Lecture 26. Lyons’ continuity theorem: Proof

We now turn to the proof of Lyons’ continuity theorem.

Theorem: Let \gamma > p \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) such that
\| S_{[p]}(x_1) \|^p_{p-var,[0,T]}+\| S_{[p]}(x_2) \|^p_{p-var,[0,T]} \le K
with K \ge 0.
Let y_1,y_2 be the solutions of the equations
y_i(t)=y(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
There exists a constant C depending only on p,\gamma and K such that for 0 \le s \le t \le T,
\|( y_2(t) -y_2(s)) - (y_1(t)-y_1(s))\| \le C  \| V \|_{\text{Lip}^{\gamma}} e^{ C  \| V \|^p_{\text{Lip}^{\gamma}}} d_{p-var,[0,T]} (  S_{[p]}(x_1), S_{[p]}(x_2)) \omega(s,t)^{1/p} ,
where \omega is the control
\omega(s,t)=\left( \frac{d_{p-var,[s,t]} (  S_{[p]}(x_1), S_{[p]}(x_2)) }{d_{p-var; [0,T]}(S_{[p]}(x_1), S_{[p]}(x_2)) } \right)^p+\left( \frac{ \| S_{[p]}(x_1) \|_{p-var,[s,t]}}{\| S_{[p]}(x_1) \|_{p-var,[0,T]} } \right)^p+\left( \frac{ \| S_{[p]}(x_2) \|_{p-var,[s,t]}}{\| S_{[p]}(x_2) \|_{p-var,[0,T]} } \right)^p.

Proof: We may assume p < \gamma <  [p]+1, and for conciseness of notations, we set \varepsilon = d_{p-var,[0,T]} (  S_{[p]}(x_1), S_{[p]}(x_2)). Let
g_i=\Delta_{\frac{1}{\omega(s,t)^{1/p}}} ( S_{[p]}(x_i)(s)^{-1}  S_{[p]}(x_i)(t)), \quad i=1,2.

We have,
d(g_1,g_2)  =\frac{1}{\omega(s,t)^{1/p}} d(S_{[p]}(x_1)(s)^{-1}  S_{[p]}(x_1)(t) , S_{[p]}(x_2)(s)^{-1}  S_{[p]}(x_2)(t))
\le \frac{1}{\omega(s,t)^{1/p}}d_{p-var,[s,t]} (  S_{[p]}(x_1), S_{[p]}(x_2))
\le \varepsilon
and, in the same way,
d(0,g_i) =\frac{1}{\omega(s,t)^{1/p}}  d (  S_{[p]}(x_i)(s), S_{[p]}(x_i)(t))
=\frac{1}{\omega(s,t)^{1/p}}  \| S_{[p]} (x_i) \|_{p-var,[s,t]} \le K.

