This an easy introduction to the theory of rough paths, a tool used to solve differential equations driven by a very rough function. It is quite surprisingly a fully analytical theory and has been the starting point for solving some of the most important singular spdes.
I will show some of the results that we obtained in a recent work.
Let us fix the dimensions $k,d>0$ and a time horizon $T>0$. Take a path $X:[0,T]\rightarrow\mathbb{R}^d$ and a continuous function $\sigma:\mathbb{R}^k\rightarrow\mathbb{R}^k\otimes(\mathbb{R}^d)^*$. We consider the controlled ODE in the form $$\dot{Z}_t=\sigma(Z_t)\dot{X}_t,$$ for a path $Z:[0,T]\rightarrow\mathbb{R}^d$.
What can we do when the driving path $X$ is continuous but not continuously differentiable?
Think of $X=B$ Brownian Motion.
For $X\in C^1,$ \begin{align*} Z_t-Z_s&=\int_s^t\sigma(Z_u)\dot{X}_u\\ &=\sigma(Z_s)(X_t-X_s)+\int_s^t(\sigma(Z_u)-\sigma(Z_s))\dot{X}_u du. \end{align*} Then, for $\sigma_2(z):=\nabla\sigma(z)\sigma(z)$ locally Lipschitz \begin{align*} \sigma(Z_u)-\sigma(Z_s)&=\sigma_2(Z_s)(X_u-X_s)+\int_s^u(\sigma_2(Z_r)-\sigma_2(Z_s))\dot{X}_r dr\\ &=\sigma_2(Z_s)(X_u-X_s)+O((u-s)^2). \end{align*} We can define the difference equation $$Z_t-Z_s=\sigma(Z_s)(X_t-X_s)+\sigma_2(Z_s)\int_s^t(X_u-X_s)\otimes\dot{X}_r dr+o(t-s).$$
Iterate!Defining $$\sigma_1(z):=\sigma(z), \qquad \sigma_{i+1}(z):=\nabla\sigma_i(z)\sigma(z),$$ and $$\mathbb{X}_{st}^{(0)}:=1 \qquad \mathbb{X}_{st}^{(i+1)}:=\int_s^t\mathbb{X}_{su}^{(i)}\otimes\dot{X}_u du,$$ write $$Z_t-Z_s=\sum_{i=1}^n\sigma_i(Z_s)\mathbb{X}_{st}^{(i)}+o(t-s).$$
$\mathbb{X}^{(i)}$ still depends on $\dot{X}$ $\implies$ need to find approximating functions:
Rough Paths!Fix $X:[0,T]\rightarrow\mathbb{R}^d$ of class $\mathcal{C}^\alpha$ for $\alpha\in(\frac{1}{n+1},\frac{1}{n}]$ and $d\in\mathbb{N}$. We call a $d$-dimensional $\alpha$-rough path over $X$ a family $\mathbb{X}=(\mathbb{X}^{(1)},...,\mathbb{X}^{(n)})$ of functions $\mathbb{X}^{(i)}:[0,T]_{\leq}^2\rightarrow\mathbb{R}^{\otimes i}$ such that, for $0\leq u\leq s\leq t\leq T$, $i\in{1,...,n}$, \begin{align*} \mathbb{X}_{st}^{(1)}&:=X_t-X_s,\\ \mathbb{X}_{st}^{(i)}&=\sum_{j=0}^i\mathbb{X}_{su}^{(i-j)}\otimes\mathbb{X}_{ut}^{(j)},\qquad (Chen's \,relation)\\ |\mathbb{X}_{st}^{(i)}|&\lesssim|t-s|^{i\alpha}. \end{align*}
We can now redefine our difference equation. Let $X\in \mathcal{C}^{\alpha}$ for $\alpha\in(\frac{1}{n+1},\frac{1}{n}]$ and let $\mathbb{X}=(\mathbb{X}^{(1)},\cdots,\mathbb{X}^{(n)})$ be an $\alpha$-rough path over $X$. Then, for $0\leq s\leq t \leq T$, we have the Rough Difference Equation $$\delta Z_{st}=\sum_{i=1}^n\sigma_i(Z_s)\mathbb{X}^{(i)}_{st}+o(t-s).$$
Define $T((\mathbb{R}^d)^*):=\bigoplus_{n\geq 0}((\mathbb{R}^d)^*)^{\otimes n}$. We have
Let us take $\mathbb{X}=(\mathbb{X}^{(1)},...,\mathbb{X}^{(n)})$, a $d$-dimensional $\alpha$-rough path over $X$. $\mathbb{X}$ is called a Geometric Rough Path if $$\langle\mathbb{X},v ⧢ w\rangle=\langle\mathbb{X},v\rangle\langle\mathbb{X},w\rangle \quad \forall v,w\in T((\mathbb{R}^d)^*),$$ where ⧢ is the shuffle product.
