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The Yamada-Watanabe-Engelbert theorem for SPDEs in Banach spaces

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Pith's one-line read A unified proof shows that for SPDEs in Banach spaces, weak existence plus pathwise uniqueness is equivalent to the existence of a strong solution plus joint weak uniqueness, and that a single measurable map produces solutions from…

desk verdict A serious, well-built Kurtz-based Yamada-Watanabe-Engelbert framework for SPDEs, with a genuinely new measurable representation of stochastic integrals, though the abstract overstates the arbitrary-Banach-space scope for analytically weak solutions. read the letter →

arxiv 2502.00189 v4 pith:ILOFN52Y submitted 2025-01-31 math.PR

Esmée Theewis This is my paper · ORCID
classification math.PR MSC 60H1560H05
keywords Yamada–WatanabetheoremYamada–Watanabe–EngelbertSPDEsinBanachspacescylindricalBrownianmotionpathwiseuniquenessjointweakmartingaletype2UMD
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 proves a Yamada–Watanabe–Engelbert theorem for stochastic partial differential equations in Banach spaces: under one set of assumptions, weak existence plus pathwise uniqueness is equivalent to the existence of a strong solution plus joint weak uniqueness, and also to the existence of a single measurable map that turns any initial value and any cylindrical Brownian motion into a solution. The result covers analytically strong, analytically weak, mild, and weakly mild solution notions in a unified way, and it holds in martingale type 2 or UMD Banach spaces, with the analytically weak case working in arbitrary Banach spaces. The value of the paper is that the classical equivalence, previously proved separately for different SPDE frameworks, follows from one abstract theorem once the stochastic integral admits a measurable representation that depends on the law of the integrand and the noise. That representation is proved here for infinite dimensions and is of independent interest.

What carries the argument

The load-bearing object is a measurable representation $I$ of the stochastic integral: for any stochastically integrable process $f$ and cylindrical Brownian motion $W$, one has $I(f(\omega), W(\omega), \mathrm{Law}(f,W)) = (\int_0^\cdot f\,dW)(\omega)$ for almost every $\omega$. Because the representation takes the joint law of integrand and noise as an argument, the condition 'u solves the SPDE' becomes a property of the joint distribution of $u$ and $W$, which is exactly what is needed to apply an abstract Yamada–Watanabe–Engelbert theorem for compatible solutions. The other central piece is the compatibility structure: a family of $\sigma$-algebras $\mathcal{B}^t$ encoding the information of paths up to time $t$, which lets adaptedness and independence of noise increments be read off from the law of $(u, u(0), W)$.

What would settle it

Test the analytically weak case in a Banach space that is neither UMD nor of martingale type 2, such as $L^1$, with a nontrivial cylindrical Wiener noise: the paper asserts the equivalence holds there, and a failure of pathwise uniqueness or of the strong-solution-map conclusion in such an example would refute the claim.

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Extended reading notes

Core claim

The central claim is Theorem 3.1. Given a Polish path space $B$ embedded continuously in the space of continuous $Z$-valued paths, and given a collection $C$ of solution conditions (chosen from analytically strong, analytically weak, mild, or weakly mild, with optional moment and a.s. constraints), for a fixed initial law $\mu$ the following are equivalent: (a) a $C$-weak solution exists and pathwise uniqueness holds; (b) a $C$-strong solution exists and joint weak uniqueness holds; (c) joint weak uniqueness holds and there is a Borel measurable map $F_\mu: Z\times W\to B$ such that, on any filtered probability space, any $U$-cylindrical Brownian motion $W$, and any initial value with law $\mu$, the pair $(F_\mu(u_0,W), W)$ is a $C$-strong solution, with paths up to time $t$ depending measurably on the paths of $(u_0,W)$ up to time $t$. The paper also derives the classical Yamada–Watanabe theorem as a corollary, giving a single map $F$ independent of the initial law under an all-initial-laws hypothesis. The proof runs through an abstract Yamada–Watanabe–Engelbert theorem for compatible solutions; the main input that makes the SPDE case work is a measurable representation of the stochastic integral in martingale type 2 or UMD spaces.

