Pith. sign in

REVIEW 1 major objections 5 minor 1 cited by

Conditional Speed and Shape Corrections for Travelling Wave Solutions to Stochastically Perturbed Reaction-Diffusion Systems

T0 review · 1 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read Small-noise corrections to travelling wave speeds are rigorous, universal, and computable to any order.

desk verdict Rigorous proof of previously formal noise-induced wave-speed corrections; heavy but coherent, and the central expansion holds together under explicit standard hypotheses. read the letter →

arxiv 2502.02164 v1 pith:DG6TCQRD submitted 2025-02-04 math.AP math.DS

Mark van den Bosch , Christian H.S. Hamster , Hermen Jan Hupkes This is my paper · ORCID
classification math.APmath.DS MSC 35K5735R60
keywords stochasticreaction-diffusionequationstravellingwavesfreezingsmallnoiseexpansionsconditionalexpectationswavespeedcorrectionsmetastabilityforwardintegrals
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

The paper sets out to make rigorous a claim that earlier work only supported numerically: when a planar travelling wave in a reaction-diffusion system is hit by small translation-invariant noise, the observed speed and shape of the wave acquire corrections that can be expanded in powers of the noise strength, with the first speed correction at order $\sigma^2$. The main theorem states that, conditional on the wave remaining stable for an exponentially long time, the expectation of any sufficiently smooth functional of the frozen perturbation has a Taylor expansion whose coefficients do not depend on the precise definition of the 'stability event'. This matters because stochastic corrections to front speeds appear throughout excitable media and chemical wave theory, and before this paper there was no rigorous way to define $\mathbb{E}[C_{\mathrm{obs}}]$ or to know which power of $\sigma$ dominates. The proof proceeds by stochastic freezing—choosing a moving coordinate $\Gamma(t)$ so that the perturbation stays small—then splitting the perturbation into ordered Taylor terms plus a residual that is controlled with high probability.

What carries the argument

The central object is the stochastic freezing decomposition $u(x+\Gamma(t),x_\perp,t)=\Phi_0(x)+V(x,x_\perp,t)$, in which the phase $\Gamma$ is chosen to keep the perturbation $V$ from drifting along the translational direction of the wave, so that $V$ stays bounded. The linearized operator $L_{\mathrm{tw}}=\partial_x^2+c_0\partial_x+Df(\Phi_0)$ carries the stability information: a simple zero eigenvalue from translation and a spectral gap $\beta_{\mathrm{tw}}>0$ ensure exponential decay once the neutral direction is projected out. The expansion writes $V=Y_1+\cdots+Y_{r-1}+Z$, with $Y_j$ defined recursively through convolutions of the evolution family with deterministic and stochastic integrands; the stochastic convolutions are forward integrals because their integrands are anticipating. Two engine results do the work: the moment bounds $\mathbb{E}\sup_{0\le t\le T}\|Y_j(t)\|_{H^{k_j}}^{2p}\le K^{2p}[(\sigma^{2p})^{jp}+(\sigma^2\ln T+\delta^2)^{jp}]$, and the convergence of multilinear expectations to explicit limits built from iterated semigroup integrals, which produces the universal coefficients. The stability event $A_{\mathrm{stb}}=\{\|V^{(j)}\|\le\sigma^{-1/2},\|Z\|\le\sigma^{r-1/2}\text{ on }[0,T_*]\}$ has probability close to one for $T_*$ exponentially long in $\sigma^{-1}$, and conditioning on it perturbs expectations only by $O(\sigma^{r-1/2})$.

What would settle it

Simulate a concrete bistable reaction-diffusion system with small translation-invariant noise at several values of $\sigma$, keep only trajectories that stay near the shifted wave for the paper's time $T(\sigma;\theta_*)$, and compare the empirical conditional mean of the observed speed with $c_0+c_2\sigma^2+c_3\sigma^3+\cdots$; a systematic gap larger than the paper's $K\sigma^{r-1/2}$ error bound would refute Corollary 2.7.

