Temporal quadratic and higher order variation for the nonlinear stochastic heat equation and applications to parameter estimation

Christian Olivera; C. Tudor

REVIEW 1 major objections 4 minor 31 references

Temporal quadratic and higher order variation for the nonlinear stochastic heat equation and applications to parameter estimation

T0 review · 1 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read The paper proves that temporal power variations of the nonlinear fractional heat equation converge to explicit functionals of the solution, and uses these limits to build consistent estimators for the drift $\theta$ and the…

desk verdict New and useful estimator for alpha in the nonlinear fractional heat equation, but a load-bearing independence claim in the proof of the variation limits is not justified as written; the paper deserves a serious referee if that gap can be closed. read the letter →

arxiv 2504.18450 v1 pith:WZZH7JQJ submitted 2025-04-25 math.PR

Christian Olivera , C. Tudor This is my paper · ORCID

classification math.PR MSC 60G1560H0560G18

keywords stochasticheatequationfractionalLaplacianBrownianmotionpowervariationquadraticparameterestimationnonlinearmultiplicativenoiseconsistency

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 establishes limit theorems for the temporal power variations of the solution to the nonlinear stochastic heat equation with a fractional Laplacian of order $\alpha\in(1,2]$, driven by spacetime white noise. Observing the solution at one fixed spatial point on an equidistant grid, it proves that a suitably renormalized quadratic variation converges in $L^1$ to an explicit constant times $\theta^{-1/\alpha}\int_0^1\sigma^2(u_\theta(s,x))\,ds$, and that, when the exponent $2\alpha/(\alpha-1)$ is an even integer, the corresponding power variation converges to a similar explicit integral. From these limits, the authors construct consistent estimators for the drift $\theta$ and for the fractional-Laplacian order $\alpha$. The point is that unknown parameters of a nonlinear SPDE can be recovered from the roughness of a single temporal path observed at discrete times.

What carries the argument

The engine is the representation of the linear solution's time process as a perturbed fractional Brownian motion (equation (28)): $u_0(t,x)=C_{0,\alpha}U_t+Y_t$, where $U$ is a fractional Brownian motion with Hurst index $H=(\alpha-1)/(2\alpha)$ and $Y$ is a self-similar Gaussian perturbation whose increments satisfy $E|Y_{i+1}-Y_i|^2\le C i^{-(\alpha+1)/\alpha}$ (bound (29)). Around this, Proposition 4 compares true nonlinear increments with modified increments of the linear solution at shifted times $t_i(\delta)=t_i-\delta^{\beta}$, $\beta=2\alpha/(2\alpha+1)$, with error $O(\delta^{4(\alpha-1)/(2\alpha+1)})$. The variation theorems then split the error into a nonlinear comparison term, a Gaussian fluctuation term controlled by the perturbed-fBm bounds in Lemmas 2 and 3, and a Riemann-sum term controlled by the solution's Hölder regularity.

What would settle it

Simulate the linear solution $u_0$ for a fixed $\alpha\in(1,2)$ and directly measure $E|Y_{i+1}-Y_i|^2$ for the perturbation term in (28); if the decay is slower than $i^{-(\alpha+1)/\alpha}$, the rates in Theorems 1–2 and the consistency proofs collapse. Alternatively, for $\alpha=3/2$, check numerically that $N^{-1/\alpha}\sum_{i=0}^{N-1}(u_\theta(t_{i+1},x)-u_\theta(t_i,x))^2$ approaches $C_{0,\alpha}^2\theta^{-1/\alpha}\int_0^1\sigma^2(u_\theta(s,x))\,ds$; a systematic mismatch would refute the main limit theorem.

