---
title: Controlled Rough Paths
url: https://www.emergentmind.com/topics/controlled-rough-paths
type: topic
---

# Controlled Rough Paths

Searching arXiv for recent and foundational papers on controlled rough paths to ground the article in the supplied literature.
Controlled rough paths are paths whose increments admit a first-order expansion against a reference rough path, with a remainder of strictly higher regularity. In the classical formulation, a path \(y\) is controlled by a rough path \(X\) if
\[
y_t-y_s=y'_sX_{s,t}+R^y_{s,t},
\]
where \(y'\) is the Gubinelli derivative and \(R^y\) is a higher-order remainder [2007.06295]. This perspective turns rough integration into a sewing problem and provides Banach spaces in which rough differential equations (RDEs) can be solved by fixed-point arguments [2210.01334, 2007.06295]. Subsequent work has extended the notion across geometric, branched, planar-branched, Sobolev, càdlàg, and mean-field settings, and has clarified that controlled rough paths are not tied to one combinatorial model but can be formulated in a general Hopf-algebraic language [2509.23148].

## 1. Classical definition and analytic role

In the standard level-2 regime, with roughness \(2\le p<3\) or Hölder exponent \(\alpha\in(1/3,1/2]\), a path \(y:[0,T]\to\mathbb R^m\) is controlled by \(X\) if there exists a path \(y':[0,T]\to \mathcal L(\mathbb R^n,\mathbb R^m)\) such that
\[
y_t-y_s = y'_s X_{s,t} + R^y_{s,t}.
\]
Here \(y'_s\) is the Gubinelli derivative and \(R^y\) is a higher-order remainder [2007.06295]. In the Hölder formulation used in the averaging literature, a controlled path over \(\mathbf X=(X^1,X^2)\in \mathscr C^\alpha(V)\) is a triple \((Y,Y',Y^\#)\) with
\[
(Y,Y',Y^\#)\in C^\beta([a,b],W)\times C^\beta([a,b],L(V,W))\times C_2^{2\beta}([a,b],W),
\]
satisfying
\[
Y_{s,t}=Y'_sX^1_{s,t}+Y^\#_{s,t} \qquad (s,t)\in \Delta_{[a,b]}.
\tag{3.2}
\]
This is the formulation used in [2210.01334].

The corresponding controlled-path spaces are Banach spaces. In [2210.01334] the space is denoted
\[
\mathcal Q^\beta_{\mathbf X}([a,b],W),
\]
with seminorm
\[
\|(Y,Y',Y^\#)\|_{\mathcal Q^\beta_{\mathbf X},[a,b]} = |Y'|_{\beta,[a,b]}+|Y^\#|_{2\beta,[a,b]},
\]
and full norm
\[
|Y_a|_W+|Y_a'|_{L(V,W)}+\|(Y,Y',Y^\#)\|_{\mathcal Q^\beta_{\mathbf X},[a,b]}.
\]
In [2007.06295], the space of controlled pairs over \(x\) is denoted \(\mathcal D_x^{2\alpha}(I)\), with norm built from \(y'\) and \(R^y\). These Banach structures are the functional-analytic core of the theory: rough integration preserves controlledness, composition by smooth maps preserves controlledness, and the RDE solution map is realized as a fixed point [2007.06295, 2210.01334].

A recurrent identification in RDEs is
\[
y'_t=g(y_t)
\]
for equations of the form
\[
dy_t=f(y_t)\,dt+g(y_t)\,dx_t.
\]
The controlled expansion of the solution then takes the explicit form
\[
y_{s,t} = f(y_s)(t-s)+g(y_s)x_{s,t}+Dg(y_s)g(y_s)\mathbb X_{s,t}+O(|t-s|^{3\alpha}),
\tag{3.46}
\]
which shows that the solution is controlled by the driver with Gubinelli derivative \(g(y)\) [2007.06295].