Watch

Extended reading notes

Core claim

The central claim is that for every sufficiently smooth functional $\varphi$ of the frozen perturbation $V(t)=U(\cdot+\Gamma(t),\cdot,t)-\Phi_0$, the conditional expectation given the stability event $A_{\mathrm{stb}}$ admits the expansion $\mathbb{E}[\varphi(V(T_*))\mid A_{\mathrm{stb}}]=h_{\infty;0}+\sigma h_{\infty;1}+\cdots+\sigma^{r-1}h_{\infty;r-1}+O(\sigma^{r-1/2})$, with coefficients that are independent of how the stability event is defined. In particular, the observed average speed satisfies $|\mathbb{E}[C_{\mathrm{obs}}\mid A_{\mathrm{stb}}]-(c_0+c_2\sigma^2+\cdots+c_{r-1}\sigma^{r-1})|\le 2K\sigma^{r-1/2}$, so the first stochastic correction enters at order $\sigma^2$. The paper proves that the expansion terms $Y_j$ are well-defined with explicit moment bounds, that multilinear limits $\lim_{t\to\infty}\mathbb{E}\Lambda[Y_{i_1}(t),\dots,Y_{i_\ell}(t)]$ exist and equal explicit iterated semigroup integrals, and that the residual $Z$ stays small with high probability on exponentially long timescales.

Load-bearing premise

The load-bearing assumption is that the wave is stable in every direction except pure side-to-side translation, with all other small disturbances decaying at a fixed exponential rate; without that spectral gap the semigroup estimates and the entire expansion framework collapse.

Editorial extensions

If this is right

  • For any smooth observable $\varphi$ of the frozen perturbation, $\mathbb{E}[\varphi(V(T_*))\mid A_{\mathrm{stb}}]$ equals a universal polynomial in $\sigma$ up to an error $O(\sigma^{r-1/2})$ (Proposition 2.6).
  • The observed average wave speed satisfies $|\mathbb{E}[C_{\mathrm{obs}}\mid A_{\mathrm{stb}}]-(c_0+c_2\sigma^2+\cdots+c_{r-1}\sigma^{r-1})|\le 2K\sigma^{r-1/2}$, identifying $\sigma^2$ as the leading noise correction (Corollary 2.7).
  • The coefficients $h_{\infty;j}$ and $c_j$ are algorithmically computable as iterated semigroup integrals, so in principle the speed and shape corrections can be obtained to any order, limited only by the smoothness of $f$ and $g$.
  • The decomposition $V=Y_1+\cdots+Y_{r-1}+Z$ holds with probability at least $1-2\exp(-\tfrac12\mu\sigma^{-2\theta/r})$ on times exponentially long in $\sigma^{-1}$, supplying a rigorous metastability statement for stochastic travelling waves (Corollary 2.5).

Reading between the lines

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

  • The paper leaves implicit that the same universal coefficients should appear under different phase-pinning conventions; comparing two conventions numerically would provide a gauge-invariant test of the expansion.
  • The effective expansion parameter $\sigma\sqrt{\ln T}$ means corrections grow logarithmically in observation time; the paper's estimates stop at $\ln T_*\ll\sigma^{-2}$, so longer-time behaviour would need a different argument.
  • Because the only structural inputs are a one-dimensional translational neutral mode and a spectral gap, the machinery should transfer to other pattern-forming systems such as spiral waves or cellular fronts, although the paper does not carry out such applications.
  • A concrete next step would be to compute the higher-order coefficients $c_3,c_4,\dots$ for a standard bistable model and benchmark them against direct simulations at moderate $\sigma$, giving a quantitative check of the metastability picture.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The manuscript develops a rigorous small-noise expansion for planar traveling-wave solutions of stochastically perturbed reaction-diffusion systems on cylindrical domains. Using a stochastic freezing procedure, the perturbation relative to the deterministic wave is decomposed as V = Y1 + ... + Y_{r-1} + Z, where the Yj are defined recursively from the SPDE and Z is the residual. The main results are moment bounds for the Yj (Proposition 2.1), exponential convergence of expectations of multilinear expressions (Proposition 2.2), Taylor expansion of expectations of smooth functionals (Proposition 2.3), an exponentially long high-probability stability event (Theorem 2.4), and a conditional expansion for the observed wave speed (Corollary 2.7). The proof is organized into convolution estimates, smoothness of nonlinearities, moment bounds for expansion terms, limiting expectations, residual estimates, and the nonlinear stability argument. The paper advertises the expansion coefficients as universal, i.e. independent of the precise definition of the conditioning stability event, and provides an algorithmic route to compute them.

