Lecture 25. The Lyons’ continuity theorem: Preliminary lemmas

We now turn to the proof of the continuity theorem. We start with several lemmas, which are not difficult but a little technical. The first one is geometrically very intuitive.

Lemma: Let $g_1,g_2 \in \mathbb{G}_N(\mathbb{R}^d)$ such that $d(g_1,g_2) \le \varepsilon$ with $\varepsilon > 0$ and $d(0,g_1), d(0,g_2) \le K$ with $K \ge 0$ . Then, there exists $x_1,x_2 \in C^{1-var}([0,1], \mathbb{R}^d)$ and a constant $C=C(N,K)$ such that $S_N(x)(1)=S_N(x)(0) g_i$ , $i=1,2$ and
$\| x_1\|_{1-var,[0,1]} +\| x_2\|_{1-var,[0,1]} \le C$
and
$\| x_1-x_2 \|_{1-var,[0,1]} \le \varepsilon C.$

Proof:
See the book by Friz-Victoir, page 161 $\square$

The next ingredient is the following estimate.

Lemma: Let $\gamma \ge 1$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x_1,\tilde{x}_1,x_2,\tilde{x}_2 \in C^{1-var}([0,T], \mathbb{R}^d)$ such that
$S_{[\gamma]}(x_1)(T)=S_{[\gamma]}(\tilde{x}_1)(T), \quad S_{[\gamma]}(x_2)(T)=S_{[\gamma]}(\tilde{x}_2)(T).$

Let $y_1,y_2,\tilde{y}_1,\tilde{y}_2$ be the solutions of the equations
$y_i(t)=y_i(0)+\sum_{j=1}^d \int_0^t V_j(y_i(s)) dx_i^j(s), \quad 0 \le t \le T, \quad i=1,2$
and
$\tilde{y}_i(t)=y_i(0)+\sum_{j=1}^d \int_0^t V_j(\tilde{y}_i(s)) d\tilde{x}_i^j(s), \quad 0 \le t \le T, \quad i=1,2.$
If
$\| x_1 \|_{1-var,[0,T]} +\| \tilde{x}_1 \|_{1-var,[0,T]} +\| x_2 \|_{1-var,[0,T]} +\| \tilde{x}_2 \|_{1-var,[0,T]} \le K$
and
$\| x_1-x_2 \|_{1-var, [0,T]}+ \| \tilde{x}_1-\tilde{x}_2 \|_{1-var, [0,T]} \le M,$
then, for some constant depending only on $\gamma$ ,
$\| (y_1(T)-\tilde{y}_1(T))-(y_2(T)-\tilde{y}_2(T)) \|$
$\le C \| y_1(0)-y_2(0) \| (\| V\|_{Lip^\gamma} K)^\gamma e^{C \| V\|_{Lip^\gamma} K}+CM \| V\|_{Lip^\gamma} (\| V\|_{Lip^\gamma} K)^\gamma e^{C \| V\|_{Lip^\gamma} K}$

Proof: Let us first observe that it is enough to prove the result when $\tilde{x}_1=\tilde{x_2}=0$ . Indeed, suppose that we can prove the result in that case. Define then the path $z$ to be the concatenation of $\tilde{x}_1(T-\cdot)$ and $x_1(\cdot)$ reparametrized so that $z:[0,T] \to \mathbb{R}^d$ . It is seen that the solution of the equation
$w(t)=\tilde{y}_1(T)+\sum_{j=1}^d \int_0^t V_j(w(s)) dz_i^j(s), \quad 0 \le t \le T$
satisfies
$w(T)-w(0)=y_1(T)-\tilde{y}_1(T).$
We thus assume that $\tilde{x}_1=\tilde{x_2}=0$ . In that case, from the assumption, we have
$S_{[\gamma]}(x_1)(T)=1, \quad S_{[\gamma]}(x_2)(T)=1.$
Taylor’s expansion gives then, with $n=[\gamma]$ ,
$y_1(T)-y_1(0)$
$=\int_{s \le r_1\le \cdots \le r_n \le t} \sum_{i_1,\cdots,i_n \in \{1,\cdots,d\}} ( V_{i_1}\cdots V_{i_n} \mathbf{I} (y_1(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y_1(s)))dx^{i_1}_{1,r_1} \cdots dx^{i_n}_{1,r_n}.$
and similarly
$y_2(T)-y_2(0)$
$=\int_{s \le r_1\le \cdots \le r_n \le t} \sum_{i_1,\cdots,i_n \in \{1,\cdots,d\}} ( V_{i_1}\cdots V_{i_n} \mathbf{I} (y_2(r_1)) -V_{i_1}\cdots V_{i_n} \mathbf{I} (y_2(s)))dx^{i_1}_{2,r_1} \cdots dx^{i_n}_{2,r_n}.$
The result is then easily obtained by using classical estimates for Riemann-Stieltjes integrals (details can be found page 230 in the book by Friz-Victoir) $\square$

Finally, the last lemma is an easy consequence of Gronwall’s lemma

Lemma: Let $\gamma \ge 1$ . Assume that $V_1, \cdots, V_d$ are $\gamma$ -Lipschitz vector fields in $\mathbb{R}^n$ . Let $x_1,x_2 \in C^{1-var}([0,T], \mathbb{R}^d)$ . Let $y_1,y_2,\tilde{y}_1,\tilde{y}_2$ be the solutions of the equations
$y_i(t)=y_i(0)+\sum_{j=1}^d \int_0^t V_j(y_i(s)) dx_i^j(s), \quad 0 \le t \le T, \quad i=1,2$
and
$\tilde{y}_i(t)=\tilde{y}_i(0)+\sum_{j=1}^d \int_0^t V_j(\tilde{y}_i(s)) dx_i^j(s), \quad 0 \le t \le T, \quad i=1,2.$
If
$\| x_1 \|_{1-var,[0,T]} +\| x_2 \|_{1-var,[0,T]} \le K$
and
$\| x_1-x_2 \|_{1-var, [0,T]} \le M,$
then, for some constant depending only on $\gamma$ ,
$\| (y_1(T) -y_1(0)) - (\tilde{y}_1(T) -\tilde{y}_1(0)) - (y_2(T) -y_2(0)) + (\tilde{y}_2(T) -\tilde{y}_2(0)) \|$
$\le C \| V\|_{Lip^\gamma} K e^{ C \| V\|_{Lip^\gamma} K }\| y_1(0) - \tilde{y}_1(0) - y_2(0) + \tilde{y}_2(0) \| + C \| V\|_{Lip^\gamma} M e^{ C \| V\|_{Lip^\gamma} K }$
$+C \| V\|_{Lip^\gamma} K e^{ C \| V\|_{Lip^\gamma} K } ( \| y_1(0) - \tilde{y}_1(0)\| + \| y_2(0) - \tilde{y}_2(0) \| )^{\min (2,\gamma)-1} \left( \| \tilde{y}^1(0)-\tilde{y}^2(0)\|+ \| V\|_{Lip^\gamma} K \right)$