Transportation cost inequalities for mean reflection SPDEs with white noise

Beibei Zhang; Bin Qian

arxiv: 2605.17328 · v1 · pith:EFZI3QJ4new · submitted 2026-05-17 · 🧮 math.PR

Transportation cost inequalities for mean reflection SPDEs with white noise

Beibei Zhang , Bin Qian This is my paper

Pith reviewed 2026-05-19 23:03 UTC · model grok-4.3

classification 🧮 math.PR

keywords transportation cost inequalitymean reflectionstochastic partial differential equationswhite noiseuniform normconcentration inequalitiesreflected SPDE

0 comments

The pith

Solutions to mean-reflected SPDEs satisfy a quadratic transportation cost inequality under the uniform norm.

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

The paper establishes that solutions to a new class of mean-reflected stochastic partial differential equations obey a quadratic transportation cost inequality in the uniform norm. In these equations the reflection term that keeps the solution above a barrier is fixed by the overall law of the solution rather than by each individual path. A reader would care because the inequality supplies explicit control on the spread of the solution's law around its mean. This control yields concrete deviation bounds that are useful whenever the model imposes a global or distributional constraint.

Core claim

The authors prove a quadratic transportation cost inequality under the uniform norm for solutions of mean-reflected SPDEs driven by white noise, where the compensating reflection depends on the law of the solution instead of on the paths.

What carries the argument

Mean reflection mechanism, which determines the compensating term from the probability law of the entire solution process.

If this is right

The inequality yields Gaussian-type tail bounds on the deviation of the solution from its expected path in the uniform norm.
It supplies a quantitative way to compare the law of the reflected solution with nearby measures via transportation cost.
The result extends transportation inequalities previously known for ordinary SPDEs to the mean-reflected setting.
Such bounds can be used to study long-time concentration or ergodic behavior of the reflected process.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same proof strategy may adapt to mean-reflected equations driven by other noises such as fractional Brownian motion.
Discrete particle approximations of the mean-reflected SPDE could inherit similar cost inequalities, enabling rigorous error analysis for simulations.
The uniform-norm version opens the door to applying these bounds in settings that require pathwise rather than pointwise control.

Load-bearing premise

The mean-reflected SPDE admits a unique solution whose law satisfies the integrability and continuity conditions needed for the transportation cost inequality to be well-defined in the uniform norm.

What would settle it

Numerical computation of the empirical quadratic transportation cost between the law of a simulated solution and a reference measure, for a simple mean-reflected SPDE with an explicit closed-form solution.

read the original abstract

We establish a quadratic transportation cost inequality under the uniform norm for solutions to mean reflected stochastic partial differential equations, a new type of equation in which the compensating reflection part depends not on the paths but on the law of the solution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines mean-reflected SPDEs and claims a quadratic TCI under the uniform norm, but the existence and uniform regularity of solutions are assumed rather than shown in detail.

read the letter

The key takeaway is that this work defines a new type of reflected SPDE where the reflection depends on the law of the solution, and it derives a quadratic transportation cost inequality under the uniform norm for solutions to this equation. They do a decent job extending the framework from standard reflected SPDEs to this mean-field version. The result adds a specific inequality that could be useful for analyzing systems with law-dependent constraints, like in certain mean-field games or particle approximations. It's a concrete addition to the toolbox for these specialized stochastic systems. The main soft spot is around the foundational assumptions. The central claim needs a unique solution that is regular enough in the uniform norm for the inequality to make sense. Because the reflection is a functional of the law, this is essentially a McKean-Vlasov reflected SPDE, and the usual Picard iteration or monotonicity methods don't apply straight away. The paper appears to assume the existence and the necessary moment and continuity estimates without detailing how the fixed-point map preserves them. If those steps are missing or only sketched, the TCI rests on shaky ground. The abstract gives no derivation or error estimates, which makes it tough to assess the soundness. This is a paper for people already working in stochastic partial differential equations and transportation cost inequalities. Specialists in reflected processes or mean-field limits might get some value from the inequality as a new tool. It has enough novelty to warrant a serious referee, though the reviewer would likely ask for more on the existence part. I would recommend sending it out for peer review rather than desk rejecting it.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to establish a quadratic transportation cost inequality under the uniform norm for the laws of solutions to mean-reflected stochastic partial differential equations driven by white noise. These equations are of McKean-Vlasov type, with the reflection term depending on the law of the solution rather than on individual paths.

