Nonlinear Young Integral

• Nonlinear Young integral is a generalized pathwise integration tool that extends classical linear integrals to handle low-regularity and multidimensional integrators.
• The methodology employs sophisticated techniques like the sewing lemma, fractional calculus, and basis expansions to ensure convergence and precise error control.
• It underpins advancements in rough differential equations, BSDEs, and SPDEs, offering rigorous solutions for systems driven by irregular signals.

The nonlinear Young integral is a generalized pathwise integration theory that extends the classical Young and Riemann–Stieltjes integrals to settings involving low regularity, nonlinear dependence, and multi-parameter or space–time integrators. It forms the analytic core for numerous modern developments in the analysis of rough differential systems, backward stochastic differential equations (BSDEs) with non-smooth drives, rough partial differential equations (PDEs), and the construction of geometric rough paths.

1. Classical Young Integration and Its Extension

Classically, the Young integral $\int_a^b f(s)\,dg(s)$ exists if $f$ and $g$ are Hölder continuous with exponents $\alpha$ and $\beta$ such that $\alpha+\beta > 1$. This framework is linear in both integrand and integrator.

The nonlinear Young integral generalizes this in multiple directions:

• The integrand may depend nonlinearly on the integrator (or on time and space collectively).
• The integrators can be multivariate functions (e.g. space–time, or rough fields), and the notion of “integration” is often defined via limits of non-uniform partitions (sewing lemma), fractional calculus, or expansion in basis functions.
• The regularity requirements may be adapted to more subtle scales, involving moduli of continuity or multi-exponent Hölder/Besov spaces.

The result is a highly flexible, pathwise integration tool that accommodates non-linear interactions, multi-scale oscillations, and rough behavior.

2. Analytical Formulation: Definitions, Existence, and Regularity

Pathwise Construction via the Sewing Lemma

For a collection of functions $y^1, y^2$, or more generally a function $\eta: [0,T]\times \mathbb{R}^d \to \mathbb{R}^m$, and a path $x: [0,T]\to \mathbb{R}^d$, the nonlinear Young integral is defined using discrete increments. For intervals $[s,t]$: $A(s,t) = y_s [ \eta(t,x_s) - \eta(s,x_s) ],$ with the requirement (from sewing lemma theory) that the triple increments $\delta A(s,u,t) = A(s,t) - A(s,u) - A(u,t)$ are controlled by a superadditive function (a “control”) elevated to an appropriate power.

If $y$ has finite $p$-variation, $x$ has finite $q$-variation, and $\eta$ is jointly Hölder (in $t$, $x$), precise conditions on the exponents and auxiliary moduli (e.g., functions $m_1,m_2$ as in fine-scale estimates) yield convergence of the Riemann-type sums and well-definition of $\int y_r\,\eta(dr,x_r)$. In higher dimensions or multi-parameter settings, extensions involve charges on BV-sets and expansion in Haar/Faber–Schauder bases.

Fractional Calculus Representations

In many works, notably (Hu et al., 2015), the nonlinear Young integral is calculated via fractional derivatives: $$\int_a^b W(dt, p(t)) = (-1)^\alpha \int_a^b (D^α_{a+} W_{b-}(t, p(t)))(D^{1-α}_{b-} p_{b-})(t) dt,$$ where $D^{α}_{a+}$, $D^{1-α}_{b-}$ are (left/right) Riemann–Liouville fractional derivatives, provided the Hölder exponents of $W$ in $t$ and of $p$ ensure $T+\eta>1$.

Finer Scale Regularity and Critical Regimes

Critical regimes, such as the endpoint $1/p + 1/q = 1$ (for paths of finite $p$- and $q$-variation), require additional finer scale control via moduli $m_1, m_2$: $\int_0^1 \frac{m_1(t)m_2(t)}{t}dt < \infty$ and local estimates of the form

$$\|y^1(t) - y^1(s)\| \leq C_1 |t-s|^{1/p} m_1(t-s), \quad \|y^2(t) - y^2(s)\| \leq C_2 |t-s|^{1/q} m_2(t-s).$$

This ensures meaningful extension of Young integration to the so-called “borderline case” and provides the mechanism for rigorous enhancement of paths in rough path theory (Yang, 2012).

3. Nonlinear Young Integrals in Differential Systems

Young Differential Equations (YDEs) and Nonlinear Flows

Abstract nonlinear Young differential equations are of the form

$x_t = x_0 + \int_0^t A(ds, x_s),$

with $A$ satisfying sufficient joint Hölder regularity, such as $C^\alpha_t C^\beta_x$, with $\alpha(1+\beta) > 1$ guaranteeing the solution map is well-defined in $C^\alpha$ and, under further regularity, generates a flow of homeomorphisms or diffeomorphisms (Galeati, 2020).

This framework subsumes differential equations driven by fractional Brownian motion, rough fields, or paths of low but controlled regularity, and supports an extensive calculus (Itô-type formulae, chain rule, substitution, flow composition, etc.) (Castrequini et al., 2014, Lima et al., 2022).

Nonlinear Young PDEs, SPDEs, and Feynman–Kac Formulae

Nonlinear Young integration also underpins mild solution theory for PDEs and SPDEs driven by rough signals: $dx_t = -Ax_t\,dt + B(dt, x_t),$ where the (nonlinear) integral with respect to $dt$ is replaced with Young integration in time (often combined with semigroup theory), or where the noise is spread in space-time as in stochastic heat equations where the forcing is only Hölder continuous (Hu et al., 2014, Bechtold et al., 2 May 2024).

