Almost Rough Paths H^(am,p)(V)

• Almost rough paths H^(am,p)(V) are a functional space that produces additive perturbations in Lyons' p-rough path space, generalizing classical Cameron–Martin-type structures.
• The defined operations, including addition via a sewing map and scalar multiplication, grant a canonical vector space structure while preserving displacement orbits.
• Applications such as pure area perturbations illustrate how this framework enables precise displacement analysis without enlarging the set of attainable rough path perturbations.

Almost Rough Paths $\mathscr H^{\mathrm{am},p}(V)$ are a space of functionals used to produce additive perturbations of $p$-rough paths in Lyons’ rough path space $\Omega^p(V)$. These almost rough paths generalize the Cameron–Martin–type subspace $\mathscr H^p(V)$, allowing for perturbations that need not be strictly multiplicative but satisfy certain regularity and control conditions. The resulting structure characterizes displacement orbits in rough path theory, establishing a canonical vector space of rough path perturbations, and demonstrates that enlarging from $\mathscr H^p(V)$ to the almost-rough path space $\mathscr H^{\mathrm{am},p}(V)$ does not enlarge the set of possible perturbations of a given rough path (Geller et al., 21 Jan 2026).

1. Definition and Structure of Almost Rough Paths

Fix $p \ge 1$, a Banach space $V$, and an interval $J = [S,T]$. Let $T^{(\lfloor p\rfloor)}(V)$ denote the truncated tensor algebra at level $\lfloor p\rfloor$, and define $\triangle_J := \{(s,t) \in J^2 \mid s \le t\}$.

A functional $$H = (H^0, H^1, \ldots, H^{\lfloor p\rfloor}): \triangle_J \to T^{(\lfloor p\rfloor)}(V)$$ is in the almost-$H$-space $\mathscr H^{\mathrm{am},p}_{\phi, \omega}(V)$, for $\phi \in (1-1/p, 1]$ and a super-additive control $\omega:\triangle_J \to [0,\infty)$, if:

• $H^0_{s,t} \equiv 1$,
• There exists $K \ge 0$ such that, for all $j=1, \ldots, \lfloor p\rfloor$ and all $(s,t) \in \triangle_J$,

$\|H^j_{s,t}\| \le K\, \omega(s,t)^\phi.$

The full almost rough path space is:

$$\mathscr H^{\mathrm{am},p}(V) = \bigcup_{\phi,\, \omega} \mathscr H^{\mathrm{am},p}_{\phi, \omega}(V), \quad \text{with } \phi \in (1-1/p,1], \, \omega \text{ a super-additive control}.$$

The subspace of genuine rough paths—denoted $\mathscr H^p(V)$—consists of those $H \in \mathscr H^{\mathrm{am},p}(V)$ which are multiplicative $p$-rough paths, i.e., those for which the multiplicative defect vanishes:

$$\Delta(H)_{s,u,t} := H_{s,u} \otimes H_{u,t} - H_{s,t} \equiv 0$$

for all $s \le u \le t$ in $J$.

2. Operations: Addition $\boxplus$ and Scalar Multiplication $\odot$

Let $\Omega^p(V)$ denote the space of multiplicative functionals—Lyons’ $p$-rough paths—controlled by some control.

• Addition $\boxplus$: Given $X \in \Omega^p(V)$ and $H \in \mathscr H^{\mathrm{am},p}(V)$, define the pointwise, unit-preserving sum

$$(X \oplus H)_{s,t} := (1, X^1_{s,t} + H^1_{s,t}, \ldots, X^{\lfloor p\rfloor}_{s,t} + H^{\lfloor p\rfloor}_{s,t}).$$

The perturbation $X \boxplus H$ is constructed by a sewing operation:

$$X \boxplus H := \mathscr S(X \oplus H) = \mathscr S(X \otimes H) \in \Omega^p(V),$$

where $\mathscr S(\cdot)$ is the sewing map producing a unique genuine rough path from an almost multiplicative path.

• Scalar Multiplication $\odot$: There is a development map $\mathrm{dev}$ and a canonical lift $\mathbb1^{(\cdot)}$ identifying $\mathscr H^p(V)$ with a linear space of increments $\mathfrak I^p(V)$, from which scalar multiplication is defined as:

$$a \odot H := \mathbb1^{\,a\,\mathrm{dev}(H)}, \quad a \in \mathbb R, H \in \mathscr H^p(V).$$

Linearity properties such as

$$(\lambda \odot H_1) \boxplus (\mu \odot H_2) = (\lambda + \mu) \odot (H_1 \boxplus H_2)$$

hold.