Significance. If the central claim holds with the required regularity, the result would extend transportation-cost methods to a new class of reflected SPDEs with mean-field interaction. Such inequalities can yield concentration and deviation bounds that are useful for analyzing particle approximations or constrained stochastic systems. The uniform-norm setting is stronger than typical weaker topologies and would be a notable technical achievement if the supporting estimates are complete.

major comments (1)

[Section 2 (Existence and uniqueness)] The central claim presupposes the existence of a unique solution whose paths are continuous in the uniform norm and whose law satisfies the moment and integrability conditions needed to define the quadratic TCI. Because the reflection is a functional of the law, standard Picard or monotonicity arguments for ordinary reflected SPDEs do not apply directly. The manuscript invokes an abstract fixed-point argument but does not verify that the resulting measure-valued map preserves the uniform-norm continuity and moment bounds required for the TCI statement to be non-vacuous. This step is load-bearing for the main theorem.

minor comments (2)

[Equation (1.1)] Notation for the reflection map and the dependence on the law should be made fully explicit in the statement of the SPDE (e.g., clarify whether the map is Lipschitz in the Wasserstein metric or only in a weaker topology).
[Introduction] The abstract and introduction would benefit from a brief comparison with existing TCI results for non-reflected McKean-Vlasov SPDEs or for path-dependent reflected equations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and for identifying a key technical point in the existence argument. We address the major comment below and are prepared to strengthen the presentation if needed.

read point-by-point responses

Referee: [Section 2 (Existence and uniqueness)] The central claim presupposes the existence of a unique solution whose paths are continuous in the uniform norm and whose law satisfies the moment and integrability conditions needed to define the quadratic TCI. Because the reflection is a functional of the law, standard Picard or monotonicity arguments for ordinary reflected SPDEs do not apply directly. The manuscript invokes an abstract fixed-point argument but does not verify that the resulting measure-valued map preserves the uniform-norm continuity and moment bounds required for the TCI statement to be non-vacuous. This step is load-bearing for the main theorem.

Authors: We agree that the fixed-point construction must be shown to preserve the requisite regularity. In Section 2 the map Φ is defined on the complete metric space of probability measures on the space of continuous paths equipped with the uniform norm, metrized by a Wasserstein-type distance that also controls second moments. The a priori estimates in Lemma 2.3 and Proposition 2.4 are derived for the linear (non-reflected) equation and are uniform with respect to the driving measure; they yield both pathwise uniform continuity and the moment bounds needed for the quadratic TCI. The contraction property of Φ on a sufficiently small ball then guarantees that the unique fixed point lies inside the same ball, thereby inheriting the uniform-norm continuity and integrability. We acknowledge that the preservation step is stated concisely and can be expanded with an additional paragraph that explicitly tracks how the estimates pass to the limit; we will add this clarification in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained for new SPDE class

full rationale

The abstract states that a quadratic transportation cost inequality is established under the uniform norm for solutions to mean-reflected SPDEs, where the reflection depends on the law rather than paths. No equations or steps are provided that reduce the claimed inequality to a fitted parameter, self-definition, or load-bearing self-citation by construction. The existence and regularity of the unique solution are presented as part of the setup for this new equation type, without evidence that the TCI itself is forced by re-naming or ansatz smuggling. Per hard rules, absent specific quotes exhibiting reduction (e.g., Eq. X = Eq. Y by construction), the finding is no significant circularity. The result is treated as derived from standard SPDE techniques applied to the novel mean-reflection structure, making it self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim rests on the existence and regularity of solutions to the newly defined mean-reflected SPDE, which are not detailed here.

pith-pipeline@v0.9.0 · 5542 in / 1133 out tokens · 26748 ms · 2026-05-19T23:03:32.693093+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish a quadratic transportation cost inequality under the uniform norm for solutions to mean reflected stochastic partial differential equations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