Significance. If the results are accepted, this is a substantial rigorous contribution to the stochastic reaction-diffusion literature. It turns the formal and numerical expansions of [11,31] into a theorem, gives explicit error bounds on exponentially long time scales, and provides a method to compute speed and shape corrections to arbitrary order under sufficient smoothness. The proof is structurally sound: the hypotheses (HNL), (HTw), (HV*), and (Hq) are explicit and standard for pushed-front stability, and the construction of the expansion terms is derived from the SPDE rather than fitted. The paper also improves on prior convolution estimates by allowing only moment bounds on integrands instead of uniform pathwise bounds. The main limitation is the heavy reliance on prior work, especially [11,16,30], for semigroup estimates, forward integrals, and the mild Itô formula; this is acceptable for a technical paper in this research line, but it means the reader must consult the companion papers to verify the foundational tools.

major comments (1)
  1. [§2.4, Corollary 2.7] The proof of Corollary 2.7 states that one may apply Proposition 2.6 to the function v ↦ aσ(Φ0+v), but this function depends on σ through the explicit terms in (A.14). Proposition 2.6, as stated, guarantees an expansion with coefficients h∞;j that are constant only for a fixed functional φ independent of σ. If the coefficients are allowed to depend on σ, the conclusion that the c_j are fixed scalars with an O(σ^{r-1/2}) remainder does not follow directly. The proof should be repaired by decomposing aσ(Φ0+v) into σ-independent components, e.g. aσ = a0 + σ² a2, applying Proposition 2.6 to each component, and then recombining the corresponding polynomials; the terms beyond order r−1 are then absorbed into the O(σ^{r-1/2}) error. This is a local fix, but as written the advertised wave-speed expansion lacks a complete proof.
minor comments (5)
  1. [Lemma 5.11] The statement begins with the typo 'Suuppose' instead of 'Suppose'.
  2. [Corollary 2.5] There is a duplicated word in 'for any any 0 < σ ≤ δσ'; it should read 'for any 0 < σ ≤ δσ'.
  3. [§2.4, Eq. (2.73)] The sentence 'Note that the first half is exluded to avoid any transients' contains a spelling error; 'exluded' should be 'excluded'.
  4. [Title page] The title contains the spacing error 'W ave'; it should be 'Wave'.
  5. [§6.1, Lemma 6.1] The exchange of the expectation with the infinite Itô correction sum is only sketched; given that the full justification appears in the cited mild Itô formula, a short sentence indicating that the sum is controlled by the Hilbert-Schmidt bound (2.20) and the pairing estimate (6.14) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the conditional expansion is derived from the SPDE itself, with stability conditioning affecting only the error term.

