Rough Partial Differential Equations

• Rough partial differential equations are evolution equations driven by irregular temporal signals modeled as rough paths, extending classical PDE analysis to infinite-dimensional spaces.
• They encompass multiple frameworks including semilinear mild solutions with controlled rough paths, viscosity formulations, and energy-based weak solutions.
• The theory supports well-posedness and practical computation through splitting methods, delay equations, and covariance-based reductions, linking RPDEs to SPDEs and modern numerical analysis.

Rough partial differential equations are evolution equations in which the temporal forcing is too irregular for classical integration and is therefore modeled by a rough path. In the semigroup-based setting, they extend rough differential equations to infinite-dimensional Banach or Hilbert spaces and take the abstract form

$$dY_t=(A Y_t+f_0(t,Y_t))\,dt+f(t,Y_t)\,d\mathbf X_t,$$

with $A$ the generator of a $C_0$-semigroup and $\mathbf X$ a Hölder or finite-$p$-variation rough path. In the broader literature represented here, the term also covers linear parabolic equations with unbounded rough drivers, fully nonlinear viscosity equations with rough first-order forcing, and obstacle problems represented by rough reflected BSDEs. Across these frameworks, “rough” refers to temporal irregularity of the driving signal; the spatial operator remains a classical PDE object (Tappe, 11 May 2026, Tappe, 2024, Diehl et al., 2014, Friz et al., 2010, Li et al., 2024).

1. Definitions and canonical forms

Several prototype classes coexist under the RPDE label.

The semilinear evolution-equation form is the mild semigroup equation

$$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$

posed on a Banach or Hilbert space $W$ or $H$, with $\mathbf X$ an $\alpha$- or $A$0-Hölder rough path, typically in the regime $A$1. This is the basic framework for semilinear rough evolution equations, semilinear RPDEs, and slow–fast rough PDEs (Tappe, 2024, Li et al., 2024, Tappe, 11 May 2026).

A second class consists of viscosity-type rough PDEs on $A$2,

$A$3

where $A$4 is degenerate elliptic and $A$5 is typically first-order or zeroth-order in space. These equations are defined pathwise by stability of viscosity solutions under smooth rough-path approximations and include linear and Hamilton–Jacobi–Bellman cases (Friz et al., 2010).

A third class is variational or energy-based linear parabolic RPDEs such as

$A$6

with $A$7 in divergence form and the rough perturbation encoded by an unbounded rough driver acting on Sobolev scales. Here the solution is formulated intrinsically in negative Sobolev spaces (Hocquet et al., 2017).

A fourth class comprises rough PDEs with obstacles,

$A$8

which arise from rough reflected BSDEs and are linked to optimal stopping and American option pricing (Li et al., 2024).

Framework	Prototype	Solution notion
Semilinear evolution RPDE	$A$9	Mild controlled rough path
Viscosity RPDE	$C_0$0	Rough viscosity solution
Variational linear RPDE	$C_0$1	Weak/energy solution
Obstacle rough PDE	$C_0$2	Viscosity via rough RBSDE

This multiplicity of forms is structural rather than contradictory. The literature does not use a single universal notion of RPDE solution; the appropriate formulation depends on whether the equation is semilinear, fully nonlinear, weakly variational, or obstacle-constrained.

2. Integration theory and solution concepts

The semilinear theory is built on controlled rough paths and the Gubinelli integral. For a rough path $C_0$3, a path $C_0$4 is controlled by $C_0$5 if

$C_0$6

with $C_0$7 of higher time regularity. The rough integral is then defined by the compensated Riemann sums

$C_0$8

and continuity of this map is obtained from the sewing lemma (Tappe, 2024, Tappe, 11 May 2026).

For evolution equations, the time singularity of the semigroup requires either a semigroup-adapted controlled structure or higher domain regularity. One approach defines the semigroup increment

$C_0$9

and asks that

$\mathbf X$0

belong to a $\mathbf X$1-Hölder class. This yields the rough convolution

$\mathbf X$2

as the basic nonlinear term in mild RPDEs (Li et al., 2024). A different route avoids analyticity of the semigroup by imposing that the integrand take values in $\mathbf X$3, allowing the same Gubinelli integral as for finite-dimensional RDEs after rewriting $\mathbf X$4 as a controlled path (Tappe, 2024, Tappe, 11 May 2026).

