Nonlinear Young differential equations: a review

Lucio Galeati

Nonlinear Young differential equations: a review

T0 review · 0 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper establishes that the nonlinear Young integral is well defined whenever $\alpha+\beta\gamma>1$, and that this single exponent condition carries a full theory of well-posed Young differential equations, flows, numerical schemes…

desk verdict A careful, genuinely useful review of nonlinear Young integration whose central construction holds up; the new extensions are natural, the soft spots are minor, and it deserves a serious referee. read the letter →

arxiv 2009.12884 v1 pith:ULBZ6DSY submitted 2020-09-27 math.CA math.APmath.PR

Lucio Galeati This is my paper · ORCID

classification math.CAmath.APmath.PR MSC 60L2060L5034A08

keywords NonlinearYoungintegraldifferentialequationssewinglemmaregularizationbynoiseflowpropertynumericalschemestransportparabolicPDEs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a pathwise integral of the form $\int_0^t A(ds,x_s)$ for a drift $A$ that is $\alpha$-Hölder in time and $\beta$-Hölder in space and a path $x$ that is $\gamma$-Hölder, under the single condition $\alpha+\beta\gamma>1$. The integral is constructed by the sewing lemma applied to the increments $A_{s,t}(x_s)$, and the condition is precisely the one that makes the sewing defect regular enough for convergence; the resulting map $(A,x)\mapsto \int_0^\cdot A(ds,x_s)$ is continuous in both arguments. Around this object the paper builds a complete theory of nonlinear Young differential equations: existence, uniqueness, stability, regularity of the flow, convergence of Euler schemes, and well-posedness for Young transport and parabolic PDEs. The reason to care is that many systems that have no classical meaning—such as modulated PDEs and additively perturbed ODEs—acquire a rigorous pathwise meaning as soon as the exponents satisfy this inequality.

What carries the argument

The carrying mechanism is the sewing lemma applied to the two-index increment $\Gamma_{s,t}=A_{s,t}(x_s)=A(t,x_s)-A(s,x_s)$. The sewing lemma turns an approximately additive two-index map into a genuine path $J(\Gamma)$ whenever the defect $\delta\Gamma_{s,u,t}=\Gamma_{s,t}-\Gamma_{s,u}-\Gamma_{u,t}$ is Hölder continuous with exponent larger than $1$; here the space-time Hölder assumption gives $|\delta\Gamma_{s,u,t}|\lesssim |t-u|^\alpha |u-s|^{\beta\gamma}\,[A]_{\alpha,\beta}\,[x]_\gamma^\beta$, so the defect is $(\alpha+\beta\gamma)$-Hölder and the strict inequality is exactly the lemma's hypothesis. The nonlinear Young integral is defined as $J(\Gamma)$, and the same identity underlies the Itô-type formula, the difference formula through averaged derivatives, and the product rule that the later sections use.

What would settle it

The direct check is to compute, for any admissible triple $\alpha,\beta,\gamma$ with $\alpha+\beta\gamma>1$, the sewing defect $\delta\Gamma_{s,u,t}=A_{u,t}(x_s)-A_{u,t}(x_u)$ and compare the limit of $\sum_i A_{t_i,t_{i+1}}(x_{t_i})$ along two explicitly different dyadic partitions of $[0,T]$. If the limits differ, or if the defect is not $O(|t-s|^{\alpha+\beta\gamma})$ uniformly over $s<u<t$, Theorem 2.7's central claim fails; if both hold for all tested pairs, the threshold passes.

Watch

Extended reading notes

Core claim

The central claim is that the nonlinear Young integral is well defined for $A\in C^\alpha_t C^\beta_{V,W,\mathrm{loc}}$ and $x\in C^\gamma_t V$ whenever $\alpha+\beta\gamma>1$, with the pathwise estimate $\|\int_s^t A(du,x_u)-A_{s,t}(x_s)\|_W \lesssim |t-s|^{\alpha+\beta\gamma}$ and local Hölder continuity in $x$. For differential equations, the main well-posedness statement is that when $A\in C^\alpha_t C^{1+\beta}_V$ and $\alpha(1+\beta)>1$, there is a unique global solution for every initial condition; the solution map is Lipschitz, the flow is a flow of diffeomorphisms whose derivative solves a variational Young equation, and the Euler scheme converges at rate $2\alpha-1$. In the presence of a continuous time derivative $\partial_t A$, the threshold drops to $\alpha+\beta>1$ and the Euler rate improves to $\alpha$. The paper also gives conditional-uniqueness criteria through averaged translations, a duality proof of uniqueness for Young transport and continuity equations, and a fixed-point theory for parabolic Young PDEs in semigroup interpolation spaces.

