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Catellier, Rémi; Harang, Fabian A.

Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation. (English) Zbl 07788715

Ann. Inst. Henri Poincaré, Probab. Stat. 59, No. 3, 1572-1609 (2023).
Summary: Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a Lévy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.

Cited in 1 Document

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H50 Regularization by noise
60L20 Rough paths

Keywords:

fractional Lévy processes; generalized parabolic Anderson model; pathwise regularization by noise; stochastic heat equation
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References:

[1] S. Athreya, O. Butkovsky, K. Lê and L. Mytnik. Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation, 2020. Available at arXiv:2011.13498 [math].
[2] H. Bahouri, J.-Y. Chemin and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 343. Springer, Heidelberg, 2011. Digital Object Identifier: 10.1007/978-3-642-16830-7 Google Scholar: Lookup Link MathSciNet: MR2768550 · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7
[3] H. Bahouri, J.-Y. Chemin and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, 2011. Digital Object Identifier: 10.1007/978-3-642-16830-7 Google Scholar: Lookup Link MathSciNet: MR2768550 · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7
[4] I. Bailleul, A. Debussche and M. Hofmanová. Quasilinear generalized parabolic Anderson model equation. Stoch. Partial Differ. Equ. Anal. Comput. 7 (1) (2019) 40-63. Digital Object Identifier: 10.1007/s40072-018-0121-1 Google Scholar: Lookup Link MathSciNet: MR3916262 · Zbl 1448.35320 · doi:10.1007/s40072-018-0121-1
[5] C. Bellingeri, P. K. Friz and M. Gerencsér. Singular Paths Spaces and Applications, 2020.
[6] J. Benedikt, V. E. Bobkov, P. Girg, L. Kotrla and P. Takac. Nonuniqueness of solutions of initial-value problems for parabolic p-Laplacian. Electron. J. Differential Equations 38 (2015), 2015. MathSciNet: MR3335768 zbMATH: 1515.35148 · Zbl 1515.35148
[7] O. Butkovsky and L. Mytnik. Regularization by noise and flows of solutions for a stochastic heat equation. Ann. Probab. 47 (1) (2019) 165-212. Digital Object Identifier: 10.1214/18-AOP1259 Google Scholar: Lookup Link MathSciNet: MR3909968 · Zbl 1447.60091 · doi:10.1214/18-AOP1259
[8] R. Catellier and M. Gubinelli. Averaging along irregular curves and regularisation of ODEs. Stochastic Process. Appl. 126 (8) (2016) 2323-2366. Digital Object Identifier: 10.1016/j.spa.2016.02.002 Google Scholar: Lookup Link MathSciNet: MR3505229 zbMATH: 1348.60083 · Zbl 1348.60083 · doi:10.1016/j.spa.2016.02.002
[9] L. Coutin, R. Duboscq and A. Réveillac. The Itô-Tanaka Trick: A non-semimartingale approach, 2019. zbMATH: 1492.60153
[10] A. M. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. AMRX 40 (2007) abm009. MathSciNet: MR2387018 zbMATH: 1163.34005 · Zbl 1163.34005
[11] A. Deya and S. T. Rough. Volterra equations. I. The algebraic integration setting. Stoch. Dyn. 9 (3) (2009) 437-477. Digital Object Identifier: 10.1142/S0219493709002737 Google Scholar: Lookup Link MathSciNet: MR2566910 · Zbl 1181.60105 · doi:10.1142/S0219493709002737
[12] P. Friz. Mini-Course on Rough Paths, 2009.
[13] P. K. Friz and M. Hairer. A Course on Rough Paths: With an Introduction to Regularity Structures. Universitext. Springer, Cham, 2014. Digital Object Identifier: 10.1007/978-3-319-08332-2 Google Scholar: Lookup Link MathSciNet: MR3289027 · Zbl 1327.60013 · doi:10.1007/978-3-319-08332-2
[14] H. Fujita and S. Watanabe. On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations. Comm. Pure Appl. Math. 21 (6) (1968) 631-652. Digital Object Identifier: 10.1002/cpa.3160210609 Google Scholar: Lookup Link MathSciNet: MR0234129 zbMATH: 0157.17603 · Zbl 0157.17603 · doi:10.1002/cpa.3160210609