full rationale

The central claim (Proposition 2.6 and Corollary 2.7) is not circular. The coefficients h_{\infty;i} and c_i are constructed from the SPDE data through the recursive definitions (2.29)-(2.32) and the limiting procedure in Section 6; no parameter is fitted to the quantity being predicted. The stability event A_stb appears only in the error estimate: Proposition 2.6 bounds E[phi(Y_tay)|A_stb] - E[phi(Y_tay)] using p_stb close to one, and the limiting polynomial h_infty(sigma) is independent of the event definition. The proof chain is self-contained modulo standard semigroup and stochastic-convolution estimates, several of which are imported from the authors' prior work [11,30,31]. These citations are load-bearing in the sense that semigroup bounds such as (3.5) are used throughout, but they are not equivalent to the expansion being proved: they are previous theorems with explicit hypotheses (HTw), (HNL), (Hq) and do not assume the target conditional-expansion result. There is no renaming of a known empirical pattern as a derivation, and no uniqueness theorem is invoked to forbid alternatives. The main hypotheses are explicit and the results are conditional on them; if the spectral gap fails, the results do not apply rather than becoming circular.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. Its assumptions are explicitly stated hypotheses on the system, and its technical tools are standard results from stochastic analysis and semigroup theory, mostly cited from the authors' prior work and the literature.

assumptions (6)
  • domain assumption Global Lipschitz and smoothness assumptions on f and g (HNL)
    Stated in Section 2.1; ensures Nemytskii operators are smooth between Sobolev spaces and that the SPDE is well-posed. The authors note f,g can be modified outside the region of interest, so this is not restrictive for the local behavior.
  • domain assumption Traveling wave existence, exponential decay, and spectral gap (HTw)
    Stated in Section 2.1; the spectral gap with simple zero eigenvalue is the main stability hypothesis. It is invoked throughout for exponential decay of the semigroup (e.g., (3.5)) and is load-bearing for all bounds.
  • domain assumption Orthogonality and normalization of initial condition (HV*)
    Stated in Section 2.1; projects out the neutral translation mode and sets the initial perturbation scale. This is needed to define the expansion terms and control the residual.
  • domain assumption Noise covariance assumptions (Hq)
    Stated in Section 2.1; translationally invariant, colored noise with non-negative definite Fourier symbol. This ensures the noise is a valid Q-Wiener process and that Hilbert-Schmidt bounds hold.
  • domain assumption Growth bound on functional φ (Hφ)
    Stated in Section 2.2; a smooth functional with polynomial growth. Required to control the remainder in the Taylor expansion of φ(Y_tay).
  • standard math Mild Itô formula for SPDEs
    Invoked in Lemma 6.1 and elsewhere, cited from Da Prato, Jentzen, and Röckner [16]. Standard but non-trivial; the paper does not reprove it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Conditional Speed and Shape Corrections for Travelling Wave Solutions to Stochastically Perturbed Reaction-Diffusion Systems." pith.science (2026). https://pith.science/paper/DG6TCQRD

@misc{pith2026250202164,
  author       = {Pith},
  title        = {Pith review of: Conditional Speed and Shape Corrections for Travelling Wave Solutions to Stochastically Perturbed Reaction-Diffusion Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DG6TCQRD}},
  note         = {Machine review of arXiv:2502.02164}
}
read the original abstract

In this work we perform rigorous small noise expansions to study the impact of stochastic forcing on the behaviour of planar travelling wave solutions to reaction-diffusion equations on cylindrical domains. In particular, we use a stochastic freezing approach that allows effective limiting information to be extracted concerning the behaviour of the stochastic perturbations from the deterministic wave. As an application, this allows us to provide a rigorous definition for the stochastic corrections to the wave speed. In addition, our approach allows their size to be computed to any desired order in the noise strength, provided that sufficient smoothness is available.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Microscopic Variability Alters Macroscopic Rotation Speed in Stochastic Spiral Waves

    nlin.PS 2025-11 conditional novelty 6.0 of 10

    Noise slows spiral-wave rotation at second order in noise strength, via an always-negative instantaneous term plus an orbital-drift term that can have either sign.