Watch

Extended reading notes

Core claim

For the solution $u_\theta$ of $\partial_t u_\theta = -\theta(-\Delta)^{\alpha/2}u_\theta + \sigma(u_\theta)\dot W$ with zero initial condition, observed at times $t_i=i/N$ and a fixed spatial point, the renormalized temporal quadratic variation $N^{-1/\alpha}\sum_{i=0}^{N-1}(u_\theta(t_{i+1},x)-u_\theta(t_i,x))^2$ converges in $L^1$ to $C_{0,\alpha}^2\theta^{-1/\alpha}\int_0^1\sigma^2(u_\theta(s,x))\,ds$. When $2\alpha/(\alpha-1)$ is an even integer, the power variation $\sum_{i=0}^{N-1}|u_\theta(t_{i+1},x)-u_\theta(t_i,x)|^{2\alpha/(\alpha-1)}$ converges to $B_{0,\alpha}\theta^{-1/(\alpha-1)}\int_0^1\sigma^{2\alpha/(\alpha-1)}(u_\theta(s,x))\,ds$. These two limits are the basis for consistent in-probability estimators of $\alpha$ and $\theta$: $\hat\alpha_N = \log N/\log A_N$, $\hat\theta_{N,1}$ from the quadratic variation, and $\hat\theta_{N,2}$ from the higher-order variation.

Load-bearing premise

All stated convergence rates inherit their exponents from the perturbation bound $E|Y_{i+1}-Y_i|^2\le C i^{-(\alpha+1)/\alpha}$ in (29), and the higher-order variation theorem additionally assumes that $2\alpha/(\alpha-1)$ is an even integer.

Editorial extensions

If this is right

The estimator $\hat\alpha_N=\log N/\log A_N$ recovers the fractional-Laplacian order $\alpha$ in probability without requiring knowledge of the drift $\theta$.
The drift estimators $\hat\theta_{N,1}$ and $\hat\theta_{N,2}$ are consistent in probability for $\theta$ when $\alpha$ is known.
When $\alpha=2$, the higher-order variation theorem yields a nontrivial quartic variation with error rate $N^{-3/20}$, matching the known standard-Laplacian case.
The limits extend to any time interval $[A_1,A_2]$, with the quadratic limit acquiring a factor $(A_2-A_1)^{2H-1}$.
The quadratic and higher-order variations converge at the same $L^1$ rate, so either observable can serve as the basis for parameter inference.

Reading between the lines

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

The even-integer condition on $2\alpha/(\alpha-1)$ appears to be technical; a central-limit or noninteger-power extension would make $\hat\theta_{N,2}$ available for all $\alpha\in(1,2)$, not only the discrete set where the exponent is even.
Because $\hat\alpha_N$ is independent of $\theta$, a natural two-step inference procedure—estimate $\alpha$ first, then $\theta$—would decouple the parameters; the asymptotic joint distribution of such a procedure is not addressed in the paper.
If $\sigma$ vanishes on a portion of the observed path, the denominators in (62)–(64) could degenerate; handling such cases would require a modified normalization that the paper does not discuss.
The same perturbed-fractional-Brownian strategy could be applied to spatial power variations at a fixed time, potentially yielding estimators that identify $\alpha$ without relying on temporal sampling rates.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

I read this paper carefully. The genuinely new result is an estimator for the fractional Laplacian index alpha in the nonlinear multiplicative-noise equation, built from a renormalized temporal quadratic variation. That estimation problem is new as far as I know, and the estimator is natural: a log-ratio of normalized variation to a Riemann sum of sigma^2. Theorems 1 and 2 also extend temporal variation limits from the linear to the nonlinear setting, and the rates check out at alpha=2 against the known Pospisil-Tribe result. The comparison-to-linear-solution strategy is the right high-level idea, and Proposition 4 gives a clean L2 bound on the replacement error.

