Set of integral curves of a vector field

Question

Let $V \colon [0,1]\times \mathbb R^d \to \mathbb R^d$ be a Borel vector field which is globally bounded, $V \in L^\infty$.

I am looking for a reference for the following result (which I suppose it is true and already been proved somewhere).

The set of integral curves of $V$, i.e.

$\displaystyle \mathcal C :=\big\{ \gamma \in C([0,1];\mathbb R^d): \gamma(s)-\gamma(t) - \int_t^s V(\tau,\gamma(\tau)) d\tau = 0, \quad \forall 0\le t \le s \le 1\big\}$

is a Borel set in the space of continuous path $C([0,1];\mathbb R^d)$ endowed with the topology inherited from the $\sup$ norm.

[The integral curves are defined for all times being the vector field bounded]

It is immediate to show that for continuous vector fields the statement holds (the set $\mathcal C$ is closed) but I do not find anything for the general case.

Answer 1

Actually this follows plainly as a consequence of an important classical result in descriptive set theory, namely, the set of all Borel real-valued functions on a metric space is the smallest class containing the continuous functions, and closed under point-wise convergence, that is, the Baire class. (And, of course, nothing changes, if you consider $\mathbb{R}^d$-valued maps instead, or also, maps valued in the unit ball of $\mathbb{R}^d$).

A suitable reference is e.g. A. S. Kechris' Classical Descriptive Set Theory, ch. 11, or this paper by R. W. Hansell available on-line, that contains all ingredients. (Alternatively, you may replace "Borel vector field" with "Baire vector field" in the statement).

For any bounded Borel map $V:[0,1]\times\mathbb{R}^d\to\mathbb{R}^d$ we can consider the operator $F_V$ on the Banach space $E:=C^0([0,1],\mathbb{R}^d), \|\cdot\|_\infty$ defined by $F_V( \gamma)(t)=\int_0^tV(\tau,\gamma(\tau))d\tau$ for any $\gamma\in E$ and $t\in[0,1]$. Two simple facts easily follow by bounded convergence:

• If $V$ is continuous on $[0,1]\times\mathbb{R}^d$, then $F_V$ is continuous on $E$. Indeed, if $\gamma_n\to\gamma$ in $E$ $$\big\|F_V(\gamma )-F_V(\gamma_n)\big\|_\infty\le\int_0^1\big\|V(\tau,\gamma(\tau))-V(\tau,\gamma_n(\tau))\big\| d\tau=o(1).$$
• If $V_n$ converges point-wise to $V$ on $[0,1]\times\mathbb{R}^d$, then $F_{V_n}$ converges point-wise to $F_V$ on $E$. Indeed, if $V_n\to V$ point-wise, then for any $\gamma\in E$ $$\big\|F_V(\gamma )-F_{V_n}(\gamma)\big\|_\infty\le\int_0^1\big\|V(\tau,\gamma(\tau))-V_n(\tau,\gamma(\tau))\big\| d\tau=o(1).$$

As a consequence, the set of all bounded Borel maps $V:[0,1]\times\mathbb{R}^d\to\mathbb{R}^d$ such that $F_V$ is a Borel map on $E$ contains all bounded continuous functions and it is closed under point-wise convergence. Therefore it is the whole class of bounded Borel maps.

To conclude, let's denote ${\mathbb 1}$ and $\mathrm{ev_0}$ the identity map on $E$, respectively, the evaluation at $0$ (seen as a bounded, linear rank-one operator on $E$). The set $$\mathcal{C}=\big\{\gamma\in E : \gamma(t)-F_V(\gamma)(t)-\gamma(0)=0,\; \forall\, t\in[0,1] \big\},$$ which is the zero-set of the Borel map ${\mathbb 1}-F_V-\mathrm{ev_0}$, is therefore a Borel set.

Rmk: From the above argument also follows by induction that, more precisely, if $V$ is in the Baire class $\alpha$ for a countable ordinal $\alpha$, then so is $F_A$ on $E$, so that $\mathcal{C}$ is in the corresponding Borel set hierarchy.