## 2. Rough integration, composition, and fixed-point calculus

Controlled rough paths are designed so that rough integrals can be reconstructed from local expansions. In the level-2 Hölder setting of [2210.01334], if \((Y,Y')\in \mathcal Q^\beta_{\mathbf X}([a,b],L(V,W))\), one defines
\[
J_{s,t}=Y_sX^1_{s,t}+Y'_sX^2_{s,t}.
\]
Using Chen’s relation and the controlled expansion, the defect satisfies
\[
J_{s,u}+J_{u,t}-J_{s,t} = Y^\#_{s,u}X^1_{u,t}+Y'_{s,u}X^2_{u,t}.
\tag{3.5}
\]
Since \(3\beta>1\), the sewing lemma applies, and the rough integral is
\[
\int_s^t Y_u\,d\mathbf X_u:=\lim_{|P|\to 0} J_{s,t}(P).
\tag{3.6}
\]
Moreover,
\[
\left(\int_a^\cdot Y_u\,d\mathbf X_u,\;Y\right)\in \mathcal Q^\beta_{\mathbf X}([a,b],W),
\tag{3.7}
\]
with estimate
\[
\left| \int_s^t Y_u\,d\mathbf X_u-\bigl(Y_sX^1_{s,t}+Y'_sX^2_{s,t}\bigr) \right|_W \le K_\beta (t-s)^{3\beta} \Big( |Y^\#|_{2\beta,[a,b]}|X^1|_{\beta,[a,b]} + |Y'|_{\beta,[a,b]}|X^2|_{2\beta,[a,b]} \Big),
\tag{3.8}
\]
where
\[
K_\beta=2^{3\beta}\zeta(3\beta).
\]
This is the main sewing-type estimate in that framework [2210.01334].

In the \(p\)-variation/Hölder formulation of [2007.06295], Gubinelli’s rough integral is
\[
\int_s^t y_u\,dx_u := \lim_{|\Pi|\to 0}\sum_{[u,v]\in\Pi} \big( y_u\,x_{u,v}+y'_u\,\mathbb X_{u,v}\big),
\]
with estimate
\[
\Big| \int_s^t y_u\,dx_u -y_s x_{s,t}-y'_s\mathbb X_{s,t} \Big| \le C_\alpha |t-s|^{3\alpha} \Big( \|x\|_{\alpha,[s,t]}\|R^y\|_{2\alpha,[s,t]^2} +\|\mathbb X\|_{2\alpha,[s,t]^2}\|y'\|_{\alpha,[s,t]} \Big).
\tag{2.7}
\]
A \(p\)-variation version is also given there [2007.06295].

Composition is equally central. In [2210.01334], if \((Y,Y')\in \mathcal Q^\beta_{\mathbf X}(W)\) and \(g\in C^2(W,W')\), then \((g(Y),g(Y)')\in \mathcal Q^\beta_{\mathbf X}(W')\) with
\[
g(Y)'_t=Vg(Y_t)Y'_t,
\]
and
\[
g(Y_t)-g(Y_s)=Vg(Y_s)Y'_sX^1_{s,t}+g(Y)^\#_{s,t}.
\tag{3.3}
\]
This is the classical controlled-path chain rule. In [2007.06295], if \(y\) is controlled with \(y'=g(y)\), then \(g(y)\) is controlled with
\[
[g(y)]'_s=Dg(y_s)g(y_s).
\]

These two operations—composition and integration—are the ingredients needed for the fixed-point method. In [2210.01334], RDEs are written as
\[
Y_t=\xi+\int_0^t f(Y_s,\vartheta_s)\,ds+\int_0^t \sigma(Y_s)\,d\mathbf X_s, \qquad Y'_t=\sigma(Y_t),
\tag{3.11}
\]
and are solved by contraction on controlled-path spaces. Global existence and uniqueness hold under the stated boundedness and Lipschitz assumptions, first for bounded globally Lipschitz drift and then for bounded locally Lipschitz drift [2210.01334]. In [2007.06295], linear diffusion coefficients are treated directly by a controlled-path fixed point, while nonlinear diffusion with unbounded drift is handled via a Doss–Sussmann transform after first analyzing the driftless rough equation [2007.06295].

