Global Lipschitz ODE flows and the C1 density formula

Fix a compact parameter set $K$. Let $C(t)={\int }_0^t{L}_K(s)ds$. By [F4] and [F12], $C$ is continuous and nondecreasing on $[0,T]$, and [F6] makes it uniformly continuous. Choose a finite partition $0=t_0<\cdots <t_N=T$ with mesh small enough that $C(t_{i+1})-C(t_i)={\int }_{t_i}^{t_{i+1}}{L}_K(s)ds\le 1\mathrm{/}2$ on each cell. On a cell $J=[t_i,t_{i+1}]$, with initial value $z_i$, consider the closed supremum-norm ball of radius $R=2(1+\mathrm{\mid }{z}_i\mathrm{\mid }+{\int }_J{a}_K)$ about the constant curve $z_i$. For $z$ in that ball define $\mathcal{T}z(t)=z_i+{\int }_{t_i}^{t}b(s,z(s),y)ds\mathrm{.}$ The integrand is continuous by [F13], so $\mathcal{T}z$ is continuous.

Use [F14] throughout to identify the continuous integrands' Riemann and Lebesgue integrals. Put $q_J={\int }_J{L}_K\le 1\mathrm{/}2$. The triangle inequality and spatial Lipschitz bound with $x^{\mathrm{\prime }}=0$ give $\mathrm{\mid }b(s,z(s),y)\mathrm{\mid }\le a_K(s)+L_K(s)\mathrm{\mid }z(s)\mathrm{\mid }$ almost everywhere. Hence $\mathrm{\parallel }\mathcal{T}z-z_i{\mathrm{\parallel }}_{\mathrm{\infty }}\le {\int }_J{a}_K+q_J(\mathrm{\mid }{z}_i\mathrm{\mid }+R)\le R,$ so the ball maps into itself. For two curves, [F3] and the Lipschitz bound give $\mathrm{\parallel }\mathcal{T}z-\mathcal{T}w{\mathrm{\parallel }}_{\mathrm{\infty }}\le q_J\mathrm{\parallel }z-w{\mathrm{\parallel }}_{\mathrm{\infty }}$. The ball is nonempty and complete by [F1], so [F2] gives a unique fixed point, which solves the integral equation and hence the ODE on $J$ by [F5]. Applying this construction successively on the finite partition, with each cell's endpoint as the next initial value, gives a solution on all of $[0,T]$.

For solutions $x(t),x^{\mathrm{\prime }}(t)$ with the same $y$ and initial values $x_0,x_0^{\mathrm{\prime }}$, subtract their integral equations and use [F3] to obtain $\mathrm{\mid }x(t)-x^{\mathrm{\prime }}(t)\mathrm{\mid }\le \mathrm{\mid }{x}_0-x_0^{\mathrm{\prime }}\mathrm{\mid }+{\int }_0^t{L}_K(s)\mathrm{\mid }x(s)-x^{\mathrm{\prime }}(s)\mathrm{\mid }ds\mathrm{.}$ The forward clause of the integrable-coefficient Grönwall lemma gives the factor $\exp ({\int }_0^t{L}_K)$, proving uniqueness and the forward Lipschitz bound. Applying the same existence and uniqueness construction to the reversed drift $\bar{b}(s,z,y)=-b(T-s,z,y)$ gives a backward solution from any terminal value; uniqueness shows that its time-zero map is the inverse of $T_y$ and the same estimate gives the inverse Lipschitz bound.