Load-bearing premise

The path space $B$ in which solutions are required to live must be Polish and embed continuously into the space of continuous $Z$-valued paths, so that Kuratowski's theorem and measurable selection arguments apply.

Editorial extensions

If this is right

  • All previously known Yamada–Watanabe-type results for SPDEs, including mild solutions, the variational framework, and analytically weak solutions, become special cases of a single theorem.
  • The reverse direction of the equivalence, from strong existence plus joint weak uniqueness to pathwise uniqueness, now applies to settings it was not available for: SPDEs with time-dependent operators, quasilinear SPDEs outside the variational framework, and semilinear SPDEs in critical spaces with transport noise.
  • The measurable representation $I$ of the stochastic integral gives a canonical, law-dependent version of the integral in infinite dimensions, usable independently of Yamada–Watanabe theory.
  • When the hypotheses hold for every initial law in a class $\mathcal{M}$, Corollary 3.15 upgrades the solution map to a single map $F$ that does not depend on the initial law.

Reading between the lines

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

  • The law-dependent representation suggests a general recipe: any stochastic equation whose solution condition can be expressed as a measurable function of the path and the joint law of the path and noise should fall under the same abstract equivalence, so the framework may be adapted to other solution notions that fit the compatibility structure.
  • The Polish path-space assumption marks a likely boundary: solution spaces such as spaces of weakly continuous paths with a non-metrizable weak topology, or non-Polish Banach spaces, would need a different representation argument, since Kuratowski's theorem and measurable selections are used essentially.
  • A natural extension would be to replace the continuous path space $C(\bar I_T;Z)$ by a Skorokhod space, which would allow discontinuous solutions and Lévy-type noise; the paper's compatibility argument is set up for continuous paths, so the stochastic integral representation would need to be rebuilt.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper proves a Yamada–Watanabe–Engelbert equivalence for a wide class of SPDE solution notions in Banach spaces. Under Assumption 2.1, which fixes a Polish path space B continuously embedded into C(I_T;Z), the paper shows that for any admissible collection C of conditions (covering analytically strong, analytically weak, mild, and weakly mild solutions) the following are equivalent: existence of a C-weak solution plus pathwise uniqueness, existence of a C-strong solution plus joint weak uniqueness, and joint weak uniqueness plus the existence of a single measurable solution map F_μ from Z×W to B (Theorem 3.1). The proof reduces the statement to Kurtz's abstract theorem after showing that all conditions in C can be expressed through the joint law of (u,W). A key tool is a measurable representation of the stochastic integral in martingale type 2 and UMD Banach spaces (Theorem 3.7). The paper also derives an analogue of the classical Yamada–Watanabe theorem (Corollary 3.15) and discusses applications to variational, critical-space, and weak/mild solution settings.

Significance. If the results stand, the paper provides a valuable unification: it covers several existing Yamada–Watanabe theorems (Ondreját, Röckner–Schmuland–Zhang, Kunze) and extends the Yamada–Watanabe–Engelbert equivalence to solution classes for which it was not previously available. The measurable representation of stochastic integrals is a useful contribution in its own right and is the technical heart of the paper. The proof is detailed and does not rely on fitted parameters or on assuming the conclusion. The main caveat is that the scope of the 'arbitrary Banach spaces' claim for analytically weak solutions is narrower than the abstract suggests, because the path space B is required to be Polish and to consist of strongly continuous paths.