When $n=2$ the definition reduces to $(\mathbb{X}^{(2)}_{st})^{ij}+(\mathbb{X}^{(2)}_{st}) ^{ji}=(\mathbb{X}^{(1)}_{st})^i(\mathbb{X}^{(1)}_{st}) ^j.$
\begin{align*} v_1v_2⧢w_1w_2&=v_1v_2w_1w_2+v_1w_1v_2w_2+v_1w_1w_2v_2\\ &+w_1v_1v_2w_2+w_1v_1w_2v_2+w_1w_2v_1v_2. \end{align*}
Analytical theory built to prove well-posedness for $X\in\mathcal{C}^{\alpha}$ Hölder continuous for $\alpha\in(\frac{1}{3},\frac{1}{2}]$ Want to extend to $X\in\mathcal{C}^{\alpha}$ Hölder continuous for $\alpha\in(\frac{1}{n+1},\frac{1}{n}]$ for generic $n$.
There is a relationship between $\sigma_n$ and rooted trees: $$\sigma_n=\sum_{|\tau|=n}\alpha(\tau)\tau,$$ where we sum over the rooted trees of order $n$ and $\alpha$ is the Connes-Moscovici coefficient.
We have $$\sigma_{n+1}=\bullet \curvearrowright \sigma_n,$$ where $\curvearrowright$ is the grafting operator.
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If $[\cdots]$ is the operator that attaches trees to a single root, the example shows that $$\sigma_3=[\sigma,\sigma]+[\sigma_2], \qquad \sigma_4=[\sigma,\sigma,\sigma]+3[\sigma,\sigma_2]+[\sigma_3].$$ This motivates the following theorem.
For $n\geq2$ $$\sigma_n=\sum_{k=1}^{n-1}\sum_{\substack{l_1 < \cdots < l_k\\\sum_im_il_i=n-1}}C(k,l,m)[\sigma_{l_k}^{m_k},\cdots,\sigma_{l_1}^{m_1}],$$ where $$C(k,l,m)=\frac{1}{m_k!\cdots m_1!}\frac{(l_km_k+\cdots+l_1m_1)!}{(l_k!)^{m_k}\cdots(l_1!)^{m_1}}.$$
With the help of the last result we can state the following fundamental theorem.
Let $\mathbb{X}=(\mathbb{X}^{(1)},\cdots,\mathbb{X}^{(n)})$ be a geometric rough path. We have for $z\in\mathbb{R}^k.$ $$\sigma_{n+1}(z)\mathbb{X}^{(n)}=\sum_{k=1}^n\frac{\nabla^k\sigma(z)}{k!}\sum_{\substack{(l_1,\cdots,l_k)\\\sum_il_i=n}}\prod_{i=1}^k\sigma_{l_i}(z)\mathbb{X}^{(l_i)}_{st}.$$
We state the following fundamental bound. Take $R\in C([0,T]_{\leq}^2,\mathbb{R}^k)$ such that $R_{st}=o(t-s)$. Then for any $\eta\in(1,\infty)$, $\tau>0$, $$\|R\|_{\eta,\tau}\leq K_{\eta}\|\delta R\|_{\eta,\tau},$$ where $\delta R_{sut}:=R_{st}-R_{su}-R_{ut}$, \quad $K_{\eta}:=(1-2^{1-\eta})^{-1}$ and $$\|R\|_{\eta,\tau}:=\sup_{0\leq s\leq t\leq T} \mathbb{1}_{\{0 < t-s\leq \tau\}}e^{-\frac{t}{\tau}}\frac{|R_{st}|}{(t-s)^{\eta}}.$$ We call $\|\cdot\|_{\eta,\tau}$ a weighted norm. It is useful to pass from local to global results.