From the integral equation and [F3], $\mathrm{\mid }{x}_y(t;x_0)\mathrm{\mid }\le \mathrm{\mid }{x}_0\mathrm{\mid }+{\int }_0^T{a}_K+{\int }_0^t{L}_K(s)\mathrm{\mid }{x}_y(s;x_0)\mathrm{\mid }ds$. The integrable Grönwall estimate therefore gives $\mathrm{\mid }{x}_y(t;x_0)\mathrm{\mid }\le (\mathrm{\mid }{x}_0\mathrm{\mid }+{\int }_0^T{a}_K)\exp ({\int }_0^T{L}_K)$, uniformly in $t$ and for $x_0,y$ in compact sets. Thus all such trajectories lie in a common compact state ball. The drift is uniformly continuous on the product of this ball, $[0,T]$, and $K$ by [F6]. Choose a compact $K$ containing a convergent sequence $y_n$ and its limit. For $(x_n,y_n)\to (x,y)$, subtract the integral equations along $x_{y_n}(\cdot ;x_n)$ and $x_y(\cdot ;x)$; the result is bounded by $\mathrm{\mid }{x}_n-x\mathrm{\mid }+T\omega (\mathrm{\mid }{y}_n-y\mathrm{\mid })+{\int }_0^t{L}_K(s)\mathrm{\mid }{x}_{y_n}(s;x_n)-x_y(s;x)\mathrm{\mid }ds$, where $\omega (r)\to 0$ is a modulus from [F6]. Grönwall proves uniform convergence of the paths. Repeating for the reversed drift proves continuity of the inverse map in initial value and parameter.

In the C${}^1$ clause, fix $y$ and an initial state $x_0$. Steps 2.1 and 4.1 give common existence and continuous dependence for nearby initial states on $[0,T]$. Their trajectories and the line segments joining corresponding states lie in a common compact state ball, so $[0,T]$ times that ball is a compact state tube in the field domain. On this tube $D_{x}b$ is bounded and uniformly continuous. The difference-quotient argument in [F7] therefore applies on $[0,T]$: the map $x_0\mapsto x_y(t;x_0)$ is $C^1$ for every $t$, and its derivative is the solution of the variational equation. Apply the same argument to the reversed drift $\bar{b}(s,x,y)=-b(T-s,x,y)$; its time-$T$ map is $T_y^{-1}$ by uniqueness in step 3.1. The two maps are consequently $C^1$ inverses, so $T_y$ is a $C^1$ diffeomorphism.

Write $B(t)=D_{x}b(t,x_y(t;x_0),y)$ and $A(t)=A_y(t,x_0)$. Step 5.1 applied on each restricted interval $[0,t]$ shows that the time-$t$ flow is a $C^1$ diffeomorphism, so $A(t)$ is invertible for every $t$ (and $A(0)=I_d$). The path $B$ is continuous. By [F5], $G(t):={\int }_0^t\mathrm{tr}B(s)ds$ is differentiable on $(0,T)$ with $G^{\mathrm{\prime }}(t)=\mathrm{tr}B(t)$ and has the corresponding one-sided endpoint derivatives. For $0<t<T$, the total-derivative chain rule [F8] and the invertible-matrix formula in [F9], applied to the variational equation $A^{\mathrm{\prime }}=BA$, give $\frac{d}{dt}\det A(t)=D\det (A(t))[A^{\mathrm{\prime }}(t)]=\det A(t)\mathrm{tr}(A(t{)}^{-1}B(t)A(t))=\det A(t)\mathrm{tr}B(t),$ using cyclicity of trace; continuity extends this identity to the endpoints. By the scalar product and chain rules in [F11], $e^{-G(t)}\det A(t)$ has derivative zero on $(0,T)$, so the mean-value theorem makes it constant on $[0,T]$. Its value at $t=0$ is $1$; hence $\det A(t)=\exp (G(t))$, the asserted Liouville formula.

If $\eta (dx)=q(x)dx$, then for every Borel $E\subseteq {\mathbb{R}}^d$, $T_y{}_{\mathrm{\#}}\eta (E)={\int }_{T_y^{-1}(E)}q(x)dx\mathrm{.}$ Apply [F10] to the diffeomorphism $T_y^{-1}$ and the nonnegative measurable integrand $x\mapsto {\mathbf{1}}_{T_y^{-1}(E)}(x)q(x)$. This rewrites the last integral as ${\int }_{E}q(T_y^{-1}(z))\mathrm{\mid }\det D{T}_y^{-1}(z)\mathrm{\mid }dz$, which proves the stated density formula. The first part of the Statement used no C${}^1$ assumption; therefore global Lipschitzness alone yields only the bi-Lipschitz flow conclusion.