Reference graph

Works this paper leans on

49 extracted references · 48 canonical work pages · cited by 1 Pith paper

  1. [1]

    R. A. Adams and J. J. Fournier (2003), Sobolev spaces. Elsevier

    work page 2003

  2. [2]

    Z. P. Adams (2024), Quasi-Ergodicity of transient patterns in s tochastic reaction-diffusion equations. Electronic Journal of Probability 29, 1–29

    work page 2024

  3. [3]

    Z. P. Adams and J. MacLaurin (2025), The isochronal phase of s tochastic pde and integral equations: Metastability and other properties. Journal of Differential Equations 414, 773–816. 46

    work page 2025

  4. [4]

    Agresti and M

    A. Agresti and M. Veraar (2023), Reaction-diffusion equations with transport noise and critical super- linear diffusion: local well-posedness and positivity. Journal of Differential Equations 368, 247–300

    work page 2023

  5. [5]

    Agresti and M

    A. Agresti and M. Veraar (2024), The critical variational sett ing for stochastic evolution equations. Probability Theory and Related Fields 188(3), 957–1015

    work page 2024

  6. [6]

    Agresti and M

    A. Agresti and M. Veraar (2024), Reaction-diffusion equations with transport noise and critical super- linear diffusion: Global well-posedness of weakly dissipative systems. SIAM Journal on Mathematical Analysis 56(4), 4870–4927

    work page 2024

  7. [7]

    Armero, J

    J. Armero, J. Sancho, J. Casademunt, A. Lacasta, L. Ramirez -Piscina, and F. Sagu´ es (1996), External fluctuations in front propagation. Physical review letters 76(17), 3045

    work page 1996

  8. [8]

    Beyn and V

    W.-J. Beyn and V. Th¨ ummler (2004), Freezing solutions of equiva riant evolution equations. SIAM Journal on Applied Dynamical Systems 3(2), 85–116

    work page 2004

Show all 49 references
  1. [9]

    L. A. Bianchi, D. Bl¨ omker and P. Wacker (2017), Pattern size in G aussian fields from spinodal decom- position. SIAM Journal on Applied Mathematics 77(4), 1292–1319

    2017

  2. [10]

    Birzu, O

    G. Birzu, O. Hallatschek and K. S. Korolev (2018), Fluctuations uncover a distinct class of traveling waves. Proceedings of the National Academy of Sciences 115(16), E3645–E3654

    2018

  3. [11]

    M. v. d. Bosch and H. J. Hupkes (2024), Multidimensional Stabilit y of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems. arXiv preprint arXiv:2406.04232

    2024 arXiv

  4. [12]

    P. C. Bressloff and Z. P. Kilpatrick (2015), Nonlinear Langevin eq uations for wandering patterns in stochastic neural fields. SIAM Journal on Applied Dynamical Systems 14(1), 305–334

    2015

  5. [13]

    P. C. Bressloff and M. A. Webber (2012), Front propagation in s tochastic neural fields. SIAM Journal on Applied Dynamical Systems 11(2), 708–740

    2012

  6. [14]

    Chow (2014), Stochastic partial differential equations

    P.-L. Chow (2014), Stochastic partial differential equations . CRC Press

    2014

  7. [15]

    Cox and J

    S. Cox and J. van Winden (2024), Sharp supremum and H \” older bounds for stochastic integrals indexed by a parameter. arXiv preprint arXiv:2409.13615

    2024 arXiv

  8. [16]

    Da Prato, A

    G. Da Prato, A. Jentzen and M. R¨ ockner (2019), A mild Itˆ o fo rmula for SPDEs. Transactions of the American Mathematical Society

    2019

  9. [17]

    Da Prato and J

    G. Da Prato and J. Zabczyk (2014), Stochastic equations in infinite dimensions , Vol. 152. Cambridge university press

    2014

  10. [18]

    M. V. Day (1990), Large deviations results for the exit problem with characteristic boundary. Journal of mathematical analysis and applications 147(1), 134–153