In the variational theory, the rough perturbation is encoded by an unbounded rough driver $\mathbf X$5 acting on a Sobolev scale $\mathbf X$6. A weak solution $\mathbf X$7 to

$\mathbf X$8

satisfies the intrinsic expansion

$\mathbf X$9

for test functions $p$0, with $p$1 a higher-order remainder (Hocquet et al., 2017).

In the viscosity setting, roughness enters through smooth approximations of the driving path. One solves the PDE with smooth drivers, proves stability in the rough path topology, and defines the rough viscosity solution as the unique locally uniform limit. This same approximation principle underlies splitting methods for RPDEs (Friz et al., 2010).

A recurrent misunderstanding is that rough integration always means a stochastic integral. In these theories the rough integral is first defined pathwise; probabilistic structure enters only when the rough path itself is random.

3. Well-posedness, regularity, and a priori bounds

Local and global well-posedness for semilinear RPDEs have been developed in several semigroup frameworks. For

$p$2

with $p$3 the generator of a $p$4-semigroup, $p$5, and $p$6, there is a unique local mild solution in a controlled-path space; under stronger domain assumptions on $p$7, $p$8, and the initial condition $p$9, the solution is global and is also a strong solution (Tappe, 2024).

Global-in-time semilinear parabolic RPDEs with analytic semigroups were obtained by working on a Banach scale and deriving a priori estimates without quadratic terms in the controlled rough path norm. For

$$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$0

with $$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$1 of linear growth and $$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$2 sufficiently smooth with bounded derivatives, one obtains a unique global solution

$$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$3

together with an exponential-in-time bound on $$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$4 (Hesse et al., 2021).

Linear parabolic RPDEs admit a sharp energy theory under the same deterministic assumptions as classical linear parabolic equations. For the equation

$$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$5

with uniformly elliptic $$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$6, $$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$7, and lower-order coefficients in the Ladyzhenskaya–Solonnikov–Ural’tseva class, there exists a unique weak solution

$$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$8

and it satisfies the energy estimate

$$Y_t=S_t\xi+\int_0^t S_{t-s}f_0(s,Y_s)\,ds+\int_0^t S_{t-s}f(s,Y_s)\,d\mathbf X_s,$$9

The proof uses intrinsic remainder estimates, a rough Gronwall lemma, tensorization, and passage to the diagonal (Hocquet et al., 2017).

Integrability results sharpen this well-posedness theory in low-regularity regimes. For non-autonomous semilinear RPDEs driven by Gaussian rough paths with $W$0 and unbounded diffusion coefficients losing $W$1 derivatives, a Banach-scale controlled rough path theory with control-function norms yields stretched-exponential tail bounds for the number of greedy intervals and, consequently,

$W$2

This is the paper’s “integrable bound” for mild solutions (Blessing et al., 6 Mar 2025).

These results support a broad principle: rough forcing does not preclude global well-posedness, but the required functional setting depends sensitively on whether one exploits semigroup smoothing, higher generator domains, or energy inequalities.

4. Relations to SPDEs, Feynman–Kac formulas, and rough BSDEs

A central theme of RPDE theory is that it recovers classical SPDEs when the rough path is a stochastic lift. For infinite-dimensional $W$3-Wiener noise, one can construct an Itô-enhanced rough path $W$4 for any $W$5, and for adapted bounded integrands the rough integral coincides almost surely with the Hilbert-space Itô integral. The same semigroup framework also treats $W$6-fractional Brownian motion with Hurst index $W$7, producing global mild solutions for fractional-noise RPDEs (Tappe, 2024).

The pathwise viewpoint is especially explicit in the linear theory. For backward and forward linear RPDEs,

$W$8

rough Feynman–Kac formulas express solutions through rough SDEs. In the forward case this yields a pathwise robustification of the Zakai equation, and the solution map is continuous in rough path topology (Diehl et al., 2014).

In the viscosity framework, the canonical Stratonovich lift of a semimartingale turns the rough PDE

$W$9

into the pathwise counterpart of the Stratonovich SPDE

$H$0

with Wong–Zakai approximations converging to the rough path solution (Friz et al., 2010).

Obstacle problems admit a parallel probabilistic representation. Reflected BSDEs with rough drivers are formulated as limits of reflected BSDEs driven by smooth approximations of a geometric rough path, and the associated Markovian solution $H$1 is the viscosity solution of the rough obstacle PDE

$H$2

This supplies a nonlinear Feynman–Kac formula for rough PDEs with obstacles and connects them to optimal stopping and American option pricing in rough-volatility settings (Li et al., 2024).