Load-bearing premise

Everything in the paper rests on the strict inequality $\alpha+\beta\gamma>1$; if the time exponent $\alpha$, the spatial smoothness $\beta$, and the path exponent $\gamma$ do not combine past this threshold, the sewing lemma cannot be applied and no nonlinear Young integral, hence no nonlinear Young differential equation, is defined by this construction.

Editorial extensions

If this is right

If $A\in C^\alpha_t C^\beta_{V,W,\mathrm{loc}}$ and $x\in C^\gamma_t V$ satisfy $\alpha+\beta\gamma>1$, the integral exists, is independent of the partition, and is locally $\delta$-Hölder in $x$ for every $\delta<(\alpha+\beta\gamma-1)/\gamma$.
If $A\in C^\alpha_t C^{1+\beta}_V$ with $\alpha(1+\beta)>1$, every initial value produces a unique global solution, and the map from initial value to solution is globally Lipschitz in $C^\alpha_t V$.
The same regime gives a flow of diffeomorphisms: the spatial derivative solves the variational Young equation $J_{s\to t}=I+\int_s^t DA(dr,\Phi_{s\to r}(x))\circ J_{s\to r}$, and in finite dimensions $\det D_x\Phi_{s\to t}(x)=\exp(\int_s^t \mathrm{div}\,A(dr,\Phi_{s\to r}(x)))$.
The $n$-step Euler scheme converges in $C^\alpha$ with rate $2\alpha-1$; if $\partial_t A$ exists continuously, the rate is $\alpha$ under the weaker condition $\alpha+\beta>1$.
Young transport and continuity equations with $A,c\in C^\alpha_t C^{1+\beta}_x$ and $\alpha(1+\beta)>1$ have unique weak solutions, given explicitly by the push-forward of the flow times the exponential of the integrated divergence minus $c$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the threshold $\alpha+\beta\gamma>1$ is sharp, then the boundary case should require genuinely higher-order data—iterated integrals or their equivalent—so this level-1 theory would be the natural benchmark for a nonlinear rough-path extension at $\alpha\le 1/2$. This is an editor's inference, not a claim of the paper.
The averaged-translation criterion suggests a cheap uniqueness test in applications: compute $\tau^{x}A(t,z)=\int_0^t A(ds,z+x_s)$ for a candidate path and check whether it lies in $C^\alpha_t \mathrm{Lip}$; applying this to modulated PDEs could produce explicit thresholds on the irregularity of the modulating path.
For parabolic Young PDEs, the condition that $B\in C^\gamma_t C^2$ needs $2\gamma+\rho-\delta>1$ has the shape of a parabolic scaling threshold; testing examples just below it would tell whether the fixed-point condition is also necessary.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: read this if you work in Young integration or regularization by noise. It is a careful, mostly self-contained review with several genuinely new extensions, and the central construction is sound. None of the new results reshape the field, but they are useful and correctly proved.

The core is Theorem 2.7: for A in C^α_t C^β_{V,W,loc} and x in C^γ_t V with α+βγ>1, the nonlinear Young integral is well-defined by sewing Γ_{s,t}=A_{s,t}(x_s). I checked the key estimate: ||δΓ_{s,u,t}|| ≤ [A]_{α,β,‖x‖∞}[x]_γ^β |t-u|^α |u-s|^{βγ}, giving Γ in C^{α,α+βγ}_2, and α+βγ>1 is exactly the sewing threshold. The continuity estimates and the fixed point arguments in Theorem 3.12 use only the stated hypotheses; no hidden compactness. The strict Young condition is the load-bearing boundary, and the paper is explicit about it.

What is actually new: Theorem 3.17 relaxes uniqueness to α+β>1 when ∂_t A exists continuous; Theorem 5.1 gives a Van Kampen-type conditional uniqueness; Theorem 5.6 is a conditional comparison principle; Theorem 7.4 gives well-posedness for parabolic nonlinear Young PDEs. These are natural extensions of the Catellier–Gubinelli / Hu–Lê program, not a new framework. That is fine: they are stated cleanly and proved correctly.