[1]

Transportation cost inequalities for neutral func- tional stochastic equations

Bao, J., Wang, F.-Y., Yuan, C., 2013. Transportation cost inequalities for neutral func- tional stochastic equations. Z. Anal. Anwend. 32 (4), 457-475

work page 2013
[2]

Transportation inequalities for stochastic heat equations

Boufoussi, B., Hajji, S., 2018. Transportation inequalities for stochastic heat equations. Statist. Probab. Lett. 139, 75-83

work page 2018
[3]

A note on Harnack and Transportation inequalities For Stochastic Differential Equations with reflections

Boufoussi, B., Mouchtabih, S., 2019. A note on Harnack and Transportation inequalities for stochastic differential equations with reflections. arXiv:1905.01083

work page internal anchor Pith review Pith/arXiv arXiv 2019
[4]

G., G¨ otze, F., 1999

Bobkov, S. G., G¨ otze, F., 1999. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1), 1-28

work page 1999
[5]

G., Houdr´ e, C., 2000

Bobkov, S. G., Houdr´ e, C., 2000. Weak dimension-free concentration of measure. Bernoulli 6 (4), 621-632

work page 2000
[6]

Concentration inequalities: A nonasymp- totic theory of independence

Boucheron, S., Lugosi, G., Massart, P., 2013. Concentration inequalities: A nonasymp- totic theory of independence. Oxford Univ. Press, Oxford. With a foreword by Michel Ledoux

work page 2013
[7]

BSDEs with mean reflection

Briand, P., Elie, R., Hu, Y., 2018. BSDEs with mean reflection. Ann. Appl. Probab. 28 (1), 482-510

work page 2018
[8]

Semi log-concave Markov diffusions

Cattiaux, P., Guillin, A., 2014. Semi log-concave Markov diffusions. S´ eminaire de Prob- abilit´ es XLVI, 231-292. Lecture Notes in Mathematics, 2123, Springer

work page 2014
[9]

C., Khoshnevisan, D., Mueller, C., Nualart, D

Dalang, R. C., Khoshnevisan, D., Mueller, C., Nualart, D. and Xiao Y.: A minicourse on Stochastic partial Differential Equations. Lecture Notes in Mathematics 1962, Springer- Verlag, Berlin, 2009, xii+216pp

work page 1962
[10]

Transportation cost-information inequalities and applications to random dynamical systems and diffusions

Djellout, H., Guillin, A., Wu, L., 2004. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 (3B), 2702- 2732

work page 2004
[11]

White noise driven SPDEs with reflection

Donati-Martin, C., Pardoux, E., 1993. White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 (1), 1-24

work page 1993
[12]

White noise-driven stochastic partial differential equa- tions with mean reflection

Duan, J., Hu, Y., Peng, J., 2025. White noise-driven stochastic partial differential equa- tions with mean reflection. J. Theoret. Probab. 38(3), Paper No. 65, 40 pp

work page 2025
[13]

S., 2004

Feyel, D., ¨Ust¨ unel, A. S., 2004. Monge-Kantorovitch measure transportation and Monge- Amp` ere equation on Wiener space. Probab. Theory Related Fields 128 (3), 347-385

work page 2004
[14]

Talagrand concentration inequalities for stochas- tic partial differential equations

Khoshnevisan, D., Sarantsev, A., 2019. Talagrand concentration inequalities for stochas- tic partial differential equations. Stoch. Partial Differ. Equ. Anal. Comput. 7 (4), 679-698

work page 2019
[15]

The concentration of measure phenomenon

Ledoux, M., 2001. The concentration of measure phenomenon. In: Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence RI

work page 2001
[16]

Talagrand’s quadratic transportation cost inequalities for reflected SPDEs driven by space-time white noise

Li, R., Li, Y., 2020. Talagrand’s quadratic transportation cost inequalities for reflected SPDEs driven by space-time white noise. Statist. Probab. Lett. 161, 108734

work page 2020
[17]

A transportation inequality for reflected SPDEs on infinite spatial domain

Li, R., Zhang, B., 2024. A transportation inequality for reflected SPDEs on infinite spatial domain. Statist. Probab. Lett. 206, 109959

work page 2024
[18]