But there is a load-bearing gap. In the proof of Theorem 1, the 'far' part l(2,2) is handled by asserting that ~Delta_i and ~Delta_j are independent of the sigma^2(u(t_i(delta))) and sigma^2(u(t_j(delta))) factors when t_j+delta <= t_i(delta). That is false. The modified increment ~Delta_j is defined using the original noise W on [t_j(delta), t_j+delta], and u(t_i(delta)) depends on W on exactly that interval. Using an independent copy ~W on the past does not remove that shared W. So the factorization E[sigma^2(u(t_i))sigma^2(u(t_j))] * E[Delta_i Delta_j] is unjustified. The same flaw appears in the a(2,2) part of Theorem 2. Without an extra estimate bounding the cross-covariance between sigma^2(u(t_i(delta))) and the modified increment of an earlier block, the stated L1 rates (35) and (52) do not follow from the written argument. This is not a typo; it is the step that converts the comparison bound into the limit theorem. It might be fixable—the cross-covariance should be small for large i-j—but the paper does not do it.

Other weaknesses are minor by comparison. The even-integer condition on 2alpha/(alpha-1) is restrictive but the authors disclose it. The heavy reliance on earlier work from the same group is noticeable but the citations are relevant and not padding. The writing is clear and the appendix contains the necessary kernel estimates.

The paper is for researchers working on high-frequency statistics for SPDEs, especially anyone trying to estimate the fractional Laplacian index from discrete temporal observations. I would send it to a serious referee, because the problem is interesting and the gap may be closable, but I would not accept it as is; the referee should focus on the independence claim and request the missing estimate.

Referee Report

1 major / 4 minor

Summary. The paper studies the temporal power variations of the solution to the nonlinear stochastic heat equation with a fractional Laplacian, ∂u/∂t = -(-Δ)^{α/2} u + σ(u) Ẇ, for α∈(1,2], observed at a fixed spatial point over an equidistant time partition. The main theoretical results are Theorem 1, which states that the renormalized quadratic variation N^{-1/α} Σ (u(t_{i+1},x)-u(t_i,x))² converges in L¹ to C_{0,α}² ∫_0^1 σ²(u(s,x)) ds with explicit rate (35), and Theorem 2, which states that, when p=2α/(α-1) is an even integer, the p-th power variation converges to B_{0,α} ∫_0^1 σ^p(u(s,x)) ds with rate (52). The proofs compare the increments of the nonlinear solution with those of the linear equation via a modified increment ~Δ₁u₀ that is independent of the past after a time shift, and use the representation of the linear solution as a perturbed fractional Brownian motion. The last section applies these limits to construct a consistent estimator for the anomality parameter α and two consistent estimators for the drift θ. The paper is clearly written and the rates are internally consistent, reproducing the known N^{-3/20} rate at α=2. However, the central decoupling step in the proof of Theorem 1, and its analogue in Theorem 2, contains a false independence assertion; the proofs of the limit theorems are therefore incomplete as written.

Significance. If the gap identified below is repaired, the paper would be a valuable contribution: it gives the first estimator for the fractional-Laplacian index α from discrete temporal observations at one spatial point, and it extends exact-variation results from the linear or standard-Laplacian setting to the nonlinear fractional setting with explicit convergence rates. The constants C_{0,α} and B_{0,α} are taken from the known linear theory and are not fitted, and the estimators are genuine functions of the data, so the statistical claims are not circular. The appendix estimates, in particular Proposition 4 and Lemma 4, appear sound and document the kernel estimates carefully. The main obstacle is a single but load-bearing independence argument in the proofs of Theorems 1 and 2; if that step cannot be justified, the stated L¹ rates and the consistency conclusions do not follow from the written proof.

major comments (1)