[15] L. Galeati. Nonlinear young differential equations: A review. J. Dynam. Differential Equations (2021). zbMATH: 07691647
[16] L. Galeati and M. Gubinelli. Prevalence of rho-irregularity and related properties, 2020.
[17] L. Galeati and M. Gubinelli. Noiseless regularisation by noise. Rev. Mat. Iberoam. (2021). Digital Object Identifier: 10.4171/rmi/1280 Google Scholar: Lookup Link MathSciNet: MR4404773 zbMATH: 1510.60066 · doi:10.4171/rmi/1280
[18] L. Galeati and F. A. Harang. Regularization of multiplicative sdes through additive noise, 2020. zbMATH: 07609686
[19] M. Gubinelli, P. Imkeller and N. Perkowski. Paracontrolled distributions and singular PDEs. Forum Math. Pi 3 (2015), e6. Digital Object Identifier: 10.1017/fmp.2015.2 Google Scholar: Lookup Link MathSciNet: MR3406823 · Zbl 1333.60149 · doi:10.1017/fmp.2015.2
[20] M. Gubinelli, P. Imkeller and N. Perkowski. A Fourier analytic approach to pathwise stochastic integration. Electron. J. Probab. 21 (2016). Digital Object Identifier: 10.1214/16-EJP3868 Google Scholar: Lookup Link MathSciNet: MR3485344 zbMATH: 1338.60139 · Zbl 1338.60139 · doi:10.1214/16-EJP3868
[21] M. Hairer and X.-M. Li. Averaging dynamics driven by fractional Brownian motion. Ann. Probab. 48 (4) (2020) 1826-1860. Digital Object Identifier: 10.1214/19-AOP1408 Google Scholar: Lookup Link MathSciNet: MR4124526 · Zbl 1453.60087 · doi:10.1214/19-AOP1408
[22] M. Hairer and N. S. Pillai. Regularity of laws and ergodicity of hypoelliptic SDES driven by rough paths. Ann. Probab. 41 (4) (2013) 2544-2598. Digital Object Identifier: 10.1214/12-AOP777 Google Scholar: Lookup Link zbMATH: 1288.60068 MathSciNet: MR3112925 · Zbl 1288.60068 · doi:10.1214/12-AOP777
[23] F. Harang and A. Mayorcas. Pathwise regularisation of singular interacting particle systems and their mean field limits, 2020. zbMATH: 1509.60173
[24] F. A. Harang and C. Ling. Regularity of local times associated with Volterra-Lévy processes and path-wise regularization of stochastic differential equations. J. Theor. Probab. (2021).
[25] F. A. Harang and N. Perkowski. C infinity regularization of odes perturbed by noise. Stoch. Dyn. (2021), 2140010. Digital Object Identifier: 10.1142/S0219493721400104 Google Scholar: Lookup Link MathSciNet: MR4342752 zbMATH: 1490.60135 · Zbl 1490.60135 · doi:10.1142/S0219493721400104
[26] F. A. Harang and S. Tindel. Volterra equations driven by rough signals. Stochastic Process. Appl. 142(C) (2021) 34-78. Digital Object Identifier: 10.1016/j.spa.2021.08.001 Google Scholar: Lookup Link MathSciNet: MR4309474 · Zbl 1481.60204 · doi:10.1016/j.spa.2021.08.001
[27] Y. Hu, D. Nualart and J. Song. A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution. In Stochastic Processes and their Applications 1083-1103, 123, 2013. Digital Object Identifier: 10.1016/j.spa.2012.11.004 Google Scholar: Lookup Link MathSciNet: MR3005015 · Zbl 1272.60043 · doi:10.1016/j.spa.2012.11.004
[28] T. Hytönen, J. van Neerven, M. Veraar and L. Weis. Analysis in Banach Spaces: Volume I: Martingales and Littlewood-Paley Theory, 1st edition. Springer, Berlin, 2016. MathSciNet: MR3617205 · Zbl 1366.46001
[29] M. Kwaśnicki. Ten equivalent definitions of the fractional Laplace operator. In Fractional Calculus and Applied Analysis 7-51. Fractional Calculus and Applied Analysis. 20. De Gruyter, Berlin, 2017. Digital Object Identifier: 10.1515/fca-2017-0002 Google Scholar: Lookup Link MathSciNet: MR3613319 · Zbl 1375.47038 · doi:10.1515/fca-2017-0002
