## Lecture 1. An overview of rough paths theory

Let us consider a differential equation that writes
$y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s),$
where the $V_i$‘s are vector fields on $\mathbb{R}^n$ and where the driving signal $x(t)=(x^1(t), \cdots, x^d(t))$ is a continuous bounded variation path. If the vector fields are Lipschitz continuous then, for any fixed initial condition, there is a unique solution $y(t)$ to the equation. We can see this solution $y$ as a function of the driving signal $x$. It is an important question to understand for which topology, this function is continuous.

A simple example shows that the topology of uniform convergence is not the correct one here. Indeed, let us consider the differential equation
$y_1(t) = x_1(t)$
$y_2(t)= x_2(t)$
$y_3(t)= -\int_0^t y_2(s) dx_1(s) +\int_0^t y_1(s) dx_2(s)$
where
$x_1(t)=\frac{1}{n} \cos (n^2 t ), \quad x_2(t)=\frac{1}{n} \sin (n^2 t).$
A straightforward computation shows that $y_3(t)=t$. When $n \to \infty$, $(x_1,x_2)$ converges uniformly to 0 whereas, of course, $(y_1,y_2,y_3)$ does not converge to 0. In this framework , a correct topology is given by the topology of convergence in 1-variation on compact sets. To fix the ideas, let us work on the interval $[0,1]$. The distance in 1-variation between two continuous bounded variation paths $x,\tilde{x}:[0,1] \to \mathbb{R}^d$ is given by
$\delta_1(x,\tilde{x})=\|x(0)-\tilde{x}(0) \|+ \sup_{\pi} \sum_{k=0}^{n-1} \| (x(t_{i+1})-\tilde{x}(t_{i+1})) -(x(t_i)-\tilde{x}(t_i)) \|,$
where the supremum is taken over all the subdivisions $\pi =\{ 0 \leq t_1 \leq \cdots \leq t_n \leq 1 \}.$

It is then a fact that is going to be proved in this class that if the $V_i$‘s are bounded and if $x^n:[0,1] \to \mathbb{R}^d$ is a sequence of bounded variation paths that converges in 1-variation to a continuous path $x$ with bounded variation, then the solutions of the differential equations
$y^n(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y^n(s)) dx^{i,n}(s),$
converge in 1-variation to the solution of
$y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s).$
This type of continuity result suggests to use a topology in $p$-variation, $p \ge 1$, to try to extend the map $x \to y$ to a larger class of driving signals $x$. More precisely, for $p \geq 1$, let us denote by $\Omega^p (\mathbb{R}^d)$ the closure of the set of continuous with bounded variation paths $x:[0,1] \rightarrow \mathbb{R}^d$ with respect to the distance in $p$-variation which is given by
$\delta_p(x,\tilde{x})=\left( \|x(0)-\tilde{x}(0) \|^p+ \sup_{\pi} \sum_{k=0}^{n-1} \| (x(t_{i+1})-\tilde{x}(t_{i+1})) -(x(t_i)-\tilde{x}(t_i)) \|^p \right)^{1/p}.$
We will then prove the following result:

Proposition: Let $p < 2$. If $x^n:[0,1] \to \mathbb{R}^d$ is a sequence of bounded variation paths that converges in $p$-variation to a path $x \in \Omega^p (\mathbb{R}^d)$, then the solutions of the differential equations
$y^n(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y^n(s)) dx^{i,n}(s),$
converge in $p$-variation to some $y \in \Omega^p (\mathbb{R}^d)$. Moreover $y$ is the solution of the differential equation
$y(t)=y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s),$
where the integrals are understood in the sense of Young’s integration.

The value $p=2$ is really a treshold: The result is simply false for $p=2$. The main idea of the rough paths theory is to introduce a much stronger topology than the convergence in $p$-variation. This topology, that we now explain, is related to the continuity of lifts of paths in free nilpotent Lie groups.