    1990

  11. [19]

    di Nunno and B

    G. di Nunno and B. Oksendal (eds.) (2011), Advanced Mathematical Methods for Finance . Springer

    2011

  12. [20]

    W. G. Faris and G. Jona-Lasinio (1982), Large fluctuations for a nonlinear heat equation with noise. Journal of Physics A: Mathematical and General 15(10), 3025

    1982

  13. [21]

    C. L. E. Franzke, T. J. O’Kane, J. Berner, P. D. Williams and V. Lu carini (2015), Stochastic climate theory and modeling. Wiley Interdisciplinary Reviews: Climate Change 6(1), 63–78

    2015

  14. [22]

    M. I. Freidlin and A. D. Wentzell (1998), Random perturbations . In: Random perturbations of dynamical systems. Springer, pp. 15–43

    1998

  15. [23]

    Garc ´ ıa-Ojalvo and J

    J. Garc ´ ıa-Ojalvo and J. Sancho (2012),Noise in spatially extended systems . Springer Science & Business Media

    2012

  16. [24]

    Gawarecki and V

    L. Gawarecki and V. Mandrekar (2010), Stochastic differential equations in infinite dimensions: w ith applications to stochastic partial differential equations . Springer Science & Business Media. 47

    2010

  17. [25]

    Goldys and J

    B. Goldys and J. Van Neerven (2003), Transition semigroups of Banach space-valued Ornstein– Uhlenbeck processes. Acta Applicandae Mathematica 76, 283–330

    2003

  18. [26]

    Gowda and C

    K. Gowda and C. Kuehn (2015), Early-warning signs for patter n-formation in stochastic partial differ- ential equations. Communications in Nonlinear Science and Numerical Simulat ion 22(1), 55–69

    2015

  19. [27]

    Hairer (2009), An Introduction to Stochastic PDEs

    M. Hairer (2009), An Introduction to Stochastic PDEs. http://www.hairer.org/notes/SPDEs.pdf

    2009

  20. [28]

    C. H. S. Hamster and H. J. Hupkes (2019), Stability of Traveling Waves for Reaction-Diffusion Equations with Multiplicative Noise. SIAM Journal on Applied Dynamical Systems 18(1), 205–278

    2019

  21. [29]

    C. H. S. Hamster and H. J. Hupkes (2020), Stability of traveling waves for systems of reaction-diffusion equations with multiplicative noise. SIAM Journal on Mathematical Analysis 52(2), 1386–1426

    2020

  22. [30]

    C. H. S. Hamster and H. J. Hupkes (2020), Stability of traveling waves on exponentially long timescales in stochastic reaction-diffusion equations. SIAM Journal on Applied Dynamical Systems 19(4), 2469– 2499

    2020

  23. [31]

    C. H. S. Hamster and H. J. Hupkes (2020), Travelling waves for reaction–diffusion equations forced by translation invariant noise. Physica D: Nonlinear Phenomena 401, 132233

    2020

  24. [32]

    Inglis and J

    J. Inglis and J. MacLaurin (2016), A general framework for st ochastic traveling waves and patterns, with application to neural field equations. SIAM Journal on Applied Dynamical Systems 15(1), 195–234

    2016

  25. [33]

    Karczewska (2005), Stochastic integral with respect to c ylindrical Wiener process

    A. Karczewska (2005), Stochastic integral with respect to c ylindrical Wiener process. arXiv preprint math/0511512

    2005 arXiv

  26. [34]

    Kuehn (2013), A Mathematical Framework for Critical Tran sitions: Normal Forms, Variance and Applications

    C. Kuehn (2013), A Mathematical Framework for Critical Tran sitions: Normal Forms, Variance and Applications. Journal of nonlinear science 23(3), 457–510

    2013

  27. [35]

    Kuehn (2013), Warning Signs for Wave Speed Transitions of N oisy Fisher–KPP invasion fronts