[Section 3.2, proof of Theorem 1, estimate of l^(2,2)_{2,N}] The decoupling of l^(2,2)_{2,N} is not justified. The text asserts that when i-j ≥ [N^{-β}+1]+2 (so that t_j+δ ≤ t_i(δ)), the modified increments ~Δ₁u₀(t_i,δ) and ~Δ₁u₀(t_j,δ) are independent of σ²(u(t_i(δ),x)) and σ²(u(t_j(δ),x)). This is false. By construction, ~Δ₁u₀(t_j,δ) uses the original noise W on [t_j(δ), t_j+δ], and because t_j+δ ≤ t_i(δ), the random variable u(t_i(δ),x) also depends on W on exactly that interval. Already for the linear field, Cov(~Δ₁u₀(t_j,δ), u₀(t_i(δ),x)) = ∫_{t_j(δ)}^{t_j+δ}∫_R G_α(t_i(δ)-a,x-y)(G_α(t_j+δ-a,x-y)-G_α(t_j-a,x-y)) dy da, which is not identically zero; the same holds for the nonlinear solution. Consequently, E[σ²(u(t_i(δ),x))σ²(u(t_j(δ),x)) Z_i Z_j] does not factor as E[σ²(u(t_i(δ),x))σ²(u(t_j(δ),x))] E[Z_i Z_j], and the bound (38) does not follow from Lemma 2. Since this term feeds directly into E|L_{2,N}|² and therefore into the rate (35), Theorem 1 is not proved as written. The identical defect appears in the treatment of a^(2,2)_{2,N} in the proof of Theorem 2 and affects (52). The limit statements may well be true, but a supplementary estimate controlling the cross-covariance between ~Δ₁u₀(t_j,δ) and the σ(u(t_i(δ),x))-dependent factors is required.

minor comments (4)

[Section 3.2, Eq. (36)] The bound E|~Δ₁u₀(t_i,δ)|² ≤ C N^{-(α-1)/(2α)} has the exponent off by a factor of 2 for a second moment; the next line uses C N^{-(α-1)/α}, which is the correct order for the second moment of a temporal increment of u₀. Please correct the displayed bound.
[Sections 3.2 and 3.3, split threshold] The condition for t_j+δ ≤ t_i(δ) is i-j ≥ N^{1-β}+1, not i-j ≥ [N^{-β}+1]+2; as printed, the 'far' set includes pairs that do not have the stated time separation. Since the subsequent estimate passes to the full double sum, this typo alone is not fatal, but it should be corrected.
[Section 3.3, proof of Theorem 2, a^(2,2)_{2,N}] In the displayed formula for a^(2,2)_{2,N}, both factors in the product are written with index j: [ (~Δ₁u₀(t_j,δ))^p - B_{0,α}/N ][ (~Δ₁u₀(t_j,δ))^p - B_{0,α}/N ]. One factor should involve ~Δ₁u₀(t_i,δ).
[Throughout] There are several typographical errors: 'spce-time' in the first paragraph of Section 3.3, 'we need need' before Lemma 3, 'Altough' in the proof of Proposition 1, and a missing parenthesis in reference [14]. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variation limits are computed from the known linear equation and standard fBm estimates; the estimators are genuine statistics whose consistency follows from the proved limits.

full rationale

The derivation chain is independent of the conclusions. Theorem 1 and Theorem 2 start from the exact Gaussian decomposition (28) of the linear solution u0 as C0,alpha times an fBm plus a smoother self-similar process Y, with the constant C0,alpha determined by the Green kernel and the increment bound (29) quoted from [4] and [29]; this is a published property of the linear equation, not a consequence of the nonlinear variation limits under proof. The perturbed-fBm Lemmas 2 and 3 are proved in the paper from this decomposition and from classical fBm quadratic and power variation estimates, and they supply bounds for the auxiliary process tilde-Delta_1 u0, not fitted values. Proposition 4 compares a nonlinear increment with sigma(u(t(delta),x)) times tilde-Delta_1 u0 and is proved directly in the appendix. The limits (35) and (52) then follow by bounding L1,N, L2,N, L3,N (respectively A1,N, A2,N, A3,N); the constants C0,alpha and B0,alpha are external benchmark values from the linear equation, and the limiting integrands are integrals of the observed nonlinear coefficient, not fitted parameters. The estimators (61), (63), and (64) are explicit functions of the observations whose consistency is deduced from these limits; no parameter is fitted to a subset and then reported as a prediction. The citations to [16], [23], and [29] involve self-citations in part, but they concern published auxiliary results for fBm variations and the linear heat equation; they are real evidence and none of them assumes the nonlinear theorem being proved. The possible gap in the l(2,2) independence factorization would be a proof-correctness issue, not a circular reduction, since it does not identify the target limit with any input of the argument.