major comments (2)
  1. [Abstract; §2, Assumption 2.1 and Definition 2.5; Theorem 3.1] The abstract's statement that for analytically weak solutions 'the results hold in arbitrary Banach spaces' is broader than what Assumption 2.1 delivers. In Definition 2.5 every C-weak solution must satisfy u ∈ B a.s., where B is Polish and B embeds continuously into C(I_T;Z), and the proof uses this path-space regularity essentially (Lemma 3.3, Kuratowski arguments in Corollaries 3.4 and 3.9, Blackwell–Dubins representations, and the measurable-selection step in Step 5 of Theorem 3.1). For a solution class whose paths are only weakly continuous, condition (5) gives only scalar continuity of t → ⟨u(t), z*⟩ and does not force strong continuity in Z. Example 2.13c encodes weak continuity only together with strong continuity in a larger space, using compactness (Rellich); in a general Banach space such as ℓ², weakly continuous but not strongly continuous paths are not covered by any B ⊂ C([0,T];Z). Thus the theorem is correct as a conditional statement, but the claimed 'arbitrary Banach spaces' scope should be qualified, e.g. 'arbitrary separable Banach spaces with solution paths constrained to a Polish subspace of strongly continuous paths', or a proof of the stronger claim should be supplied.
  2. [Corollary 3.15] The map F in Corollary 3.15(ii) is only shown to be separately measurable: for each z ∈ Z0, F(z,·) is B_t(W)^{P∞}/B_t(B)-measurable, but joint measurability of F: Z0 × W → B is not established. The proof sets F(z,w) = F_{δ_z}(z,w) and does not show that z ↦ F_{δ_z}(z,w) is measurable in z for a fixed w. Consequently, F(u0,W) is not proved to be a random variable for a general random initial condition u0, and Corollary 3.15 does not deliver the 'unique strong solution' notion of [24, Def. 1.6 Chap. 4] or [42, Def. 1.9], which require a measurable solution map in both variables. Either joint measurability should be proved, or the statement should explicitly say that F is only a separately measurable selection and the claim that this is the classical Yamada–Watanabe theorem should be weakened accordingly.
minor comments (3)
  1. [Notation, p. 4] The definition of ar I_T is not stated clearly enough: since ar I_T is used throughout as the domain of continuous paths and contains 0, it should be defined explicitly as [0,T] for T < ∞ and [0,∞) for T = ∞.
  2. [Theorem 3.7 and Corollary 3.9] The UMD version of the representation theorem is stated only for T ∈ (0,∞), while Theorem 3.1 allows T = ∞; the proof handles T = ∞ through the local conditions (2) and (9), but a sentence in the statement of Theorem 3.7 or Corollary 3.9 explaining this localisation would prevent confusion.
  3. [Corollary 3.15] The identification of μ ∈ P(Z0) with its trivial extension in P(Z) is made in the text immediately before the corollary, but it is easy to miss; since F_μ is a priori defined only for μ ∈ P(Z), this identification should be stated as part of the corollary's assumptions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.1 is deduced from Kurtz's external abstract theorem and an independently constructed measurable stochastic-integral representation; no fitted parameter, definitional collapse, or self-citation chain is load-bearing.

full rationale

The paper's central claim is not assumed or introduced through a fitted input. Theorem 3.1 is obtained by specializing Kurtz's abstract Yamada–Watanabe–Engelbert theorem (Theorem 1.6) to the SPDE setting: the proof sets S1 = B, S2 = Z × W, defines the compatibility structure C and the set Γ so that membership in Γ exactly expresses the chosen solution conditions, and then transfers Kurtz's equivalence through Steps 1–5. The key auxiliary result, the measurable representation I of the stochastic integral, is proved directly from the functional representation of limits in probability and the Itô isomorphism, not assumed. Pathwise uniqueness, joint weak uniqueness, and strong solutions are shown to correspond to, respectively, pointwise uniqueness, joint uniqueness in law, and strong solutions in the abstract framework; this correspondence is proven rather than built into the definitions. No parameters are fitted to data, no prediction is renamed as an outcome, and no load-bearing claim rests on a self-citation. The cited results of Kurtz, Ondreját, Kunze, and others are external and independent of the present paper. The criticism that the Polish path-space assumption restricts the scope of the 'arbitrary Banach spaces' claim is a legitimate accuracy/scope concern, but it is not a circularity: it concerns whether all intended solution classes fit Assumption 2.1, not whether the theorem's conclusion is equivalent to its hypotheses by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no fitted parameters or invented physical entities. The theorem is conditional on structural hypotheses on the Banach spaces, the path space, the coefficients, and the abstract Yamada-Watanabe theorem of Kurtz; these are standard mathematical tools rather than circular inputs.