## 3. Higher-order and combinatorial formulations

For \(\beta\in(0,1/2)\), the classical level-2 picture is no longer sufficient once \(\beta\le 1/3\). In [2009.13084], the driving signal is a \(\beta\)-Hölder geometric rough path with
\[
0<\beta<\frac12,\qquad N=[1/\beta],\qquad \frac{1}{N+1}<\alpha<\beta.
\]
A controlled rough path over \(X\) is then a hierarchy
\[
Y=(Y^0,Y^1,\dots,Y^{N-1}),
\]
where for \(0\le i\le N-2\),
\[
R^{Y,i}_{s,t} := Y^i_t-Y^i_s-\sum_{j=1}^{N-1-i} Y^{i+j}_s X^j_{s,t},
\]
and for \(i=N-1\),
\[
R^{Y,N-1}_{s,t}:=Y^{N-1}_t-Y^{N-1}_s.
\]
Controlledness means
\[
\sup_{0\le s<t\le T}\frac{|R^{Y,i}_{s,t}|}{|t-s|^{(N-i)\alpha}}<\infty.
\]
The resulting Banach space is denoted \(D^\alpha_X(U)\) [2009.13084].

The central result of [2009.13084] is a higher-order composition theorem. If \(F=(F^0,\dots,F^N)\) is \(\gamma\)-Lipschitz in the Stein sense with \(\gamma\in(N,N+1]\), then \(F(Y)\) is again controlled, with transformed coefficients
\[
Z^r_s := \sum_{j=1}^{N-1} \frac1{j!} F^j(Y^0_s) \!\!\!\sum_{i_1+\cdots+i_j=r}\!\!\! (Y^{i_1}_s\boxtimes\cdots\boxtimes Y^{i_j}_s)\circ \delta_j\big|_{V^{\otimes r}}.
\tag{4.2}
\]
The proof relies on the coproduct structure on the truncated tensor algebra and the characterization of geometric rough paths by
\[
\delta_k(\xi)= \sum_{0\le l_1+\cdots+l_k\le N} \xi^{l_1}\boxtimes\cdots\boxtimes \xi^{l_k}
\]
for \(\xi\in G^{(N)}(V)\) [2009.13084]. This establishes closure under composition below the threshold \(\beta=1/3\), identified there as the main missing ingredient in extending controlled rough path theory to arbitrary \(\beta\in(0,1/2)\).

A distinct combinatorial development appears in [2112.08034], which treats weakly geometric rough paths of bounded \(1<p\)-variation without smooth approximation. There a controlled path \(H\) is an element of
\[
\mathscr D_{\mathbf X}(U),
\]
encoded as a graded element in \(\mathcal L(T^{\lfloor p\rfloor-1}(V),U)\). In coordinates,
\[
H^k_{\mathbf i;t} \approx_{\lfloor p\rfloor-|\mathbf i|} \sum_{|\mathbf j|=0}^{\lfloor p\rfloor-1-|\mathbf i|} H^k_{(\mathbf j,\mathbf i);s}\,\mathbf X^{\mathbf j}_{st}.
\tag{2}
\]
The paper’s main controlled-path contribution is an explicit lift of a controlled path to a genuine weakly geometric rough path:
\[
(\upharpoonleft_{\mathbf X}H)^m_{st} = \langle H_s^{\boxtimes m},\, \widetilde\Delta_{\overline\shuffle}^{\,m}\mathbf X_{st}\rangle, \qquad 2\le m\le \lfloor p\rfloor,
\tag{5}
\]
or in coordinates,
\[
(\upharpoonleft_{\mathbf X}H)^{(k_1,\ldots,k_m)}_{st} = \sum_{\substack{|\mathbf i^1|,\ldots,|\mathbf i^m|\ge 1\\ |\mathbf i^1|+\cdots+|\mathbf i^m|\le \lfloor p\rfloor\\ \mathbf j\in \overline{Sh}(\mathbf i^1,\ldots,\mathbf i^m)}} H^{k_1}_{\mathbf i^1;s}\cdots H^{k_m}_{\mathbf i^m;s}\, \mathbf X^{\mathbf j}_{st}.
\tag{7}
\]
This yields the genuine lift
\[
(\uparrow_{\mathbf X}H)_{st}
=
\lim_{n\to\infty}
\Big(\bigotimes_{[u,v]\in \pi_n}(\upharpoonleft_{\mathbf X}H)_{uv}\Big).
\tag{8}
\]
That construction supports associativity of the rough integral, the adjunction between pushforwards and pullbacks, and change-of-variables formulas for RDEs on manifolds [2112.08034].