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

The paper introduces no new entities and no fitted parameters. It relies on standard SPDE existence theory and prior linear-equation variation results. The only ad hoc restriction is the even-integer condition for the higher-order variation theorem.

assumptions (5)

domain assumption Unique mild solution exists with uniform Lp bounds and Holder continuity, Eqs. (7) and (8).
Taken from [3] and [15]; standard existence theory for Lipschitz-coefficient SPDEs.
domain assumption The temporal linear solution decomposes as C_{0,alpha} fBm with H=(alpha-1)/(2 alpha) plus a self-similar Gaussian perturbation Y satisfying E|Y_{i+1}-Y_i|^2 <= C i^{-(alpha+1)/alpha}, Eq. (29).
Cited from [4] and [29]; this decomposition drives Lemmas 2 and 3 and hence Theorems 1 and 2.
standard math Green kernel bounds (66) and integral estimates, including Lemma 4, hold for the fractional heat kernel.
Appendix 5.1 and [20]; used in the proof of Proposition 4.
standard math Dalang-Walsh stochastic integral isometry (6) and Gaussian hypercontractivity (26) are valid.
Standard tools used throughout the proofs.
ad hoc to paper The power 2 alpha/(alpha-1) is an even integer in Theorem 2.
Assumption (44) is needed for Lemma 3 and the estimate of A_{1,N}; it restricts alpha to a discrete set of values.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Temporal quadratic and higher order variation for the nonlinear stochastic heat equation and applications to parameter estimation." pith.science (2026). https://pith.science/paper/WZZH7JQJ

@misc{pith2026250418450,
  author       = {Pith},
  title        = {Pith review of: Temporal quadratic and higher order variation for the nonlinear stochastic heat equation and applications to parameter estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZZH7JQJ}},
  note         = {Machine review of arXiv:2504.18450}
}

abstract

We consider the stochastic heat equation which includes a fractional power of the Laplacian of order $\alpha \in (1, 2]$ and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the solution to this equation: the renormalized quadratic variation and the power variation of order $\frac{2\alpha}{\alpha -1}$, both over an equidistant partition of the unit interval. We prove that these two sequences admit nontrivial limits when the mesh of the partition goes to zero. We apply these results to identify certain parameters of the stochastic heat equation.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 30 canonical work pages

[1]

Altmeyer

R. Altmeyer. I. Cialenco and G. Pasemann (2023): Parameter estimation for semi- linear SPDEs from local measurements. Bernoulli 29 (3), 2035–2061

work page 2023
[2]

Assaad and C

O. Assaad and C. A. Tudor (2021): Pathwise analysis and parameter estimation for the stochastic Burgers equation. Bull. Sci. Math. 170, Paper No. 102995, 26 pp

work page 2021
[3]

Assaad, D

O. Assaad, D. Nualart, C.A. Tudor and L. Viitasaari (2021 ): Quantitative normal approximations for the stochastic fractional heat equatio n. Stoch. Partial Diﬀer. Equ. Anal. Comput. 10 (2022), no. 1, 223–254

work page 2022
[4]

Bajja, K

S. Bajja, K. Es-Sebaiy, and L. Viitasaari (2018): Limit theorems for quadratic vari- ations of the Lei-Nualart process. Stochastic processes and applications, 105–121, Springer Proc. Math. Stat., 271, Springer, Cham

work page 2018
[5]

Bibinger, M

M. Bibinger, M. Trabs (2020): Volatility estimation for stochastic PDEs using high- frequency observations. Stochastic Process. Appl., 130 (5), 3005-3052

work page 2020
[6]

Bibinger and M

M. Bibinger and M. Trabs (2019): On central limit theorems for power variations of the solution to the stochastic heat equation. Stochastic models, statistics and their applications, 69–84, Springer Proc. Math. Stat., 294, Spri nger, Cham

work page 2019
[7]