assumptions (5)
  • standard math Kurtz's abstract Yamada-Watanabe-Engelbert theorem for C-compatible solutions and its supporting lemmas (Theorem 1.6 here; Kurtz 2007 and 2014).
    Used as the black box that converts weak existence plus pathwise uniqueness into strong existence plus joint uniqueness in law; the paper proves the SPDE application on top of it.
  • standard math Stochastic integral theory in M-type 2 and UMD Banach spaces: definition, isometries and Burkholder bounds, characterization of stochastic integrability, and progressive measurability (van Neerven-Veraar-Weis, Ondrejat-Seidler).
    Needed to make conditions (1)-(12) meaningful and to prove the measurable representation Theorem 3.7.
  • standard math Functional representation for limits in probability (Kallenberg, Prop. 5.32) and Blackwell-Dubins almost sure representation.
    Used multiple times in Lemma 3.6 to construct the measurable stochastic-integral representation, and in Theorem 1.6 to couple arbitrary laws.
  • domain assumption Assumption 2.1: finite or infinite time horizon, separable Banach spaces Y and Z with Y embedding into Z, Polish path space B embedding continuously into C(I_T; Z), separable Hilbert space U with fixed basis, and measurable adapted coefficients b and sigma.
    Defines the class of SPDEs to which the theorem applies; not proved because it is the model setting.
  • standard math Borel isomorphism and Kuratowski theorems for Polish and Suslin spaces, including the statements used in the appendices.
    Used throughout Lemma 3.3, Corollary 3.9, and the proof of the additional measurability property in Theorem 3.1.

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Pith. "Pith review of The Yamada-Watanabe-Engelbert theorem for SPDEs in Banach spaces." pith.science (2026). https://pith.science/paper/ILOFN52Y

@misc{pith2026250200189,
  author       = {Pith},
  title        = {Pith review of: The Yamada-Watanabe-Engelbert theorem for SPDEs in Banach spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ILOFN52Y}},
  note         = {Machine review of arXiv:2502.00189}
}
abstract

We give a unified proof of the Yamada-Watanabe-Engelbert theorem for various notions of solutions for SPDEs in Banach spaces with cylindrical Wiener noise. We use Kurtz' generalization of the theorems of Yamada, Watanabe and Engelbert. In addition, we deduce the classical Yamada-Watanabe theorem for SPDEs, with a slightly different notion of `unique strong solution' than that corresponding to the result of Kurtz. Our setting includes analytically strong solutions, analytically weak solutions and mild solutions. Moreover, our approach offers flexibility with regard to the function spaces and integrability conditions that are chosen in the solution notion (and affect the meaning of existence and uniqueness). All results hold in Banach spaces which are either martingale type 2 or UMD. For analytically weak solutions, the results hold in arbitrary Banach spaces. In particular, our results extend the Yamada-Watanabe theorems of Ondrej\'at for mild solutions in 2-smooth Banach spaces, of R\"ockner et al. for the variational framework and of Kunze for analytically weak solutions, and cover many new settings. As a tool, and of interest itself, we construct a measurable representation $I$ of the stochastic integral in a martingale type 2 or UMD Banach space, in the sense that for any stochastically integrable process $f$ and cylindrical Brownian motion $W$, we have $I(f(\omega),W(\omega),\mathrm{Law}(f,W)) = (\int_0^{\cdot} f\, \mathrm{d}W)(\omega)$ for almost every $\omega$.