3. Algebraic Properties and Theorems

The addition $\boxplus$ and scalar multiplication $\odot$ endow $\mathscr H^p(V)$ with a (vector space) structure which interacts with rough path space $\Omega^p(V)$ by translation. The following properties are key:

• Associativity: For $X \in \Omega^p(V)$ and $H, \tilde H \in \mathscr H^p(V)$,

$$(X\boxplus H)\boxplus \tilde H = X\boxplus (H\boxplus \tilde H).$$

• Trivial Kernel: For $X \in \Omega^p(V)$ and $H \in \mathscr H^p(V)$,

$X\boxplus H = X \iff H = \mathbb1,$

where $\mathbb1$ is the additive zero in $\mathscr H^p(V)$.

• Equality of Displacement Sets:

$$\{X\boxplus H: H\in \mathscr H^p(V)\} = \{X\boxplus H: H\in \mathscr H^{\mathrm{am},p}(V)\}.$$

Thus allowing almost–$H$-paths as perturbations does not enlarge the set of possible displacements of any $X$.

This establishes $\Omega^p(V)$ as a torsor for $\mathscr H^p(V)$ under $\boxplus$, with perturbations exhaustively described by genuine $p$-rough paths.

4. Sewing Lemma and Construction Methodology

Central to the structure is the sewing operation $\mathscr S(\cdot)$. Any map $Z: \triangle_J \to T^{(\lfloor p\rfloor)}(V)$ which is $\theta$–almost multiplicative with $\theta > 1$ and finite $p$-variation admits a unique genuine rough path sewing:

$$\|\mathscr S(Z)^j_{s,t} - Z^j_{s,t}\| = O(\omega(s,t)^\theta).$$

This enables the extension of addition and perturbation to almost rough paths by correcting the multiplicative defect via sewing.

Given an almost-$H$-path $\tilde H \in \mathscr H^{\mathrm{am},p}(V)$, the corresponding genuine $p$-rough path is $H := \mathscr S(\tilde H) \in \mathscr H^p(V)$. For $X \in \Omega^p(V)$,

$X\boxplus \tilde H = X\boxplus H.$

All possible perturbations induced by almost–$H$-paths are thus realized already within $\mathscr H^p(V)$.

5. Examples and Geometric Interpretation

A notable example is the pure area perturbation for $p \in [2,3)$: Let $A^2: \triangle_J \to V^{\otimes 2}$ be any additive map satisfying

$\|A^2_{s,t}\| \le K\,|t-s|^{2/p}$

for some $K>0$. The functional

$H = (1, 0, A^2)$

lies in $\mathscr H^{\mathrm{am},p}(V)$ with $\phi = 2/p > 1-1/p$. Its sewing $\mathscr S(H)$ produces a “pure area” rough path, and $X\boxplus H$ produces the classical area perturbation to $X$. This is beyond the reach of classical Cameron-Martin space, which requires more regular first-level increments, indicating the utility of allowing $\phi < 1$ but $> 1-1/p$.

Geometrically, $\mathscr H^p(V)$ is the minimal linear space of genuine rough paths such that their unit-preserving addition (via sewing) endows $\Omega^p(V)$ with a torsor structure. Enlarging to almost–$H$–paths, despite the relaxed regularity, does not create new displacement orbits for any base rough path.

6. Summary and Significance

The construction of almost rough paths $\mathscr H^{\mathrm{am},p}(V)$ provides a rigorous mechanism for additive perturbations in rough path space, generalizing Cameron-Martin-type structures. The canonical vector space $\mathscr H^p(V)$ enables translation of rough paths under a torsor structure in $\Omega^p(V)$, with $\boxplus$ and $\odot$ supplying the required algebraic properties (associativity, trivial kernel, scalar multiplication). The inclusion of almost–$H$–paths does not expand the attainable perturbation set for any $X\in\Omega^p(V)$, confirming $\mathscr H^p(V)$ as the canonical space of displacements in rough path geometry (Geller et al., 21 Jan 2026).