Chong (2020): High-frequency analysis of parabolic stochastic PDEs

C. Chong (2020): High-frequency analysis of parabolic stochastic PDEs. Ann. Statist., 48 1143-1167

work page 2020
[8]

Chong and R

C. Chong and R. Dalang (2023): Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs. Bernoulli 29 (3), 1792-1820

work page 2023

Show all 31 references

[9]

Cialenco (2018): Statistical inference for spdes: an overview

I. Cialenco (2018): Statistical inference for spdes: an overview. Statistical Inference for Stochastic Processes, 21(2), 309-329

2018
[10]

Cialenco, F

I. Cialenco, F. Delgado-Vences, H.-J. Kim (2020): Drift estimation for discretely sampled SPDEs. Stoch. PDE: Anal. Comp., 8 (2020), 895-920

2020
[11]

Cialenco and Y

I. Cialenco and Y. Huang (2020): A note on parameter estimation for discretely sampled spdes. Stoch. Dyn. 20 (2020), no. 3, 2050016, 28 pp

2020
[12]

Cialenco, H-J

I. Cialenco, H-J. Kim (2022): Parameter estimation for discretely sampled stochastic heat equation driven by space-only noise. Stochastic Process. Appl. 143, 1–30

2022
[13]

Cialenco, H.-J

I. Cialenco, H.-J. Kim, and G. Pasemann (2024): Statistical analysis of discretely sampled semilinear SPDEs: a power variation approach. Stoch. Partial Diﬀer. Equ. Anal. Comput. 12 (1), 326-351. 34

2024
[14]

Dalang (1999: Extending the Martingale Measure Stochastic Integral With A ppli- cations to Spatially Homogeneous S.P.D.E.’s

R. Dalang (1999: Extending the Martingale Measure Stochastic Integral With A ppli- cations to Spatially Homogeneous S.P.D.E.’s. Electron. J. Probab. Volume 4, paper no. 6, 29 pp

1999
[15]

Debbi and M

L. Debbi and M. Dozzi (2005): On the solutions of nonlinear stochastic fractional partial diﬀerential equations in one spatial dimension. Stoch. Proc. Appl., 115: 1761- 1781

2005
[16]

Gamain and C

J. Gamain and C. A. Tudor (2023): Exact variation and drift parameter estimation for the nonlinear fractional stochastic heat equation. Jpn. J. Stat. Data Sci. 6 (1), 381-406

2023
[17]

Gaudlitz (2023): Non-parametric estimation of the reaction term in semi-lin ear SPDEs with spatial ergodicity

S. Gaudlitz (2023): Non-parametric estimation of the reaction term in semi-lin ear SPDEs with spatial ergodicity. Preprint arXiv:2307.05457

2023 arXiv
[18]

Gaudlitz, M

S. Gaudlitz, M. Reiss (2023): Estimation for the reaction term in semi-linear SPDEs under small diﬀusivity. Bernoulli, 29 (4), 3033 - 3058

2023
[19]

Hildebrandt and M

F. Hildebrandt and M. Trabs (2021): Parameter estimation for SPDEs based on discrete observations in time and space. Electron. J. Stat. 15 (1), 2716-2776

2021
[20]

Jacob and H.G

N. Jacob and H.G. Leopold (1993): Pseudo diﬀerential operators with variable order of diﬀerentiation generating Feller semigroups. Integral Equations Operator Theory, 17, 544-553

1993
[21]

Kaino and M

Y. Kaino and M. Uchida (2021): Parametric estimation for a parabolic linear SPDE model based on sampled data. J. Statist. Plann. Inference, 211, 190-220

2021
[22]

Khalil-Mahdi and C

Z. Khalil-Mahdi and C. A. Tudor (2019): On the distribution and q-variation of the solution to the heat equation with Fractional Laplacian . Probability Theory and Mathematical Statistics, 39 (2), 315–335

2019
[23]

Mahdi Khalil and C

Z. Mahdi Khalil and C. A. Tudor (2019). Estimation of the drift parameter for the fractional stochastic heat equation via power variation, M od. Stoch. Theory Appl. 6 (4), 397-417

