Lecture 22. Davie’s estimate (1)

In this Lecture, we prove one of the fundamental estimates of rough paths theory. This estimate is due to Davie. It provides a basic estimate for the solution of the differential equation
y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s)
in terms of the p-variation of the lift of x in the free Carnot group of step [p].

We first introduce the somehow minimal regularity requirement on the vector fields V_i‘s to study rough differential equations.

Definition. A vector field V on \mathbb{R}^n is called \gamma-Lipschitz if it is [\gamma] times continuously differentiable and there exists a constant M \ge 0 such that the supremum norm of its kth derivatives k=0, \cdots, [\gamma] and the \gamma-[\gamma] Holder norm of its [\gamma]th derivative are bounded by M. The smallest M that satisfies the above condition is the \gamma-Lipschitz norm of V and will be denoted \| V \|_{\text{Lip}^\gamma}.

The fundamental estimate by Davie is the following;

Definition: Let \gamma > p \ge 1. Assume that V_1, \cdots, V_d are (\gamma-1)-Lipschitz vector fields in \mathbb{R}^n. Let x \in C^{1-var}([0,T], \mathbb{R}^d). Let y be the solution of the equation
y(t)=y(0)+\sum_{i=1}^d \int_0^t V_i(y(s)) dx^i(s), \quad 0 \le t \le T.
There exists a constant C depending only on p and \gamma such that for every 0 \le s < t \le T,
\| y \|_{p-var, [s,t]} \le C \left(\| V \|_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|_{p-var,[s,t]} +\| V \|^p_{\text{Lip}^{\gamma-1}} \| S_{[p]} (x) \|^p_{p-var,[s,t]}   \right),
where S_{[p]} (x) is the lift of x in \mathbb{G}_{[p]}(\mathbb{R}^d).

We start with two preliminary lemmas, the first one being interesting in itself.

Lemma: Let \gamma > 1. Assume that V_1, \cdots, V_d are (\gamma-1)-Lipschitz vector fields in \mathbb{R}^n. Let x \in C^{1-var}([s,t], \mathbb{R}^d). Let y be the solution of the equation
y(v)=y(s)+\sum_{i=1}^d \int_s^v V_i(y(u)) dx^i(u), \quad s \le v \le t.
There exists a constant C depending only on \gamma such that,
\left\| y(t)-y(s)-\sum_{k=1}^{[\gamma]} \sum_{i_1,\cdots,i_k \in \{1,\cdots,d\}} V_{i_1}\cdots V_{i_k} \mathbf{I} (y(s)) \int_{\Delta^k[s,t]} dx^{i_1,\cdots,i_k} \right\|
\le  C \left(\| V \|_{\text{Lip}^{\gamma-1}} \int_s^t \| dx_r\|  \right)^\gamma,
where \mathbf{I} is the identity map.

Proof: For notational simplicity, we denote n=[\gamma]. An iterative use of the change of variable formula leads to
y(t)-y(s)-\sum_{k=1}^{n} \sum_{i_1,\cdots,i_k \in \{1,\cdots,d\}} V_{i_1}\cdots V_{i_k} \mathbf{I} (y(s)) \int_{\Delta^k[s,t]} dx^{i_1,\cdots,i_k}
=\int_{s < r_1 < \cdots < r_n < t}  \sum_{i_1,\cdots,i_n \in \{1,\cdots,d\}} ( V_{i_1}\cdots V_{i_n} \mathbf{I} (y(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y(s)))dx^{i_1}_{r_1} \cdots dx^{i_n}_{r_n}.
Since V_1, \cdots, V_d are (\gamma-1)-Lipschitz, we deduce that
\| V_{i_1}\cdots V_{i_n} \mathbf{I} (y(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y(s))\| \le \| V \|_{\text{Lip}^{\gamma-1}}^n \|y(r_1)-y(s)\|^{\gamma-n}.
Since,
\|y(r_1)-y(s)\|\le \| V \|_{\text{Lip}^{\gamma-1}} \int_s^{r_1} \| dx_r\|,
we deduce that
\| V_{i_1}\cdots V_{i_n} \mathbf{I} (y(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y(s))\| \le \| V \|_{\text{Lip}^{\gamma-1}}^\gamma \left( \int_s^{t} \| dx_r\|  \right)^{\gamma-n}.
The result follows then easily by plugging this estimate into the integral
\int_{s < r_1 < \cdots < r_n < t}   ( V_{i_1}\cdots V_{i_n} \mathbf{I} (y(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y(s)))dx^{i_1}_{r_1} \cdots dx^{i_n}_{r_n} \square

The second lemma is an analogue of a result already used in previous lectures (Young-Loeve estimate, estimates on iterated integrals).

Lemma: Let \Gamma: \{ 0 \le s \le t \le T \} \to \mathbb{R}^n. Let us assume that:

  • There exists a control \tilde{\omega} such that
    \lim_{r \to 0} \sup_{(s,t)\in \Gamma, \tilde{\omega}(s,t) \le r } \frac{\| \Gamma_{s,t} \|}{r}=0;
  • There exists a control \omega and \theta > 1, \xi > 0, K \ge 0, \alpha > 0 such that for 0 \le s \le t \le u\le T,
    \| \Gamma_{s,u} \| \le \left( \| \Gamma_{s,t} \|+ \| \Gamma_{t,u} \| +\xi \omega(s,u)^\theta\right)\exp( K \omega(s,t)^\alpha).

Then, for all 0 \le s < t \le  T,
\| \Gamma_{s,t} \| \le \frac{\xi}{1-2^{1-\theta}} \omega(s,t)^\theta \exp\left( \frac{2K}{1-2^{-\alpha}}  \omega(s,u)^\alpha\right).

Proof:
For \varepsilon > 0, consider then the control
\omega_\varepsilon (s,t)= \omega(s,t) +\varepsilon \tilde{\omega}(s,t)
Define now
\Psi(r)= \sup_{s,u, \omega_\varepsilon (s,u)\le r}  \| \Gamma_{s,u}\|.
If s,u is such that \omega_\varepsilon (s,u) \le r, we can find a t such that \omega_\varepsilon(s,t) \le \frac{1}{2} \omega_\varepsilon(s,u), \omega_\varepsilon(t,u) \le \frac{1}{2} \omega_\varepsilon(s,u). Indeed, the continuity of \omega_\varepsilon forces the existence of a t such that \omega_\varepsilon(s,t)=\omega_\varepsilon(t,u) . We obtain therefore
\| \Gamma_{s,u}\|\le \left( 2 \Psi(r/2) + \xi r^\theta \right) \exp (K r^\alpha) ,
which implies by maximization,
\Psi(r)\le  \left( 2 \Psi(r/2) + \xi r^\theta \right) \exp (K r^\alpha).
We have \lim_{r \to 0} \frac{\Psi (r)}{r} =0 and an iteration easily gives
\Psi (r) \le \frac{\xi}{1-2^{1-\theta}}r^\theta \exp\left( \frac{2K}{1-2^{-\alpha}}  r^\alpha\right).
We deduce
\| \Gamma_{s,t} \| \le \frac{\xi}{1-2^{1-\theta}} \omega_\varepsilon (s,t)^\theta \exp\left( \frac{2K}{1-2^{-\alpha}}  \omega_\varepsilon (s,u)^\alpha\right),
and the result follows by letting \varepsilon \to 0 \square

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