Invariant Distributions via Vector Field Flow

Motivation: Invariant Points

Consider a smooth vector field $X$ on a manifold $M$, generating a flow ${\varphi }_t:M\to M$. A point $p\in M$ is invariant under the flow if ${\varphi }_t(p)=p$ for all $t\in \mathbb{R}$. Equivalently, $p$ is a stationary point where $X(p)=0$.
This concept captures points that remain fixed as the system evolves according to the vector field.
But a point can be identified by its corresponding Dirac delta distribution $T_p$, given by

How would we define the action of the flow on the distribution $T_p$? An idea would be to define

This way, ${\varphi }_t^{\ast }(T_p)$ corresponds to the Dirac delta of the point ${\varphi }_t(p)$, as expected.

Invariant "Smeared Out Points" - Distributions

Now we generalize from points to "smeared out points": distributions.
We say a distribution $T$ is invariant under the flow ${\varphi }_t$ if ${\varphi }_t^{\ast }(T)=T$, that is, if for every test function $\phi$,

or, in other words,

Particular case: $L^1$ Functions

When the distribution $T$ corresponds to an $L^1$ function $f$ (i.e., $⟨T,\phi ⟩={\int }_{\mathrm{\Omega }}f\phi d\lambda$), the invariance condition becomes:

Using the change of variables formula, differentiating $f({\varphi }_t(x))|detD{\varphi }_t(x)|=f(x)$ at $t=0$ gives:

This is equivalently written as ${\mathcal{L}}_X(fd\lambda )=0$, since the Lie derivative of the density $fd\lambda$ expands to $(X(f)+f\cdot \mathrm{div}(X))d\lambda$.

Thus, invariant $L^1$ functions satisfy a steady-state continuity equation $\mathrm{div}(fX)=0$.

Remark (Jacobi last multiplier). A positive function $M$ satisfying $\mathrm{div}(MX)=0$ is called a Jacobi last multiplier of $X$. The condition is identical: an invariant $L^1$ density for the flow of $X$ with respect to Lebesgue measure is exactly a Jacobi last multiplier of $X$. Jacobi multipliers are useful for finding first integrals and invariant volume forms of ODEs.