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

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    math.PR 2025-01 conditional novelty 4.0 of 10

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Reference graph

Works this paper leans on

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

  1. [1]

    Agresti and M

    A. Agresti and M. Sauerbrey. Well-posedness of the stochastic thin-film equation with an interface potential. 2024. arXiv: 2403.12652

    work page arXiv 2024

  2. [2]

    Nonlinear parabolic stochastic evolution equations in critical spaces part I. Stochastic maximal regularity and local existence

    A. Agresti and M. Veraar. “Nonlinear parabolic stochastic evolution equations in critical spaces part I. Stochastic maximal regularity and local existence”. In: Nonlinearity 35.8 (2022), pp. 4100–4210

    work page 2022

  3. [3]

    Nonlinear parabolic stochastic evolution equations in critical spaces part II. Blow-up criteria and instantaneous regularization

    A. Agresti and M. Veraar. “Nonlinear parabolic stochastic evolution equations in critical spaces part II. Blow-up criteria and instantaneous regularization”. In: J. Evol. Equ. 22.56 (2022)

    work page 2022

  4. [4]

    Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity

    A. Agresti and M. Veraar. “Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity”. In: J. Differ. Equ. 368 (2023), pp. 247–300

    work page 2023

  5. [5]

    Stochastic Navier–Stokes equations for turbulent flows in critical spaces

    A. Agresti and M. Veraar. “Stochastic Navier–Stokes equations for turbulent flows in critical spaces”. In: Commun. Math. Phys. 405.43 (2024)

    work page 2024

  6. [6]

    An extrapolation result in the variational setting: improved regularity, compactness, and applications to quasilinear systems

    S. Bechtel and M. Veraar. An extrapolation result in the variational setting: improved regu- larity, compactness, and applications to quasilinear systems . 2023. arXiv: 2311.01271

    work page Pith review arXiv 2023

  7. [7]

    An extension of Skorohod’s almost sure representation theorem

    D. Blackwell and L. E. Dubins. “An extension of Skorohod’s almost sure representation theorem”. In: 89.4 (1983), pp. 691–692

    work page 1983

  8. [8]

    Uniqueness of the nonlinear Schr¨ odinger equation driven by jump processes

    A. de Bouard, E. Hausenblas, and M. Ondrej´ at. “Uniqueness of the nonlinear Schr¨ odinger equation driven by jump processes”. In: Nonlinear Differ. Equ. Appl. 26.22 (2019)

    work page 2019

Show all 48 references
  1. [9]

    On stochastic convolution in Banach spaces and applications

    Z. Brzeźniak. “On stochastic convolution in Banach spaces and applications”. In: Stoch. Stoch. Rep. 61.3 (1997), pp. 245–295

    1997

  2. [10]

    Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces

    Z. Brzeźniak and D. Gątarek. “Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces”. In: Stoch. Process. Their Appl. 84.2 (1999), pp. 187– 225. THE YAMADA–WATANABE–ENGELBERT THEOREM 39

    1999

  3. [11]

    Uniqueness of martingale solutions for the stochastic nonlinear Schr¨ odinger equation on 3d compact manifolds

    Z. Brzeźniak, F. Hornung, and L. Weis. “Uniqueness of martingale solutions for the stochastic nonlinear Schr¨ odinger equation on 3d compact manifolds”. In: Stoch. Partial Differ. Equ.: Anal. Comput. 10.3 (2022), pp. 828–857

    2022

  4. [12]

    Invariant measures for stochastic nonlinear beam and wave equations

    Z. Brzeźniak, M. Ondrej´ at, and J. Seidler. “Invariant measures for stochastic nonlinear beam and wave equations”. In: J. Differ. Equ. 260.5 (2016), pp. 4157–4179

    2016

  5. [13]

    Cazenave and A

    T. Cazenave and A. Haraux. An introduction to semilinear evolution equations . Rev. ed. Oxford lecture series in mathematics and its applications 13. Oxford: Clarendon Press, 1998

    1998

  6. [14]

    On decoupling in Banach spaces

    S. Cox and S. Geiss. “On decoupling in Banach spaces”. In: J. Theor. Probab. 34.3 (2021), pp. 1179–1212

    2021

  7. [15]