The soft spots are minor. Three proofs are omitted with references (Lemma 4.3, Theorem 4.7, Theorem 7.2), so the self-contained claim is a bit strong. Several localization arguments are sketched, though the reader can fill them. In Theorem 7.4 the set E with the extra spatial-norm constraint is not obviously closed in the stated metric; the fix is to run the fixed point in the closed Hölder ball and recover the extra regularity a posteriori. That does not threaten the main results.

I have no objection to the citation pattern. The author cites [20] and [22] for some results, but in a review that is normal; the material is either reproved or used as an external benchmark, and the central claim is independent.

Who it is for: specialists in rough paths, Young differential equations, and pathwise regularization by noise. It will be a convenient reference and a good entry point for graduate students. I would cite it if I wrote in this area, and I would send it to a serious referee. It does not need major surgery, just a light pass on the omitted proofs and the Theorem 7.4 technicality.

Referee Report

0 major / 4 minor

Summary. The manuscript develops a self-contained theory of nonlinear Young integrals of the form ∫_0^t A(ds,x_s), for a drift A with joint time-space Hölder regularity and an integrand x of Hölder exponent γ, under the Young threshold α+βγ>1 (Theorem 2.7). It then proves well-posedness of nonlinear Young differential equations using Euler schemes and fixed-point arguments (Theorems 3.2 and 3.12), studies flow regularity and differentiability of the Itô map (Section 4), gives conditional uniqueness criteria (Section 5), treats Young transport equations (Section 6), and extends the framework to parabolic nonlinear Young PDEs (Section 7). The appendix collects supporting lemmas and topological properties of solution sets. Several statements are new extensions, while others are collected from the literature; proofs of Lemma 4.3, Theorem 4.7, and Theorem 7.2 are explicitly deferred to other papers.

Significance. The central construction is sound and the paper has clear reference value: the sewing lemma is applied to Γ_{s,t}=A_{s,t}(x_s), the decisive estimate is ||δΓ_{s,u,t}|| ≤ [A][x]^β |t-u|^α |u-s|^{βγ}, and the condition α+βγ>1 is exactly the sewing threshold. The fixed-point and Euler-scheme arguments in Sections 2 and 3 are internally consistent, and the paper provides explicit continuity estimates and convergence rates that are directly usable in applications. The paper also offers a systematic bridge between Young integration and regularization-by-noise problems, and it develops applications to flows, transport equations, and semilinear parabolic equations in Banach spaces. The main limitation is that the self-contained claim is slightly stronger than what is actually proved: several technical results are cited from the literature, and one fixed-point domain in Section 7 needs a small correction. These issues are local and do not affect the core Young/YDE well-posedness results.

minor comments (4)

[Theorem 7.4 (proof)] The fixed-point domain E = {x ∈ C^κ([0,τ];V_δ) : x(0)=x_0, [x]_{κ,Vδ}≤M, sup_t ||x_t||_{Vδ+κ}≤M} is not closed in the stated metric d_E(x1,x2)=[x1−x2]_{κ,Vδ}, because convergence in this seminorm does not control sup_t ||x_t||_{Vδ+κ}. I suggest running the contraction in the closed ball of C^κ([0,τ];Vδ) with the complete C^κ norm (which is equivalent for paths with fixed initial value), and recovering the extra Vδ+κ regularity of the fixed point a posteriori from the mild formula (7.9). This is a local fix and does not affect the main existence and uniqueness argument.
[Introduction, Section 4, Section 7] The abstract promises a self-contained account, but the proofs of Lemma 4.3, Theorem 4.7, and Theorem 7.2 are explicitly deferred to [33], [20], and [25]. For a review this is acceptable, but please qualify the self-contained claim accordingly, or include the missing arguments in an appendix.
[Definition 2.5] In the display defining [A]_{α,β} and ||A||_{α,β} for the local spaces C^α_t C^β_{V,W,loc}, the radius R appears on the right-hand side but not on the left-hand side; the notation should be [A]_{α,β,R} and ||A||_{α,β,R}, as is used later in the proofs.
[Throughout] There are several typographical errors such as 'Riemann-Stjeltes', 'Frechét', 'constractivity', and 'inﬁnte'; please correct them. They do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central integral construction and well-posedness theorems are self-contained via the sewing lemma.