Therefore, there exist x^{s,t}_1,x^{s,t}_2 \in C^{1-var}([s,t], \mathbb{R}^d) and a constant C_1=C_1([p],K) such that
S_{[p]}(x^{s,t}_i)(s)^{-1}  S_{[p]}(x^{s,t}_i)(t)  =S_{[p]}(x_i)(s)^{-1}  S_{[p]}(x_i)(t) , i=1,2
\| x^{s,t}_1\|_{1-var,[s,t]} +\| x^{s,t}_2\|_{1-var,[s,t]} \le C_1\omega(s,t)^{1/p}
\| x^{s,t}_1-x^{s,t}_2 \|_{1-var,[s,t]} \le \varepsilon C_1\omega(s,t)^{1/p}.
We define then x_i^{s,t,u} as the concatenation of x_i^{s,t} and x_i^{t,u}. As in the proof of Davie’s lemma, we denote by y_i^{s,t} the solution of the equation
y^{s,t}_i(r)=y_i(s)+\sum_{j=1}^d \int_s^r V_j(y^{s,t}_i(v)) dx_i^j(v), \quad s \le r \le t, \quad i=1,2
and consider the functionals
\Gamma^i_{s,t}=(y_i(t)-y_i(s))-(y^{s,t}_i(t) -y^{s,t}_i(s))= y_i(t)-y^{s,t}_i(t),
\bar{\Gamma}_{s,t} =\Gamma^1_{s,t}-\Gamma^2_{s,t}
From the proof of Davie’s estimate, it is seen that
\| \Gamma^i_{s,t} \| \le \frac{1}{2} C_2 \left(  \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} \right)^{[p]+1},
and thus
\| \bar{\Gamma}_{s,t} \| \le  C_2 \left(  \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} \right)^{[p]+1}.
On the other hand, by estimating
\bar{\Gamma}_{s,u}- \bar{\Gamma}_{s,t}- \bar{\Gamma}_{t,u},
as in the proof of Davie’s lemma, that is by inserting y_i^{s,t,u} which is the solution of the equation driven by the concatenation of x_i^{s,t} and x_i^{t,u}, and then by using the two lemmas of the previous lecture, we obtain the estimate
\| \bar{\Gamma}_{s,u} \|
\le \|  \bar{\Gamma}_{s,t}  \| e^{C_3 \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p}} + \| \bar{\Gamma}_{t,u} \|+C_3( \| y_1-y_2\|_{\infty, [s,t]} +\varepsilon)\left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p} \right)^\gamma  e^{C_3 \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p}}
\le \left( \| \bar{\Gamma}_{s,t} \| + \| \bar{\Gamma}_{t,u} \|+C_3( \| y_1-y_2\|_{\infty, [s,t]} +\varepsilon) \left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p} \right)^{\gamma} \right) e^{C_3 \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p}}.
It remains to bound \| y_1-y_2\|_{\infty, [s,t]}. For this let us observe that
\| (y_1(t)-y_2(t))-(y_1(s)-y_2(s))-\bar{\Gamma}_{s,t} \|  =\|( y^{s,t}_1(t) -y^{s,t}_2(t) )-( y^{s,t}_1(s) -y^{s,t}_2(s) )\|.
\|( y^{s,t}_1(t) -y^{s,t}_2(t) )-( y^{s,t}_1(s) -y^{s,t}_2(s) )\| can then be estimated by using classical estimates on differential equations driven by bounded variation paths. This gives,
\|( y^{s,t}_1(t) -y^{s,t}_2(t) )-( y^{s,t}_1(s) -y^{s,t}_2(s) )\| \le C_4 \left( \| y_1(s) -y_2(s) \| +\varepsilon \right) \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} e^{ C_4\| V \|_{\text{Lip}^{\gamma}}  \omega(s,t)^{1/p}}.
By denoting z=y_1-y_2, we can summarize the two above estimates as follows:
\| \bar{\Gamma}_{s,u} \| \le  \left( \| \bar{\Gamma}_{s,t} \| + \| \bar{\Gamma}_{t,u} \|+C_3( \| z\|_{\infty, [s,t]} +\varepsilon) \left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p} \right)^{\gamma} \right) e^{C_3 \| V \|_{\text{Lip}^{\gamma}} \omega(s,u)^{1/p}}
\| z(t)-z(s) -\bar{\Gamma}_{s,t} \| \le C_4 \left( \| z \|_{\infty,[0,s]} +\varepsilon \right) \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} e^{ C_4\| V \|_{\text{Lip}^{\gamma}}  \omega(s,t)^{1/p}}.
From a lemma already used in the proof of Davie’s estimate, the first estimate implies
\| \bar{\Gamma}_{s,t} \| \le  C_5 \left( \varepsilon + \| z \|_{\infty,[0,t]} \right) \left( \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p} \right)^{\gamma} e^{ C_5 \| V \|_{\text{Lip}^{\gamma}} \omega(s,t)^{1/p}}.
Using now the second estimate we obtain that for any interval [a,b] included in [0,T],
\sup_{s,t \in [a,b]} \| z(t)-z(s) \| \le C_6 (\varepsilon + \| z \|_{\infty, [0,b]} )  \| V \|_{\text{Lip}^{\gamma}} \omega(a,b)^{1/p} e^{C_6  \| V \|_{\text{Lip}^{\gamma}} \omega(a,b)^{1/p}}.
Using the fact that z(0)=0 and picking a subdivision 0 = \tau_0 \le \tau_1 \le \cdots \le \tau_N \le T such that
C_6  \| V \|_{\text{Lip}^{\gamma}} e^{C_6  \| V \|_{\text{Lip}^{\gamma}} } \omega(\tau_i , \tau_{i+1} )^{1/p} \le 1/2
we see that it implies
\| z \|_{\infty,[0,T]} \le C_7 \varepsilon  e^{C_7  \| V \|^p_{\text{Lip}^{\gamma}} }.
Coming back to the estimate
\sup_{s,t \in [a,b]} \| z(t)-z(s) \| \le C_6 (\varepsilon + \| z \|_{\infty, [0,b]} )  \| V \|_{\text{Lip}^{\gamma}} \omega(a,b)^{1/p} e^{C_6  \| V \|_{\text{Lip}^{\gamma}} \omega(a,b)^{1/p}}.
concludes the proof \square

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