A further geometric development, but now at the level of the family of all controlled-path spaces, is given in [2203.05946]. For branched rough paths of arbitrary order, the fibers \(\mathscr D_X^{\alpha,\beta}\) assemble into a continuous field of Banach spaces over the base rough path space. The total space
\[
E^{\alpha,\beta} := \bigsqcup_{X\in \mathrm{BRP}^\alpha}\mathscr D_X^{\alpha,\beta}
\]
carries an intrinsic tube topology, and in the geometric case
\[
E_g^{\alpha,\beta}
:=
\bigsqcup_{X\in \mathrm{GRP}^\alpha}\mathscr D_X^{\alpha,\beta}
\]
is Polish [2203.05946]. This shows that controlled rough paths are naturally organized as Banach fibers varying with the driver, rather than as a single fixed Banach space.

## 4. Hopf-algebraic unification

A major unifying development is the formulation of controlled rough paths on a general class of combinatorial Hopf algebras [2509.23148]. The paper works with an \(N\)-truncated connected graded Hopf algebra
\[
\mathcal H=\bigoplus_{n=0}^N \mathcal H_n,
\]
with finite-dimensional homogeneous components, truncated characters, convolution
\[
\langle \xi\star \eta,x\rangle = \langle \xi\otimes \eta,\Delta x\rangle,
\]
and Chen relation
\[
\mathbf X_{st}=\mathbf X_{su}\star \mathbf X_{ut}.
\]
This framework simultaneously encompasses the shuffle Hopf algebra, the Butcher–Connes–Kreimer Hopf algebra, and the Munthe-Kaas–Wright Hopf algebra [2509.23148].

In this language, an \(\mathbf X\)-controlled rough path is a path
\[
\mathbf Z=(\mathbf Z^1,\dots,\mathbf Z^e):[0,T]\to \mathcal H^e
\]
such that for homogeneous \(h\in\mathcal H^\star\),
\[
\langle h,R_{\mathbf X}(\mathbf Z)_{st}\rangle := \langle h,\mathbf Z_t\rangle - \langle \mathbf X_{st}\star h,\mathbf Z_s\rangle
\tag{CRP}
\]
satisfies
\[
\|\langle h,R_{\mathbf X}(\mathbf Z)_{st}\rangle\|_{\mathbb R^e} \le C \|h\|_{\mathcal H^\star}\,\omega(s,t)^{(N-|h|)\gamma}.
\]
The degree-zero component recovers the underlying path:
\[
Z_t:=\langle \mathbf 1^\star,\mathbf Z_t\rangle.
\]
This is the exact Hopf-algebraic controlled expansion [2509.23148].

The associated Banach norm is defined through the two-parameter remainder:
\[
\|R\|_{\omega,\gamma} := \sup_{j=0,\dots,N}\sup_{(s,t)\in \Delta_T} \|\pi_j(R_{st})\|_{\mathcal H^e}\, \omega(s,t)^{-(N-j)\gamma},
\]
and
\[
\|\mathbf Z\| = \|\mathbf Z_0\|_{\mathcal H^e} + \|R_{\mathbf X}(\mathbf Z)\|_{\omega,\gamma}.
\]
This makes the space \(\mathcal V_{\mathbf X}^{[0,T]}\) a Banach space [2509.23148].

Smooth functions are treated through a reduced coproduct. For \(\varphi\in C^N(\mathbb R^e,\mathbb R^e)\),
\[
\langle \mathbf 1^\star,\varphi(\mathbf Z)_s\rangle=\varphi(Z_s),
\]
and for \(h\in \ker \epsilon_\star\),
\[
\langle h,\varphi(\mathbf Z)_s\rangle = \sum_{n=1}^N \frac{D^n\varphi(Z_s)}{n!} \left\langle \Delta_\star^{\mathrm{red}(n-1)}(h),\, \mathbf Z_s^{\otimes n} \right\rangle.
\tag{Phi}
\]
Rough integration is encoded by degree-\(+1\) cocycles \(L_i\). If
\[
\Xi^i_{st}:=\langle \mathbf X_{st},L_i(\mathbf Z_s)\rangle,
\]
then sewing yields the rough integral, and the lifted integral is
\[
\int_0^t \mathbf Z_r\,dX_r^i := L_i(\mathbf Z_t)+\left(\int_0^t Z_r\,dX_r^i\right)\mathbf 1.
\tag{IntLift}
\]
This allows local and global well-posedness of the lifted integral equation
\[
\mathbf Z_t = \mathbf Z_0+\sum_{i=1}^d \int_0^t \varphi_i(\mathbf Z)_s\,dX_s^i.
\tag{LIVP}
\]
A universal limit theorem is also proved in this general setting [2509.23148].