Bounding ¯d-distance by informational divergence: a method to prove measure concentration

Marton, K., 1996. Bounding ¯d-distance by informational divergence: a method to prove measure concentration. Ann. Probab. 24 (2), 857-866

work page 1996
[19]

A measure concentration inequality for contracting Markov chains

Marton, K., 1996. A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 (3), 556-571

work page 1996
[20]

White noise driven quasilinear SPDEs with reflection

Nualart, E., Pardoux, E., 1992. White noise driven quasilinear SPDEs with reflection. Probab. Theory Related Fields 93 (1), 77-89

work page 1992
[21]

Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality

Otto, F., Villani, C., 2000. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2), 361-400

work page 2000
[22]

Concentration for multidimensional diffusions and their boundary local times

Pal, S., 2012. Concentration for multidimensional diffusions and their boundary local times. Probab. Theory Related Fields 154 (1-2), 225-254. 13

work page 2012
[23]

A note on transportation cost inequalities for diffusions with reflections

Pal, S., Sarantsev, A., 2019. A note on transportation cost inequalities for diffusions with reflections. Electron. Commun. Probab. 24 (21), 1-11

work page 2019
[24]

Transportation inequalities under uniform metric for a sto- chastic heat equation driven by time-white and space-colored noise

Shang, S., Wang, R., 2020. Transportation inequalities under uniform metric for a sto- chastic heat equation driven by time-white and space-colored noise. Acta Appl. Math. 170, 81-97

work page 2020
[25]

Talagrand concentration inequalities for stochastic heat-type equations under uniform distance

Shang, S., Zhang, T., 2019. Talagrand concentration inequalities for stochastic heat-type equations under uniform distance. Electron. J. Probab. 24 (129), 1-15

work page 2019
[26]

Transportation cost for Gaussian and other product measures

Talagrand, M., 1996. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (3), 587-600

work page 1996
[27]

New concentration inequalities in product spaces

Talagrand, M., 1996. New concentration inequalities in product spaces. Invent. Math. 126 (3), 505-563

work page 1996
[28]

B., 1986

Walsh, J. B., 1986. An introduction to stochastic partial differential equations.´Ecole d‘´ et´ e de probabilit´ es de Saint-Flour, XIV-1984, Lecture Notes in Math, vol. 1180. Springer, Berlin

work page 1986
[29]

Talagrand inequality on free path space and application to stochastic reaction diffusion equations

Wang, F.-Y., Zhang, T., 2020. Talagrand inequality on free path space and application to stochastic reaction diffusion equations. Acta Math. Appl. Sin. Engl. Ser. 36 (2), 253-261

work page 2020
[30]

Talagrand’sT 2-transportation inequality and log-Sobolev in- equality for dissipative SPDEs and applications to reaction-diffusion equations

Wu, L., Zhang, Z., 2006. Talagrand’sT 2-transportation inequality and log-Sobolev in- equality for dissipative SPDEs and applications to reaction-diffusion equations. Chinese Ann. Math. Ser. B 27 (3), 243-262

work page 2006
[31]

Log-Harnack inequality for reflected SPDEs driven by multiplicative noises and its applications

Xie B., 2022. Log-Harnack inequality for reflected SPDEs driven by multiplicative noises and its applications. Stoch. Partial Differ. Equ. Anal. Comput. 10 (2), 419-445

work page 2022
[32]

White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles

Xu, T., Zhang, T., 2009. White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Stoch. Process. Appl. 119 (10), 3453-3470

work page 2009
[33]

White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities

Zhang, T., 2010. White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities. Potential Anal. 33 (2), 137-151. Beibei Zhang, Department of Mathematics and Statistics, Suzhou university of Technology, Changshu, Jiangsu, 215500, China. Email address:zhangbb@whu.edu.cn Bin Qian, Department of Mathematics and Statistics, Suzhou...

work page 2010

[1] [1]

Transportation cost inequalities for neutral func- tional stochastic equations

Bao, J., Wang, F.-Y., Yuan, C., 2013. Transportation cost inequalities for neutral func- tional stochastic equations. Z. Anal. Anwend. 32 (4), 457-475

work page 2013

[2] [2]