    C. Kuehn (2013), Warning Signs for Wave Speed Transitions of N oisy Fisher–KPP invasion fronts. Theoretical Ecology 6(3), 295–308

    2013

  28. [36]

    Kuehn (2020), Travelling waves in monostable and bistable sto chastic partial differential equations

    C. Kuehn (2020), Travelling waves in monostable and bistable sto chastic partial differential equations. Jahresbericht der Deutschen Mathematiker-Vereinigung 122, 73–107

    2020

  29. [37]

    Kuske, C

    R. Kuske, C. Lee and V. Rottsch¨ afer (2017), Patterns and coherence resonance in the stochastic Swift- Hohenberg equation with Pyragas control: The Turing bifurcation c ase. Physica D: Nonlinear Phe- nomena

    2017

  30. [38]

    MacLaurin (2023), Phase Reduction of Waves, Patterns, a nd Oscillations Subject to Spatially Ex- tended Noise

    J. MacLaurin (2023), Phase Reduction of Waves, Patterns, a nd Oscillations Subject to Spatially Ex- tended Noise. SIAM Journal on Applied Mathematics 83(3), 1215–1244

    2023

  31. [39]

    Prato and J

    G. Prato and J. Zabczyk (1992), Stochastic equations in infinite dimensions . Cambridge University Press, Cambridge New York

    1992

  32. [40]

    Pr´ evˆ ot and M

    C. Pr´ evˆ ot and M. R¨ ockner (2007),A concise course on stochastic partial differential equatio ns, Vol

    2007

  33. [41]

    Russo and P

    F. Russo and P. Vallois (1991), Int´ egrales progressive, r´ et rograde et sym´ etrique de processus non adapt´ es.Comptes rendus de l’Acad´ emie des sciences. S´ erie 1, Math´ematique 312(8), 615–618

    1991

  34. [42]

    Salins and K

    M. Salins and K. Spiliopoulos (2021), Metastability and exit problem s for systems of stochastic reaction– diffusion equations. The Annals of Probability 49(5), 2317–2370

    2021

  35. [43]

    Schimansky-Geier and C

    L. Schimansky-Geier and C. Z¨ ulicke (1991), Kink propagation in duced by multiplicative noise. Zeits- chrift f¨ ur Physik B Condensed Matter 82(1), 157–162

    1991

  36. [44]

    Shardlow (2005), Numerical simulation of stochastic PDEs fo r excitable media

    T. Shardlow (2005), Numerical simulation of stochastic PDEs fo r excitable media. Journal of compu- tational and applied mathematics 175(2), 429–446. 48

    2005

  37. [45]

    Talagrand (2005), The generic chaining: upper and lower bounds of stochastic p rocesses

    M. Talagrand (2005), The generic chaining: upper and lower bounds of stochastic p rocesses. Springer Science & Business Media

    2005

  38. [46]

    Veraar and L

    M. Veraar and L. Weis (2011), A note on maximal estimates for s tochastic convolutions. Czechoslovak mathematical journal 61(3), 743

    2011

  39. [47]

    Vi˜ nals, E

    J. Vi˜ nals, E. Hern´ andez-Garc ´ ıa, M. San Miguel and R. Toral (1991), Numerical study of the dynamical aspects of pattern selection in the stochastic Swift-Hohenberg e quation in one dimension. Physical Review A 44(2), 1123

    1991

  40. [48]

    Zhang, A

    J. Zhang, A. Holden, O. Monfredi, M. Boyett and H. Zhang (200 9), Stochastic vagal modulation of cardiac pacemaking may lead to erroneous identification of cardiac c haos. Chaos: An Interdisciplinary Journal of Nonlinear Science 19(2), 028509

  41. [49]

    Zumbrun and P

    K. Zumbrun and P. Howard (1998), Pointwise Semigroup Method s and Stability of Viscous Shock Waves. Indiana Univ. Math. J. 47(3), 741–871. 49

    1998

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.