full rationale

The paper's central claim, Theorem 2.7, constructs the nonlinear Young integral by applying the sewing lemma to the increment field Γ_{s,t}=A_{s,t}(x_s). The decisive estimate is ‖δΓ_{s,u,t}‖ ≤ [A]_{α,β,‖x‖∞}[x]_γ^β |t-u|^α |u-s|^{βγ}, giving Γ ∈ C^{α,α+βγ}_2 with α+βγ>1; this is precisely the sewing threshold, and the integral is characterized as the unique limit of Riemann sums. No parameter is fitted to the conclusion, and the subsequent existence, uniqueness, stability, and flow results (Theorems 3.12, 3.14, 4.5, and 4.9) are derived from this integral via explicit Banach fixed-point arguments, a priori estimates, and Comparison Principles, with constants depending only on the stated Hölder norms. The paper does omit proofs of some auxiliary results, notably Lemma 4.3, Theorem 4.7, and Theorem 7.2, but it explicitly refers to external references ([33], [25]) alongside the author's own works; the cited results are not used to define the central object, and the central Young integral and YDE well-posedness do not depend on an unverified self-citation chain. Self-citations such as [20] and [22] appear mostly as background or as results that are reproved in the text (e.g., Corollary 2.12 gives an alternative proof of Lemma 6 from [20]; Theorem A.6 is proved rather than merely quoted). I find no equation whose definition is equivalent to its supposed output, no fitted input relabeled as a prediction, and no uniqueness theorem imported solely from the authors' prior work. The derivation chain is internally consistent and self-contained for the main claims.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no free parameters and no new entities; it relies on standard analytic and probabilistic tools.

assumptions (7)

standard math Sewing lemma for Hölder increments with α<1<β
Used throughout to construct nonlinear Young integrals and to estimate differences; stated with proof reference to [18].
standard math Banach fixed point theorem
Used to prove existence and uniqueness in Theorems 3.12, 3.17, 7.4.
standard math Ascoli-Arzelà theorem
Used to extract convergent subsequences in Theorems 3.2, 3.25 and Appendix A.3.
standard math Schaefer's fixed point theorem
Used in Proposition 3.25 for existence in fractional YDEs.
domain assumption Analytic semigroup regularity estimates (7.2)-(7.3)
Assumed in Section 7 for parabolic nonlinear Young PDEs; standard from [39].
domain assumption Compact embedding W ↪ V in Theorem 3.2
Explicitly assumed for Euler scheme convergence; in infinite dimensions this fails in general.
standard math Fubini theorem for Bochner integrals and the sewing map
Used in Corollary 2.12 and Lemma A.1.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Nonlinear Young differential equations: a review." pith.science (2026). https://pith.science/paper/ULBZ6DSY

@misc{pith2026200912884,
  author       = {Pith},
  title        = {Pith review of: Nonlinear Young differential equations: a review},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ULBZ6DSY}},
  note         = {Machine review of arXiv:2009.12884}
}

read the original abstract

Nonlinear Young integrals have been first introduced in [Catellier,Gubinelli, SPA 2016] and provide a natural generalisation of classical Young ones, but also a versatile tool in the pathwise study of regularisation by noise phenomena. We present here a self-contained account of the theory, focusing on wellposedness results for abstract nonlinear Young differential equations, together with some new extensions; convergence of numerical schemes and nonlinear Young PDEs are also treated. Most results are presented for general (possibly infinite dimensional) Banach spaces and without using compactness assumptions, unless explicitly stated.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 44 canonical work pages

[20]

Noiseless regularisation by noise

L. Galeati and M. Gubinelli. Noiseless regularization by noise. arxiv preprint arXiv:2003.14264 , 2020

work page Pith review arXiv 2003
[22]

Regularization of multiplicative SDEs through additive noise

L. Galeati and F. A. Harang. Regularization of multipli cative SDEs through additive noise. arXiv preprint arXiv:2008.02335, 2020

work page Pith review arXiv 2008
[1]

Ambrosio

L. Ambrosio. Transport equation and Cauchy problem for B V vector ﬁelds. Inventiones mathe- maticae, 158(2):227–260, 2004

work page 2004
[2]

Bailleul

I. Bailleul. Flows driven by rough paths. Revista Matemática Iberoamericana , 31(3):901–934, 2015

work page 2015
[3]

Bailleul and M

I. Bailleul and M. Gubinelli. Unbounded rough drivers. Ann. Fac. Sci. Toulouse Math. (6) , 26(4):795–830, 2017

work page 2017
[4]

Bailleul and S

I. Bailleul and S. Riedel. Rough ﬂows. Journal of the Mathematical Society of Japan , 71(3):915– 978, 2019

work page 2019
[5]