[30] K. Lê. A stochastic sewing lemma and applications. Electron. J. Probab. 25 (2020) 1-55. Digital Object Identifier: 10.1214/20-ejp442 Google Scholar: Lookup Link MathSciNet: MR4089788 · Zbl 1480.60162 · doi:10.1214/20-ejp442
[31] M. Lifshits and T. Simon. Small deviations for fractional stable processes. Ann. Inst. Henri Poincaré Probab. Stat. 41 (4) (2005) 725-752. Digital Object Identifier: 10.1016/j.anihpb.2004.05.004 Google Scholar: Lookup Link MathSciNet: MR2144231 · Zbl 1070.60042 · doi:10.1016/j.anihpb.2004.05.004
[32] W. Liu and M. Röckner. Stochastic Partial Differential Equations: An Introduction. Universitext. Springer, Berlin, 2015. Digital Object Identifier: 10.1007/978-3-319-22354-4 Google Scholar: Lookup Link MathSciNet: MR3410409 · Zbl 1361.60002 · doi:10.1007/978-3-319-22354-4
[33] J.-C. Mourrat and H. Weber. Global well-posedness of the dynamic \(\mathit{\phi}^4\) model in the plane. Ann. Probab. 45 (4) (2017) 2398-2476. Digital Object Identifier: 10.1214/16-AOP1116 Google Scholar: Lookup Link MathSciNet: MR3693966 · Zbl 1381.60098 · doi:10.1214/16-AOP1116
[34] C. Mueller, L. Mytnik and E. Perkins. Nonuniqueness for a parabolic SPDE with \(\frac{3}{4}-\varepsilon \)-Hölder diffusion coefficients. Ann. Probab. 42 (5) (2014) 2032-2112. Digital Object Identifier: 10.1214/13-AOP870 Google Scholar: Lookup Link MathSciNet: MR3262498 zbMATH: 1301.60080 · Zbl 1301.60080 · doi:10.1214/13-AOP870
[35] L. Mytnik and E. A. Perkins. Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: The white noise case. Probab. Theory Related Fields 149 (2008) 1-96. Digital Object Identifier: 10.1007/s00440-009-0241-7 Google Scholar: Lookup Link MathSciNet: MR2773025 · Zbl 1233.60039 · doi:10.1007/s00440-009-0241-7
[36] L. Mytnik, E. A. Perkins and A. Sturm. On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients. Ann. Probab. 34 (2005) 1910-1959. Digital Object Identifier: 10.1214/009117906000000331 Google Scholar: Lookup Link MathSciNet: MR2271487 · Zbl 1108.60057 · doi:10.1214/009117906000000331
[37] E. Neuman. Pathwise uniqueness of the stochastic heat equation with spatially inhomogeneous white noise. Ann. Probab. 46 (6) (2018) 3090-3187. Digital Object Identifier: 10.1214/17-AOP1239 Google Scholar: Lookup Link MathSciNet: MR3857853 · Zbl 1426.60089 · doi:10.1214/17-AOP1239
[38] D. Nualart and Y. Ouknine. Regularization of quasilinear heat equations by a fractional noise. Stoch. Dyn. 4 (2) (2004) 201-221. Digital Object Identifier: 10.1142/S0219493704001012 Google Scholar: Lookup Link MathSciNet: MR2069689 zbMATH: 1080.60065 · Zbl 1080.60065 · doi:10.1142/S0219493704001012
[39] I. Pinelis. Optimum Bounds for the Distributions of Martingales in Banach Spaces. Ann. Probab. 22 (4) (1994) 1679-1706. MathSciNet: MR1331198 zbMATH: 0836.60015 · Zbl 0836.60015
[40] T. Rippl and A. Sturm. New results on pathwise uniqueness for the heat equation with colored noise. Electron. J. Probab. 18 (2012) 1-46. Digital Object Identifier: 10.1214/EJP.v18-2506 Google Scholar: Lookup Link MathSciNet: MR3101643 · Zbl 1411.60100 · doi:10.1214/EJP.v18-2506
[41] G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Random Processes. Stochastic Modeling: Stochastic Models with Infinite Variance. Chapman & Hall, New York, 1994. MathSciNet: MR1280932 · Zbl 0925.60027
[42] H. Triebel. Theory of Function Spaces. III. Monographs in Mathematics 100. Birkhäuser Verlag, Basel, 2006. MathSciNet: MR2250142 · Zbl 1104.46001
[43] X. Yang and X. Zhou. Pathwise uniqueness for an SPDE with Hölder continuous coefficient driven by \(α\)-stable noise. Electron. J. Probab. 22 (2017). Digital Object Identifier: 10.1214/16-EJP23 Google Scholar: Lookup Link MathSciNet: MR3613697 · Zbl 1357.60067 · doi:10.1214/16-EJP23
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