=\lim_{t\to0}\frac{U_tf-f}{t},\qquad D(L)=\{f:\text{this limit exists in }L^2\}.

By Stone's theorem, $L$ is densely defined and skew-adjoint, $L=iA$ for a self-adjoint $A$ , and $U_t=e^{tL}=e^{itA}$ .

Smooth dynamics

If a smooth vector field $V$ generates a flow on a compact smooth manifold with an invariant probability measure, smooth observables belong to this domain and

Lf=df(V)=V\cdot\nabla f

in local coordinates. Consequently $u(t,x)=f(\Phi^t x)$ solves the linear equation $\partial_tu=Lu$ . The inverse-pullback representation has generator $-L$ .

This is the flow case of the derived pullback representation. For general Lie-group actions, the smooth-vector reading path explains the common domain, its Fréchet topology, and the relation to distribution vectors.

Domain matters

For noncompact spaces or nonsmooth observables the displayed derivative need not belong to $L^2$ . Formal differentiation alone does not specify the domain or establish skew-adjointness.

Search Knowlpedia

Infinitesimal generator of a Koopman flow

Continuity and the hypotheses

Smoothness and the infinitesimal action

Generalized vectors and nuclearity

The concrete Heisenberg model

The stronger operator-norm framework