Transport and continuity equations with (very) rough noise

C. Bellingeri, A. Djurdjevac, P. K. Friz, and N. Tapia. Tr ansport and continuity equations with (very) rough noise. arXiv preprint arXiv:2002.10432 , 2020

work page Pith review arXiv 2002
[6]

F. E. Browder and C. P. Gupta. Topological degree and nonl inear mappings of analytic type in Banach spaces. Journal of Mathematical Analysis and Applications , 26(2):390–402, 1969

work page 1969

Show all 45 references

[7]

Castaing and M

C. Castaing and M. Valadier. Convex analysis and measurable multifunctions , volume 580. Springer, 2006

2006
[8]

Catellier

R. Catellier. Rough linear transport equation with an ir regular drift. Stoch. Partial Diﬀer. Equ. Anal. Comput. , 4(3):477–534, 2016

2016
[9]

Catellier and M

R. Catellier and M. Gubinelli. A veraging along irregula r curves and regularisation of ODEs. Stochastic Process. Appl., 126(8):2323–2366, 2016

2016
[10]

Chouk and M

K. Chouk and M. Gubinelli. Nonlinear PDEs with modulate d dispersion II: Korteweg–de Vries equation. arXiv preprint arXiv:1406.7675 , 2014

2014 arXiv
[11]

Chouk and M

K. Chouk and M. Gubinelli. Nonlinear PDEs with modulate d dispersion I: Nonlinear Schrödinger equations. Comm. Partial Diﬀerential Equations , 40(11):2047–2081, 2015

2015
[12]

Coghi and T

M. Coghi and T. Nilssen. Rough nonlocal diﬀusions. arXiv preprint arXiv:1905.07270 , 2019

1905 arXiv
[13]

N. D. Cong, L. H. Duc, and P. T. Hong. Nonautonomous Young diﬀerential equations revisited. Journal of dynamics and diﬀerential equations , 30(4):1921–1943, 2018

1921
[14]

A. M. Davie. Diﬀerential equations driven by rough path s: an approach via discrete approxima- tion. Applied Mathematics Research eXpress , 2008, 2008

2008
[15]

Diehl, P

J. Diehl, P. K. Friz, and W. Stannat. Stochastic partial diﬀerential equations: a rough paths view on weak solutions via Feynman–Kac. In Annales de la Faculté des sciences de Toulouse: Mathématiques, volume 26, pages 911–947, 2017

2017
[16]

R. J. DiPerna and P. L. Lions. Ordinary diﬀerential equa tions, transport theory and Sobolev spaces. Invent. Math. , 98(3):511–547, 1989

1989
[17]

Feyel and A

D. Feyel and A. de La Pradelle. Curvilinear integrals al ong enriched paths. Electronic Journal of Probability, 11:860–892, 2006. 54

2006
[18]

P. K. Friz and M. Hairer. A course on rough paths . Universitext. Springer, Cham, 2014. With an introduction to regularity structures

2014
[19]

P. K. Friz and N. B. Victoir. Multidimensional stochastic processes as rough paths , volume 120 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2010. Theory and applications

2010
[21]

Galeati and M

L. Galeati and M. Gubinelli. Prevalence of ρ-irregularity and related properties. arXiv preprint arXiv:2004.00872, 2020

2004 arXiv
[23]

Górniewicz

L. Górniewicz. Topological ﬁxed point theory of multivalued mappings , volume 4. Springer, 2006

2006
[24]

Gubinelli

M. Gubinelli. Controlling rough paths. J. Funct. Anal. , 216(1):86–140, 2004

2004
[25]

Gubinelli, A

M. Gubinelli, A. Lejay, and S. Tindel. Young integrals a nd SPDEs. Potential Analysis, 25(4):307– 326, 2006

2006
[26]

F. A. Harang and C. Ling. Regularity of Local times assoc iated to Volterra-L \’evy processes and path-wise regularization of stochastic diﬀerential equat ions. arXiv preprint arXiv:2007.01093 , 2020

2007 arXiv
[27]

F. A. Harang and A. Mayorcas. In preparation. 2020

2020
[28]

F. A. Harang and N. Perkowski. C-inﬁnity regularizatio n of ODEs perturbed by noise. arXiv preprint arXiv:2003.05816, 2020

2003 arXiv
[29]

Hu and K

Y. Hu and K. Lê. Nonlinear Young integrals via fractiona l calculus. In Stochastics of Environ- mental and Financial Economics , pages 81–99. Springer, Cham, 2016

2016
[30]

Hu and K

Y. Hu and K. Lê. Nonlinear Young integrals and diﬀerenti al systems in Hölder media. Trans. Amer. Math. Soc. , 369(3):1935–2002, 2017