2019
[24]

Nourdin, D

I. Nourdin, D. Nualart and C. A. Tudor (2010): Central and non-central limit theorems for weighted power variations of fractional Brown ian motion. Ann. Inst. Henri Poincar´ e Probab. Stat. 46 (4), 1055–1079

2010
[25]

Nourdin and G

I. Nourdin and G. Peccati (2012): Normal Approximation s with Malliavin Calculus From Stein’s Method to Universality. Cambridge University Press

2012
[26]

Pospisil and R

J. Pospisil and R. Tribe (2007): Parameter estimates and exact variations for stochas- tic heat equation heat equations driven by space-time white noise. Stochastic Analysis and Applications 25(3), 593-611. 35

2007
[27]

Swanson (2007): Variations of the solution to a stochastic heat equation

J. Swanson (2007): Variations of the solution to a stochastic heat equation. The Annals of Probability, 35(6), 2122-2159

2007
[28]

Tonaki, Y

Y. Tonaki, Y. Kaino and M. Uchida (2023): Parameter estimation for a linear parabolic SPDE model in two space dimensions with a small noi se. Stat. Inference Stoch. Process. 27 (1), 123-179

2023
[29]

C. A. Tudor (2022): Stochastic Partial Diﬀerential Equa tions with Additive Gaussain noise: Analysis and Inference. World Scientiﬁc

2022
[30]

W Walsh (1986): An introduction to stochastic partial diﬀerential equatio ns

J. W Walsh (1986): An introduction to stochastic partial diﬀerential equatio ns. ´Ecole d’ ´ et´ e de probabilit´ es de Saint-Flour, XIV—1984, 265–439, LNM 1180, Springer

1986
[31]

Zili and E

M. Zili and E. Zougar (2019): Exact variations for stochastic heat equations with piecewise constant coeﬃcients and applications to paramet er estimation. Teor. ˘Imov ¯ ır. Mat. Stat. No. 100 (2019), 75–101. 36

2019

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Altmeyer

R. Altmeyer. I. Cialenco and G. Pasemann (2023): Parameter estimation for semi- linear SPDEs from local measurements. Bernoulli 29 (3), 2035–2061

work page 2023

[2] [2]

Assaad and C

O. Assaad and C. A. Tudor (2021): Pathwise analysis and parameter estimation for the stochastic Burgers equation. Bull. Sci. Math. 170, Paper No. 102995, 26 pp

work page 2021

[3] [3]

Assaad, D

O. Assaad, D. Nualart, C.A. Tudor and L. Viitasaari (2021 ): Quantitative normal approximations for the stochastic fractional heat equatio n. Stoch. Partial Diﬀer. Equ. Anal. Comput. 10 (2022), no. 1, 223–254

work page 2022

[4] [4]

Bajja, K

S. Bajja, K. Es-Sebaiy, and L. Viitasaari (2018): Limit theorems for quadratic vari- ations of the Lei-Nualart process. Stochastic processes and applications, 105–121, Springer Proc. Math. Stat., 271, Springer, Cham

work page 2018

[5] [5]

Bibinger, M

M. Bibinger, M. Trabs (2020): Volatility estimation for stochastic PDEs using high- frequency observations. Stochastic Process. Appl., 130 (5), 3005-3052

work page 2020

[6] [6]

Bibinger and M

M. Bibinger and M. Trabs (2019): On central limit theorems for power variations of the solution to the stochastic heat equation. Stochastic models, statistics and their applications, 69–84, Springer Proc. Math. Stat., 294, Spri nger, Cham

work page 2019

[7] [7]

Chong (2020): High-frequency analysis of parabolic stochastic PDEs

C. Chong (2020): High-frequency analysis of parabolic stochastic PDEs. Ann. Statist., 48 1143-1167

work page 2020

[8] [8]

Chong and R

C. Chong and R. Dalang (2023): Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs. Bernoulli 29 (3), 1792-1820

work page 2023