This suggests that controlled rough paths are best regarded not as a construction tied specifically to words, signatures, or rooted trees, but as an analytic structure determined by three ingredients: a graded commutative Hopf algebra, characters satisfying Chen’s relation, and degree-raising cocycles \(L_i\) governing rough integration [2509.23148].

## 5. Variants: Sobolev, càdlàg, and controlled drivers

The classical theory has been extended in several non-equivalent directions.

In the Sobolev setting, [2006.03322] defines a Sobolev rough path as a path
\[
\mathbf X:[0,1]\to G^{[1/\alpha]}(\mathbb R^d)
\]
with
\[
\|\mathbf X\|_{W_p^\alpha} := \left( \int_{[0,1]^2} \frac{d_{cc}(\mathbf X_s,\mathbf X_t)^p}{|t-s|^{\alpha p+1}}\,ds\,dt \right)^{1/p}<\infty,
\qquad \alpha>\frac1p.
\]
A controlled Sobolev path above \(\mathbf X\in W_p^\alpha([0,1];G^2(\mathbb R^d))\) is a pair \((Y,Y')\) with
\[
Y\in W_p^\alpha([0,1];\mathbb R^n),\qquad Y'\in W_p^\alpha([0,1];\mathcal L(\mathbb R^d;\mathbb R^n)),
\]
and
\[
R^Y_{s,t}:=Y_{s,t}-Y_s'X_{s,t}
\]
satisfying
\[
R^Y\in V_p^{2\alpha}(\Delta;\mathbb R^n)\cap W_{p/2}^{2\alpha}(\Delta;\mathbb R^n).
\]
This stronger remainder condition is needed to ensure that rough integration preserves the same Sobolev regularity. The corresponding Itô–Lyons map is locally Lipschitz continuous with respect to the initial value, vector field, and driver, measured by the Sobolev rough path distance \(\rho_{W_p^\alpha}\) together with the mixed distance \(\rho_{\alpha,p}\) [2006.03322].

In the càdlàg setting, [2403.17573] extends controlled rough paths to level-2 càdlàg rough paths \(\mathbf X=(X,\mathbb X)\) with \(p\in(2,3)\). A controlled path is again defined by
\[
Y_{s,t}=Y'_sX_{s,t}+R^Y_{s,t},
\]
but now in càdlàg \(p\)-variation spaces:
\[
(Y,Y')\in V_X^p([0,T];E).
\]
The forward rough integral is
\[
\int_s^t Y_r\,dX_r := \lim_{|\mathcal P|\to 0} \sum_{[u,v]\in\mathcal P} \big(Y_uX_{u,v}+Y'_u\mathbb X_{u,v}\big),
\]
with remainder estimate
\[
\bigg| \int_s^t Y_r\,dX_r - Y_sX_{s,t} - Y'_s\mathbb X_{s,t} \bigg| \le C\Big( \|R^Y\|_{p,[s,t)}\|X\|_{p,[s,t]} + \|Y'\|_{p,[s,t)}\|\mathbb X\|_{p,[s,t]} \Big).
\]
The central innovation there is that the coefficient can be a non-anticipative functional on the Banach space of controlled paths:
\[
Y_t = y_t+\int_0^t F_s(Y)\,dX_s,\qquad Y'=y'+F(Y).
\]
This gives a general rough functional differential equation framework that includes classical RDEs, controlled RDEs, and delay equations [2403.17573].