Typical application: for a parabolic SPDE with rough coefficients,

$\partial_t u + L u + c(t,x) u = 0,$

the Feynman–Kac representation,

$$u(r,x) = \mathbb{E} [ u_T(X_T^{r,x}) \exp\{ \int_r^T W(ds, X_s^{r,x}) \}],$$

relies crucially on defining and controlling the nonlinear Young integral $\int_r^T W(ds, X_s^{r,x})$ (Hu et al., 2014, Hu et al., 2015).

Backward Stochastic Differential Equations (BSDEs) with Nonlinear Young Drivers

Recent research has focused on existence, uniqueness, and comparison principles for BSDEs with nonlinear Young integral terms, particularly when the drift is a highly irregular function of space and time: $$Y_t = \xi + \int_t^T f(r, X_r, Y_r, Z_r)dr + \sum_{i=1}^M \int_t^T g_i(Y_r) \eta_i(dr, X_r) - \int_t^T Z_r dW_r.$$ The driver $\eta$ may be unbounded and possess limited regularity, requiring new localization techniques and careful sewing arguments for the Young integral (Song et al., 25 Apr 2025, Song et al., 5 Sep 2025). Markov properties, Feynman–Kac formulae for Young PDEs, and exponential decay of localization errors in non-Lipschitz settings are rigorously established.

4. Advanced Regularity Theory and Multidimensional Extensions

Set-valued and Multidimensional Young Integrals

For multifunctions or set-valued maps (with applications to stochastic differential inclusions), the nonlinear Young integral is defined over a compact, convex subset of Hölder-continuous selections, leading to a compact, convex, and closed set of possible integral values. The resulting theory enjoys strong stability and continuity properties, crucial for existence results in infinite-dimensional systems (Coutin et al., 2021).

In dimensions $d\geq 2$, the Young integral is defined with respect to Hölder “charges,” which are finitely additive set functions on BV-sets, generalizing the scalar case to integration against measures such as $\mathrm{d}g_1 \wedge \cdots \wedge \mathrm{d}g_d$ for Hölder functions $g_i$. Convergence is established under a multidimensional Hölder-type condition, e.g., $\beta + d\gamma > d$ for integrand and charge exponents $\beta,\gamma$ (Bouafia, 27 Jan 2024, Alberti et al., 2019).

Iterated integrals, essential for describing higher-level rough paths and regularity structures, are defined recursively; careful use of basis expansions (e.g. Haar or Faber–Schauder) and sewing lemma arguments is critical. The theory covers integration of nonsmooth 2-forms and extends Stratonovich vs. Itô-type summations to two-dimensional settings, with sharp error bounds and optimal regularity thresholds.

5. Geometric, Chain Rule, and Invariance Properties

The nonlinear Young integral framework supports a robust stochastic/geometric calculus extending Itô, Kunita, and Ventzel formulae to low regularity signals (Lima et al., 2022, Galeati, 2020). The chain rule for functions of Young-integrated processes,

$$F(t, y(t)) - F(s, y(s)) = \int_s^t F_t(u, y(u)) du + \int_s^t \langle F_x(u, y(u)), Ay(u) \rangle du + \int_s^t \langle F_x(u, y(u)), \sigma(y(u)) \rangle dx(u),$$

enables invariance and symmetry results for nonlinear Young systems and provides necessary conditions for conservation laws and preserved subspaces in infinite-dimensional PDEs (Addona et al., 2021, Castrequini et al., 2014).

Geometric decompositions of flows, horizontal lifts, parallel transport, and development maps—classical tools of stochastic differential geometry—are obtainable for α-Hölder paths within this rigorous analytic structure.

6. Localization, Comparison Results, and Quantitative Error Bounds

For BSDEs with non-Lipschitz coefficients and unbounded drivers, a new localization technique—truncating the domain and the data—is introduced to establish well-posedness and control convergence to the global solution (Song et al., 5 Sep 2025). Quantitative localization-error estimates, e.g.,

$$|u^n(t,x) - u(t,x)| \leq C \exp\{ - n^2/C + C|x|^2 \},$$

for Young PDEs, provide sharp exponential decay rates, essential for numerical approximation and analysis.

Markov properties and flow properties for the solutions of BSDEs and corresponding Young PDEs are established via tower-type rules and continuity of exit times, crucial for probabilistic representations and viscosity solutions in rough or pathwise analysis.

7. Novel Directions and Implications

Recent developments point toward:

• The extension of nonlinear Young integration to even rougher regimes (lower regularity, e.g., Besov spaces with edge regularity just above Brownian motion (Bechtold et al., 2 May 2024)).
• The interplay of fractional calculus, sewing lemma methodology, and stochastic/multiscale analysis.
• New perspectives on stochastic regularization by noise, pathwise SPDE existence, and non-absolute integration frameworks including Pfeffer and Lebesgue–Stieltjes–type integrals (Hanung et al., 2018, Bouafia, 27 Jan 2024).

This unified analytical paradigm provides a rigorous, flexible framework for nonlinear, pathwise stochastic and deterministic analysis in rough signal settings, forming the basis for modern advances in stochastic analysis, rough path theory, SPDEs, geometric PDEs, and related fields.