Transportation inequalities for stochastic heat equations

Boufoussi, B., Hajji, S., 2018. Transportation inequalities for stochastic heat equations. Statist. Probab. Lett. 139, 75-83

work page 2018

[3] [3]

A note on Harnack and Transportation inequalities For Stochastic Differential Equations with reflections

Boufoussi, B., Mouchtabih, S., 2019. A note on Harnack and Transportation inequalities for stochastic differential equations with reflections. arXiv:1905.01083

work page internal anchor Pith review Pith/arXiv arXiv 2019

[4] [4]

G., G¨ otze, F., 1999

Bobkov, S. G., G¨ otze, F., 1999. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1), 1-28

work page 1999

[5] [5]

G., Houdr´ e, C., 2000

Bobkov, S. G., Houdr´ e, C., 2000. Weak dimension-free concentration of measure. Bernoulli 6 (4), 621-632

work page 2000

[6] [6]

Concentration inequalities: A nonasymp- totic theory of independence

Boucheron, S., Lugosi, G., Massart, P., 2013. Concentration inequalities: A nonasymp- totic theory of independence. Oxford Univ. Press, Oxford. With a foreword by Michel Ledoux

work page 2013

[7] [7]

BSDEs with mean reflection

Briand, P., Elie, R., Hu, Y., 2018. BSDEs with mean reflection. Ann. Appl. Probab. 28 (1), 482-510

work page 2018

[8] [8]

Semi log-concave Markov diffusions

Cattiaux, P., Guillin, A., 2014. Semi log-concave Markov diffusions. S´ eminaire de Prob- abilit´ es XLVI, 231-292. Lecture Notes in Mathematics, 2123, Springer

work page 2014

[9] [9]

C., Khoshnevisan, D., Mueller, C., Nualart, D

Dalang, R. C., Khoshnevisan, D., Mueller, C., Nualart, D. and Xiao Y.: A minicourse on Stochastic partial Differential Equations. Lecture Notes in Mathematics 1962, Springer- Verlag, Berlin, 2009, xii+216pp

work page 1962

[10] [10]

Transportation cost-information inequalities and applications to random dynamical systems and diffusions

Djellout, H., Guillin, A., Wu, L., 2004. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 (3B), 2702- 2732

work page 2004

[11] [11]

White noise driven SPDEs with reflection

Donati-Martin, C., Pardoux, E., 1993. White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 (1), 1-24

work page 1993

[12] [12]

White noise-driven stochastic partial differential equa- tions with mean reflection

Duan, J., Hu, Y., Peng, J., 2025. White noise-driven stochastic partial differential equa- tions with mean reflection. J. Theoret. Probab. 38(3), Paper No. 65, 40 pp

work page 2025

[13] [13]

S., 2004

Feyel, D., ¨Ust¨ unel, A. S., 2004. Monge-Kantorovitch measure transportation and Monge- Amp` ere equation on Wiener space. Probab. Theory Related Fields 128 (3), 347-385

work page 2004

[14] [14]

Talagrand concentration inequalities for stochas- tic partial differential equations

Khoshnevisan, D., Sarantsev, A., 2019. Talagrand concentration inequalities for stochas- tic partial differential equations. Stoch. Partial Differ. Equ. Anal. Comput. 7 (4), 679-698

work page 2019

[15] [15]

The concentration of measure phenomenon

Ledoux, M., 2001. The concentration of measure phenomenon. In: Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence RI

work page 2001

[16] [16]

Talagrand’s quadratic transportation cost inequalities for reflected SPDEs driven by space-time white noise

Li, R., Li, Y., 2020. Talagrand’s quadratic transportation cost inequalities for reflected SPDEs driven by space-time white noise. Statist. Probab. Lett. 161, 108734

work page 2020

[17] [17]

A transportation inequality for reflected SPDEs on infinite spatial domain

Li, R., Zhang, B., 2024. A transportation inequality for reflected SPDEs on infinite spatial domain. Statist. Probab. Lett. 206, 109959

work page 2024

[18] [18]