Accordingly, rough PDE theory is not a competitor to SPDE theory so much as a pathwise reformulation of large classes of SPDEs, especially when continuity with respect to the noise path is essential.

5. Multiscale limits, delay equations, and random dynamical systems

Rough PDE methods have been extended beyond single-scale well-posedness to multiscale and structural problems.

For semilinear slow–fast systems on a Hilbert space,

$H$3

with the slow component driven by a $H$4-Hölder rough path, $H$5, and the fast component driven by a Brownian rough path, the averaging principle holds in a mild Hölder topology. Writing

$H$6

for the averaged drift induced by the invariant measure of the frozen fast dynamics, the averaged RPDE

$H$7

satisfies

$H$8

The proof combines controlled rough path theory with Khasminskii’s time discretization scheme (Li et al., 2024).

Delay rough evolution equations introduce delayed Lévy areas $H$9 and delayed controlled paths $\mathbf X$0, allowing equations of the form

$\mathbf X$1

Under analytic semigroup assumptions and Banach-scale regularity of $\mathbf X$2, there exists a unique global mild solution. For the special delay equation

$\mathbf X$3

and the non-delay limit

$\mathbf X$4

one has convergence in a controlled-path metric as $\mathbf X$5 (Qu et al., 2024).

Random dynamical systems enter the theory once the driving rough path is itself a cocycle. Under the assumptions for global semilinear parabolic RPDEs, the solution operator generates a continuous random dynamical system on the state space (Hesse et al., 2021). On this basis, center manifold theory has been established for semilinear parabolic RPDEs

$\mathbf X$6

with analytic semigroup, spectral center–stable splitting, and $\mathbf X$7-Hölder rough forcing. The resulting center manifold is a random manifold in the sense of random dynamical systems, and examples include reaction–diffusion equations and the Swift–Hohenberg equation (Kuehn et al., 2021).

These developments show that rough forcing is compatible with averaging, delay limits, and invariant-manifold constructions rather than obstructing them.

6. Splitting methods, exact reduction, and computational viewpoints

Computation in the RPDE setting exploits pathwise stability. A Lie-type splitting-up method has been developed for rough PDEs of the form

$\mathbf X$8

The method alternates the deterministic PDE flow and the pure rough-noise flow, implemented by time changes $\mathbf X$9 and $\alpha$0. Stability of rough viscosity solutions then implies locally uniform convergence of the splitting approximation to the exact RPDE solution. The theory covers linear equations and Hamilton–Jacobi–Bellman rough PDEs, but the paper does not provide an explicit rate of convergence (Friz et al., 2010).

For linear parabolic RPDEs, spatial discretization yields large rough differential equations amenable to exact dimension reduction. After discretizing a rough heat equation with transport and multiplicative rough terms, the resulting large-order RDE

$\alpha$1

can be reduced exactly by identifying covariance-based invariant subspaces from associated linear Itô SDEs. A first reduction uses the image of

$\alpha$2

to obtain a lower-dimensional system containing the full solution trajectory. A second reduction, in the linear-output case, uses a dual Gramian

$\alpha$3

to remove state directions that do not influence the quantity of interest $\alpha$4. In the rough heat-equation experiment reported there, a discretization of size $\alpha$5 was reduced first to dimension $\alpha$6 and then to $\alpha$7, with relative $\alpha$8-error approximately $\alpha$9 for the target output (Redmann et al., 2023).

The current computational picture is therefore twofold. On one side, pathwise splitting exploits rough-viscosity stability. On the other, covariance-based reductions exploit exact subspace structure after discretization. Both rely on the defining RPDE feature that the solution map is continuous in the rough path input, making approximation of the driver and approximation of the state dynamics compatible.

Within the literature surveyed here, rough partial differential equations have evolved from pathwise formulations of linear SPDEs and viscosity equations to a technically differentiated theory encompassing semilinear global well-posedness, energy solutions, obstacle problems, delay equations, averaging principles, invariant manifolds, and structure-preserving numerical reduction. The unifying idea is that once the driving signal is enhanced to a rough path, the temporal irregularity is handled pathwise, and the remaining analysis becomes a deterministic problem in PDE theory, semigroup theory, viscosity theory, or random dynamical systems, depending on the equation class (Tappe, 11 May 2026).