1935
[31]

Hu and D

Y. Hu and D. Nualart. Diﬀerential equations driven by Hö lder continuous functions of order greater than 1/2. In Stochastic analysis and applications , pages 399–413. Springer, 2007

2007
[32]

Hu and D

Y. Hu and D. Nualart. Rough path analysis via fractional calculus. Transactions of the American Mathematical Society, 361(5):2689–2718, 2009

2009
[33]

A. Lejay. Controlled diﬀerential equations as Young in tegrals: a simple approach. J. Diﬀerential Equations, 249(8):1777–1798, 2010

2010
[34]

T. Lyons. Diﬀerential equations driven by rough signal s (I): An extension of an inequality of L.C. Young. Mathematical Research Letters, 1(4):451–464, 1994

1994
[35]

T. Lyons. Diﬀerential equations driven by rough signal s. Revista Matemática Iberoamericana , 14(2):215–310, 1998

1998
[36]

Maurelli

M. Maurelli. Regularization by noise in ﬁnite dimension . PhD thesis, Scuola Normale Superiore di Pisa
[37]

Nualart and A

D. Nualart and A. Rascanu. Diﬀerential equations drive n by fractional Brownian motion. Collect. Math., 53(1):55–81, 2002

2002
[38]

Nualart and P

D. Nualart and P. Xia. On nonlinear rough paths. arXiv preprint arXiv:1904.11526 , 2019

1904 arXiv
[39]

A. Pazy. Semigroups of linear operators and applications to partial di ﬀerential equations , vol- ume 44. Springer Science & Business Media, 2012. 55

2012
[40]

S. G. Samko, A. A. Kilbas, and O. I. Marichev. Fractional integrals and derivatives , volume 1. Gordon and Breach Science Publishers, Yverdon Yverdon-les -Bains, Switzerland, 1993

1993
[41]

A. V. Shaposhnikov. Some remarks on DavieâĂŹs uniquene ss theorem. Proceedings of the Edin- burgh Mathematical Society , 59(4):1019–1035, 2016

2016
[42]

E. R. Van Kampen. Remarks on systems of ordinary diﬀeren tial equations. American Journal of Mathematics, 59(1):144–152, 1937

1937
[43]

Vidossich

G. Vidossich. On the structure of the set of solutions of nonlinear equations. Journal of Mathe- matical Analysis and Applications , 34(3):602–617, 1971

1971
[44]

L. C. Young. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math., 67(1):251–282, 1936

1936
[45]

M. Zähle. Integration with respect to fractal function s and stochastic calculus. I. Probability theory and related ﬁelds , 111(3):333–374, 1998. 56

1998

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [20]

Noiseless regularisation by noise

L. Galeati and M. Gubinelli. Noiseless regularization by noise. arxiv preprint arXiv:2003.14264 , 2020

work page Pith review arXiv 2003

[2] [22]

Regularization of multiplicative SDEs through additive noise

L. Galeati and F. A. Harang. Regularization of multipli cative SDEs through additive noise. arXiv preprint arXiv:2008.02335, 2020

work page Pith review arXiv 2008

[3] [1]

Ambrosio

L. Ambrosio. Transport equation and Cauchy problem for B V vector ﬁelds. Inventiones mathe- maticae, 158(2):227–260, 2004

work page 2004

[4] [2]

Bailleul

I. Bailleul. Flows driven by rough paths. Revista Matemática Iberoamericana , 31(3):901–934, 2015

work page 2015

[5] [3]

Bailleul and M

I. Bailleul and M. Gubinelli. Unbounded rough drivers. Ann. Fac. Sci. Toulouse Math. (6) , 26(4):795–830, 2017

work page 2017

[6] [4]

Bailleul and S

I. Bailleul and S. Riedel. Rough ﬂows. Journal of the Mathematical Society of Japan , 71(3):915– 978, 2019

work page 2019

[7] [5]

Transport and continuity equations with (very) rough noise

C. Bellingeri, A. Djurdjevac, P. K. Friz, and N. Tapia. Tr ansport and continuity equations with (very) rough noise. arXiv preprint arXiv:2002.10432 , 2020

work page Pith review arXiv 2002

[8] [6]

F. E. Browder and C. P. Gupta. Topological degree and nonl inear mappings of analytic type in Banach spaces. Journal of Mathematical Analysis and Applications , 26(2):390–402, 1969

work page 1969