    Vector-valued decoupling and the Burkholder–Davis–Gundy inequal- ity

    S. Cox and M. Veraar. “Vector-valued decoupling and the Burkholder–Davis–Gundy inequal- ity”. In: Illinois J. Math. 55.1 (2011)

    2011

  8. [16]

    On the theorem of T. Yamada and S. Watanabe

    H. J. Engelbert. “On the theorem of T. Yamada and S. Watanabe”. In: Stoch. Stoch. Rep. 36.3 (1991), pp. 205–216

    1991

  9. [17]

    Fahim, E

    K. Fahim, E. Hausenblas, and K. H. Karlsen. Yamada-Watanabe uniqueness results for SPDEs driven by Wiener and pure jump processes . 2025. arXiv: 2501.02924

    2025 arXiv

  10. [18]

    Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier–Stokes equations

    F. Flandoli, L. Galeati, and D. Luo. “Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier–Stokes equations”. In: J. Evol. Equ. 21.1 (2021), pp. 567–600

    2021

  11. [19]

    Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure

    F. Flandoli and D. Luo. “Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure”. In: Ann. Probab. 48.1 (2020)

    2020

  12. [20]

    High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations

    F. Flandoli and D. Luo. “High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations”. In: Probab. Theory Relat. Fields 180.1 (2021), pp. 309–363

    2021

  13. [21]

    D. H. Fremlin. Measure theory. Vol. 2 . Second edition. Colchester: Torres Fremlin, 2010

    2010

  14. [22]

    Hyt¨ onen, J

    T. Hyt¨ onen, J. van Neerven, M. Veraar, and L. Weis. Analysis in Banach spaces Vol. I. Martingales and Littlewood-Paley theory . Cham: Springer, 2016

    2016

  15. [23]

    Hyt¨ onen, J

    T. Hyt¨ onen, J. van Neerven, M. Veraar, and L. Weis. Analysis in Banach spaces Vol. II. Probabilistic methods and operator theory. Cham: Springer, 2017

    2017

  16. [24]

    Ikeda and S

    N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes . North- Holland Mathematical Library 24. Amsterdam: North-Holland Pub. Co., 1981

    1981

  17. [25]

    Weak and strong solutions of stochastic differential equations

    J. Jacod. “Weak and strong solutions of stochastic differential equations”. In: Stochastics 3.1 (1980), pp. 171–191

    1980

  18. [26]

    Kallenberg

    O. Kallenberg. Foundations of modern probability. Cham: Springer, 2021

    2021

  19. [27]

    Karatzas and S

    I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus. New York: Springer, 1998

    1998

  20. [28]

    A. S. Kechris. Classical descriptive set theory . New York: Springer, 1995

    1995

  21. [29]

    On a class of martingale problems on Banach spaces

    M. Kunze. “On a class of martingale problems on Banach spaces”. In: Electron. J. Probab. 18.104 (2013), pp. 1–30

    2013

  22. [30]

    The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities

    T. G. Kurtz. “The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities”. In: Electron. J. Probab. 12.33 (2007)

    2007

  23. [31]

    Weak and strong solutions of general stochastic models

    T. G. Kurtz. “Weak and strong solutions of general stochastic models”. In: Electron. Com- mun. Probab. 19.58 (2014)

    2014

  24. [32]

    Liu and M

    W. Liu and M. R¨ ockner. Stochastic partial differential equations: an introduction. New York: Springer, 2015

    2015

  25. [33]

    Stochastic integration in Banach spaces – a survey

    J. van Neerven, M. Veraar, and L. Weis. “Stochastic integration in Banach spaces – a survey”. In: Stochastic analysis: a series of lectures . Vol. 68. Basel: Springer, 2015, pp. 297–332

    2015

  26. [34]