We define $Z^{[m]}:=\delta Z-\sum_{i=1}^{m-1}\sigma_i(Z)\mathbb{X}^{(i)}.$ $Z$ solution $\iff Z^{[n+1]}=o(t-s)$ By the Weighted Sewing Bound $$\|Z^{[n+1]}\|_{(n+1)\alpha,\tau}\leq\|\delta Z^{[n+1]}\|_{(n+1)\alpha,\tau}.$$ \textbf{Idea}: find bounds on $\|\delta Z^{[n+1]}\|_{(n+1)\alpha,\tau}$ to get some a priori estimates.
We have $$\delta Z^{[n+1]}_{sut}=\sum_{i=1}^n\bigg(\sigma_i(Z_u)-\sigma_i(Z_s)-\sum_{j=i+1}^n\sigma_j(Z_s)\mathbb{X}_{su}^{(j-i)}\bigg)\otimes \mathbb{X}_{ut}^{(i)}.$$ $\implies$Need to bound the RHS For $n=2$, $$Z^{[3]}_{sut}=\bigg(\sigma(Z_u)-\sigma(Z_s)-\sigma_2(Z_s)\mathbb{X}_{su}^{(1)}\bigg)\otimes \mathbb{X}_{ut}^{(1)}+\bigg(\sigma_2(Z_u)-\sigma_2(Z_s)\bigg)\otimes \mathbb{X}_{ut}^{(2)}.$$
$$\sigma(Z_u)-\sigma(Z_s)-\sigma_2(Z_s)\mathbb{X}_{su}^{(1)}=\nabla \sigma(Z_s)Z^{[2]}_{su}+\int_0^1[(\nabla\sigma(Z_s+r_1\delta Z_{su})-\nabla\sigma(Z_s))\delta Z_{su}]dr_1,$$ and
\begin{align*} \sigma(Z_u)-\sigma(Z_s)-\sigma_2(Z_s)&\mathbb{X}_{su}^{(1)}=\int_0^1[(\sigma_2(Z_s+r_1\delta Z_{su})-\sigma_2(Z_s))\mathbb{X}_{su}^{(1)}]dr_1\\ &+\int_0^1[\nabla \sigma(Z_s+r_1\delta Z_{su})Z^{[2]}_{su}]dr_1\\ &-\int_0^1\nabla \sigma(Z_s+r_1\delta Z_{su})\bigg(\int_0^{r_1}[\nabla \sigma(Z_s+r_2\delta Z_{su})]\delta Z_{su}\mathbb{X}^{(1)}_{su}dr_2\bigg)dr_1. \end{align*}
What for generic n?Using the Theorem of the Geometric Rough Path decomposition we can write the following identity.
Let us take a geometric rough path $\mathbb{X}=(\mathbb{X}^{(1)},\cdots,\mathbb{X}^{(n)})$. We have \begin{align*} \sigma(Z_u)-&\sigma(Z_s)-\sum_{j=2}^n\sigma_j(Z_s)\mathbb{X}_{su}^{(j-1)}\\ &=\sum_{k=1}^n\sum_{\substack{(l_1,\cdots, l_k)\\\sum l_i=n}}\int_0^1\int_0^{r_1}\cdots\int_0^{r_{k-1}}\nabla^k\sigma(Z_s+r_k\delta Z_{su})Z_{su}^{[l_1]}\otimes\prod_{j=2}^k\sigma_{l_j}(Z_s)\mathbb{X}_{su}^{(l_j)}dr_k\cdots r_1. \end{align*}
The main result of the project is uniqueness of solution for the general rough difference equation.
Let $X$ be a path in $\mathcal{C}^\alpha$, $\alpha\in(\frac{1}{n+1},\frac{1}{n}]$, let $\mathbb{X}=(\mathbb{X}^{(1)},\cdots,\mathbb{X}^{(n)})$ be an $\alpha$-geometric rough path over $X$, let $\sigma$ be of class $C^n$ with $\nabla^n\sigma$ locally Lipshitz. Then, for every $z_0\in\mathbb{R}^k$, the rough difference equation admits a unique solution $Z$ such that $Z_0=z_0$.