Let $\mathbb{G}_N (\mathbb{R}^d)$ be the free $N$-step nilpotent Lie group with $d$ generators $X_1,\cdots,X_d$. If $x:[0,1] \rightarrow \mathbb{R}^d$ is continuous with bounded variation, the solution $x^*$ of the equation
$x^*(t)=\sum_{i=1}^d \int_0^t X_i (x^*(s)) dx^i(s),$
is called the lift of $x$ in $\mathbb{G}_N (\mathbb{R}^d)$. For $p \geq 1$, let us
denote $\Omega^p \mathbb{G}_N (\mathbb{R}^d)$ the closure of the set of lifted paths $x^*:[0,1] \rightarrow \mathbb{G}_N (\mathbb{R}^d)$ with respect to the distance in $p$-variation which is given by
$\delta^N_p (x^*,y^*) =\sup_{\pi} \left( \sum_{i=1}^{n-1} d_N \left( y_{t_i}^* (x_{t_i}^*)^{-1}, y_{t_{i+1}}^* (x_{t_{i+1}}^*)^{-1} \right)^p \right)^{\frac{1}{p}},$
where $d_N$ denotes the Carnot-Caratheodory distance on the group $\mathbb{G}_N (\mathbb{R}^d)$. This is a distance that will be explained in details later. Its main property is that it is homogeneous with respect to the natural dilation of $\mathbb{G}_N (\mathbb{R}^d)$.

Consider now the map $\mathcal{I}$ which associates to a continuous with bounded variation path
$x: [0,1] \rightarrow \mathbb{R}^d$ the continuous path with bounded variation $y : [0,1] \rightarrow \mathbb{R}^d$ that solves the ordinary differential equation
$y(t) =y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s).$
It is clear that there exists a unique map $\mathcal{I}^{*}$ from the set of continuous with bounded variation lifted paths $[0,1] \rightarrow \mathbb{G}_N (\mathbb{R}^d)$ onto the set of continuous with bounded variation lifted paths $[0,1] \rightarrow \mathbb{G}_N (\mathbb{R}^n)$ which makes the following diagram commutative
$\begin{array}{lll} & \mathcal{I}^{*} & \\ x^{*} & \longrightarrow & y^{*} \\ \uparrow & & \uparrow \\ x & \longrightarrow & y \\ & \mathcal{I} & \end{array}.$
The fundamental theorem of Lyons is the following:

Theorem: If $N \geq [p]$, then in the topology of $\delta^N_p$-variation, there exists a continuous extension of $\mathcal{I}^{*}$ from $\Omega^p \mathbb{G}_N (\mathbb{R}^d)$ into $\Omega^p \mathbb{G}_N (\mathbb{R}^n)$.

In particular, we can now give a sense to differential equations driven by some continuous paths with finite $p$-variation, for any $p \ge 1$. Indeed, let $x:[0,1] \to \mathbb{R}^d$ which is continuous with a fnite $p$-variation and assume that there exists $x^* \in \Omega^p \mathbb{G}_N (\mathbb{R}^d)$ whose projection onto $\mathbb{R}^d$ is $x$. The projection onto $\mathbb{R}^d$ of $\mathcal{I}^{*}(x^*)$ is then understood as being a solution of
$y(t) =y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dx^i(s).$

An important example of application is given by the case where the driving signal is a Brownian motion $(B(t))_{t \ge 0}$. Brownian motion has a $p$-finite variation for any $p > 2$ and, as we will see, admits a canonical lift in $\Omega^p \mathbb{G}_2 (\mathbb{R}^d)$. As a conclusion, we can consider in the rough paths sense, solutions to the equation
$y(t) =y_0+\sum_{i=1}^d \int_0^t V_i (y(s)) dB^i(s).$

It turns out that this notion of solution is exactly equivalent to solutions that are obtained by using the Stratonovitch integration theory. Therefore, the theory of stochastic differential equations appears as a very special case of the rough paths theory !

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