A third extension concerns RDEs driven not by the reference rough path \(\mathbf X\) itself but by another path \(\mathbf Z\) controlled by \(\mathbf X\). In [2603.09158], in the level-2 regime \(\alpha\in(1/3,1/2]\), one considers controlled paths
\[
\mathbf Y=(Y,Y'),\qquad \mathbf Z=(Z,Z')
\]
with
\[
Y_{s,t}=Y'_sX_{s,t}+R^{0,\mathbf Y}_{s,t},\qquad Z_{s,t}=Z'_sX_{s,t}+R^{0,\mathbf Z}_{s,t}.
\]
The rough integral is defined by
\[
\int_s^t Y_r\,d\mathbf Z_r
:=
\lim_{|\pi|\to 0}
\sum_{[u,v]\in\pi}
\big(Y_uZ_{u,v}+Y'_uZ'_u\,\mathbb X_{u,v}\big),
\]
and an a priori estimate is proved:
\[
\left| \int_s^tY_r\,d\mathbf Z_r - \big(Y_sZ_{s,t}+Y'_sZ'_s\mathbb X_{s,t}\big) \right|
\le
C_\alpha
(|Y'_0|+\|\mathbf Y\|_{\mathbf X;\alpha})
(|Z'_0|+\|\mathbf Z\|_{\mathbf X;\alpha})
(1+\|\mathbf X\|_\alpha)
|t-s|^{3\alpha}.
\]
This supports a universal limit theorem for equations of the form
\[
dY_t=F(Y_t)\,d\mathbf Z_t
\]
driven by controlled rough paths [2603.09158].

## 6. Applications, extensions, and interpretation

Controlled rough paths serve as the deterministic backbone of several applied and structural developments.

In averaging for slow–fast rough systems, [2210.01334] uses controlled-path spaces to adapt Khas’minskii’s time-discretizing method to rough differential equations. The slow component \(X^\varepsilon\) is shown to be a controlled rough path over the slow rough driver \(\mathbf B\), and the perturbation/stability estimate is formulated directly in controlled norms. The strong averaging theorem proves
\[
\lim_{\varepsilon\downarrow 0}\mathbb E\bigl[|X^\varepsilon-X|_\beta^p\bigr]=0
\]
under the stated assumptions [2210.01334].

In slow–fast large deviations under mixed fractional Brownian motion, [2401.00673] uses the continuity of the Itô–Lyons map in controlled rough path topology as the deterministic input for a variational weak-convergence argument. The slow equation is posed as a controlled rough differential equation, and the large deviation principle is proved on \(C^{\beta\text{-hld}}([0,T],\mathbb R^m)\) with the stated good rate function [2401.00673].

In reflected rough differential equations, the supplied data for [1604.02562] does not include the paper text, so no paper-specific theorem statements can be extracted. The supplied material only supports the general background statement that controlled paths can be used to prove existence for reflected rough differential equations under weaker assumptions than an earlier Euler-approximation approach, but the exact assumptions and formulas are unavailable from the provided notice [1604.02562].

In stochastic control, [1303.7160] formulates control problems for rough differential equations driven by geometric rough paths. The paper primarily uses Lyons-style RDEs rather than Gubinelli’s controlled paths, but it explicitly remarks that if one allows the diffusion coefficient to depend on the control, one may need controls “controlled by \(\eta\) in the Gubinelli sense.” A plausible implication is that controlled rough path regularity can serve as a compatibility condition when the control enters the rough channel [1303.7160].

A more abstract mean-field extension is the theory of random controlled rough paths in [2203.01185]. There the jet of a controlled object is indexed by Lions forests in a coupled Hopf algebra, and rough integration and composition by smooth functions on the Wasserstein space are both shown to be closed on this class. This suggests that the controlled rough path paradigm is flexible enough to accommodate law-dependent expansions, though the combinatorics become substantially more elaborate [2203.01185].

A common misconception is that controlled rough paths are only a level-2 device or only a convenient notation for \(Y'_sX_{s,t}\). The higher-order theory in [2009.13084], the arbitrary-\(p\) combinatorial lift in [2112.08034], and the general Hopf-algebraic formulation in [2509.23148] show otherwise. Another common misconception is that controlled rough paths are inseparable from the shuffle algebra of signatures. The Hopf-algebraic and branched formulations show that geometric, branched, and planarly branched theories share the same analytic mechanism once the correct combinatorial algebra is identified [2509.23148, 2203.05946].

Taken together, these developments present controlled rough paths as a general analytic principle: a path is controlled when its local behavior can be transported by the increment structure of a rough driver, with a remainder regular enough for sewing. In the classical geometric case this is the Gubinelli decomposition; in branched and Hopf-algebraic settings it becomes a character-based transport identity; in Sobolev and càdlàg settings the ambient topology changes; and in mean-field settings the jet is indexed by Lions forests. The unifying content is the same: rough integration, composition, stability, and RDE well-posedness are consequences of a controlled expansion with a higher-order remainder [2509.23148].

Source: https://www.emergentmind.com/topics/controlled-rough-paths