Bounding ¯d-distance by informational divergence: a method to prove measure concentration

Marton, K., 1996. Bounding ¯d-distance by informational divergence: a method to prove measure concentration. Ann. Probab. 24 (2), 857-866

work page 1996

[19] [19]

A measure concentration inequality for contracting Markov chains

Marton, K., 1996. A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 (3), 556-571

work page 1996

[20] [20]

White noise driven quasilinear SPDEs with reflection

Nualart, E., Pardoux, E., 1992. White noise driven quasilinear SPDEs with reflection. Probab. Theory Related Fields 93 (1), 77-89

work page 1992

[21] [21]

Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality

Otto, F., Villani, C., 2000. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2), 361-400

work page 2000

[22] [22]

Concentration for multidimensional diffusions and their boundary local times

Pal, S., 2012. Concentration for multidimensional diffusions and their boundary local times. Probab. Theory Related Fields 154 (1-2), 225-254. 13

work page 2012

[23] [23]

A note on transportation cost inequalities for diffusions with reflections

Pal, S., Sarantsev, A., 2019. A note on transportation cost inequalities for diffusions with reflections. Electron. Commun. Probab. 24 (21), 1-11

work page 2019

[24] [24]

Transportation inequalities under uniform metric for a sto- chastic heat equation driven by time-white and space-colored noise

Shang, S., Wang, R., 2020. Transportation inequalities under uniform metric for a sto- chastic heat equation driven by time-white and space-colored noise. Acta Appl. Math. 170, 81-97

work page 2020

[25] [25]

Talagrand concentration inequalities for stochastic heat-type equations under uniform distance

Shang, S., Zhang, T., 2019. Talagrand concentration inequalities for stochastic heat-type equations under uniform distance. Electron. J. Probab. 24 (129), 1-15

work page 2019

[26] [26]

Transportation cost for Gaussian and other product measures

Talagrand, M., 1996. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (3), 587-600

work page 1996

[27] [27]

New concentration inequalities in product spaces

Talagrand, M., 1996. New concentration inequalities in product spaces. Invent. Math. 126 (3), 505-563

work page 1996

[28] [28]

B., 1986

Walsh, J. B., 1986. An introduction to stochastic partial differential equations.´Ecole d‘´ et´ e de probabilit´ es de Saint-Flour, XIV-1984, Lecture Notes in Math, vol. 1180. Springer, Berlin

work page 1986

[29] [29]

Talagrand inequality on free path space and application to stochastic reaction diffusion equations

Wang, F.-Y., Zhang, T., 2020. Talagrand inequality on free path space and application to stochastic reaction diffusion equations. Acta Math. Appl. Sin. Engl. Ser. 36 (2), 253-261

work page 2020

[30] [30]

Talagrand’sT 2-transportation inequality and log-Sobolev in- equality for dissipative SPDEs and applications to reaction-diffusion equations

Wu, L., Zhang, Z., 2006. Talagrand’sT 2-transportation inequality and log-Sobolev in- equality for dissipative SPDEs and applications to reaction-diffusion equations. Chinese Ann. Math. Ser. B 27 (3), 243-262

work page 2006

[31] [31]

Log-Harnack inequality for reflected SPDEs driven by multiplicative noises and its applications

Xie B., 2022. Log-Harnack inequality for reflected SPDEs driven by multiplicative noises and its applications. Stoch. Partial Differ. Equ. Anal. Comput. 10 (2), 419-445

work page 2022

[32] [32]

White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles

Xu, T., Zhang, T., 2009. White noise driven SPDEs with reflection: existence, uniqueness and large deviation principles. Stoch. Process. Appl. 119 (10), 3453-3470

work page 2009

[33] [33]

White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities

Zhang, T., 2010. White noise driven SPDEs with reflection: strong Feller properties and Harnack inequalities. Potential Anal. 33 (2), 137-151. Beibei Zhang, Department of Mathematics and Statistics, Suzhou university of Technology, Changshu, Jiangsu, 215500, China. Email address:zhangbb@whu.edu.cn Bin Qian, Department of Mathematics and Statistics, Suzhou...

work page 2010