    Stochastic integration in UMD Banach spaces

    J. M. A. M. van Neerven, M. C. Veraar, and L. Weis. “Stochastic integration in UMD Banach spaces”. In: Ann. Probab. 35.4 (2007)

    2007

  27. [35]

    Stochastic evolution equations in UMD Banach spaces

    J. M. A. M. van Neerven, M. C. Veraar, and L. Weis. “Stochastic evolution equations in UMD Banach spaces”. In: J. Funct. Anal. 255.4 (2008), pp. 940–993

    2008

  28. [36]

    van Neerven

    J. van Neerven. Functional analysis. 1st ed. Cambridge University Press, 2022

    2022

  29. [37]

    Conditions for stochastic integrability in UMD Banach spaces

    J. van Neerven, M. Veraar, and L. Weis. “Conditions for stochastic integrability in UMD Banach spaces”. In: Banach Spaces and their Applications in Analysis . Berlin, Boston: De Gruyter, 2007. THE YAMADA–WATANABE–ENGELBERT THEOREM 40

    2007

  30. [38]

    Uniqueness for stochastic evolution equations in Banach spaces

    M. Ondrej´ at. “Uniqueness for stochastic evolution equations in Banach spaces”. In: Diss. Math. 426 (2004), pp. 1–63

    2004

  31. [39]

    On existence of progressively measurable modifications

    M. Ondrej´ at and J. Seidler. “On existence of progressively measurable modifications”. In: Electron. Commun. Probab. 18 (2013)

    2013

  32. [40]

    Martingales with values in uniformly convex spaces

    G. Pisier. “Martingales with values in uniformly convex spaces”. In: Israel J. Math. 20.3 (1975), pp. 326–350

    1975

  33. [41]

    On Cherny’s results in infinite dimensions: a theorem dual to Yamada–Watanabe

    M. Rehmeier. “On Cherny’s results in infinite dimensions: a theorem dual to Yamada–Watanabe”. In: Stoch. Partial Differ. Equ.: Anal. Comput. 9.1 (2021), pp. 33–70

    2021

  34. [42]

    Yamada-Watanabe theorem for stochastic evo- lution equations in infinite dimensions

    M. R¨ ockner, B. Schmuland, and X. Zhang. “Yamada-Watanabe theorem for stochastic evo- lution equations in infinite dimensions”. In: Condens. Matter Phys. 54 (2008), pp. 247–259

    2008

  35. [43]

    Local existence and non-explosion of solutions for sto- chastic fractional partial differential equations driven by multiplicative noise

    M. R¨ ockner, R. Zhu, and X. Zhu. “Local existence and non-explosion of solutions for sto- chastic fractional partial differential equations driven by multiplicative noise”. In: Stoch. Process. Their Appl. 124.5 (2014), pp. 1974–2002

    2014

  36. [44]

    Restricted Markov uniqueness for the stochastic quanti- zation of P (φ)2 and its applications

    M. R¨ ockner, R. Zhu, and X. Zhu. “Restricted Markov uniqueness for the stochastic quanti- zation of P (φ)2 and its applications”. In: J. Funct. Anal. 272.10 (2017), pp. 4263–4303

    2017

  37. [45]

    Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

    Y. Tahraoui and F. Cipriano. “Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions”. In: Stoch. Partial Differ. Equ.: Anal. Comput. 12.3 (2024), pp. 1699–1744

    2024

  38. [46]

    N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan. Probability distributions on Banach spaces. Dordrecht: Springer Netherlands, 1987

    1987

  39. [47]

    Stochastic integration in Banach spaces and applications to parabolic evo- lution equations

    M. C. Veraar. “Stochastic integration in Banach spaces and applications to parabolic evo- lution equations”. Dissertation (TU Delft). 2006

    2006

  40. [48]

    On the uniqueness of solutions of stochastic differential equations

    S. Watanabe and T. Yamada. “On the uniqueness of solutions of stochastic differential equations”. In: Kyoto J. Math. 11.1 (1971), pp. 155–167. (E. S. Theewis) Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands ...

    1971

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