##
**Regularization by random translation of potentials for the continuous PAM and related models in arbitrary dimension.**
*(English)*
Zbl 1501.35466

Summary: We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem – without the need of renormalization – in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear Young integration.

### MSC:

35R60 | PDEs with randomness, stochastic partial differential equations |

35K10 | Second-order parabolic equations |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60H50 | Regularization by noise |

### Keywords:

regularization by noise; parabolic Anderson model; nonlinear Young integral; multiplicative noise; stochastic partial differential equations; Feynman-Kac formula
PDFBibTeX
XMLCite

\textit{F. Bechtold}, Electron. Commun. Probab. 27, Paper No. 47, 13 p. (2022; Zbl 1501.35466)

### References:

[1] | R. Allez and K. Chouk, The continuous Anderson hamiltonian in dimension two, 2015, 1511.02718. |

[2] | F. Bechtold and M. Hofmanová, Weak solutions for singular multiplicative SDEs via regularization by noise, 2022, 2203.13745. |

[3] | C. Bellingeri, A. Djurdjevac, P. K. Friz, and N. Tapia, Transport and continuity equations with (very) rough noise, Partial Differential Equations and Applications 2 (2021), no. 4. · Zbl 1479.35966 |

[4] | R. Catellier, Rough linear transport equation with an irregular drift, Stochastics and Partial Differential Equations: Analysis and Computations 4 (2016), no. 3, 477-534. · Zbl 1356.60097 |

[5] | R. Catellier and M. Gubinelli, Averaging along irregular curves and regularisation of ODEs, Stochastic Processes and their Applications 126 (2016), no. 8, 2323-2366. · Zbl 1348.60083 |

[6] | R. Catellier and F. Harang, Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation, 2021, 2101.00915. |

[7] | J. Diehl, P. Friz, and W. Stannat, Stochastic partial differential equations: a rough paths view on weak solutions via Feynman-Kac, Annales de la Faculté des sciences de Toulouse: Mathématiques Ser. 6, 26 (2017), no. 4, 911-947 (en). · Zbl 1392.35335 |

[8] | P. Friz and M. Hairer, A course on rough paths, Universitext, Springer, Cham, 2014. · Zbl 1327.60013 |

[9] | L. Galeati, Nonlinear young differential equations: A review, Journal of Dynamics and Differential Equations (2021). |

[10] | L. Galeati and M. Gubinelli, Noiseless regularisation by noise, Revista Matemática Iberoamericana 38 (2021), no. 2, 433-502. · Zbl 1510.60066 |

[11] | L. Galeati and F. Harang, Regularization of multiplicative sdes through additive noise, 2020, 2008.02335. · Zbl 1529.60039 |

[12] | L. Galeati, F. A. Harang, and A. Mayorcas. Distribution dependent SDEs driven by additive fractional brownian motion. Probability Theory and Related Fields, May 2022. · Zbl 1492.60168 |

[13] | L. Galeati, F. Harang, and A. Mayorcas, Distribution dependent SDEs driven by additive continuous noise, Electronic Journal of Probability 27 (2022), 1-38. · Zbl 1492.60168 |

[14] | D. Geman and J. Horowitz, Occupation densities, The Annals of Probability 8 (1980), no. 1, 1-67. · Zbl 0499.60081 |

[15] | M Gubinelli, Controlling rough paths, J. Func. Anal. 216 (2004), no. 1, 86-140. · Zbl 1058.60037 |

[16] | M. Gubinelli, P. Imkeller, and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum of Mathematics, Pi 3e6 (2015), 1-75. · Zbl 1333.60149 |

[17] | M. Hairer and C. Labbé, A simple construction of the continuum parabolic Anderson model on \[{\mathbf{R}^2} \], Electronic Communications in Probability 20 (2015), no. none, 1-11. · Zbl 1332.60094 |

[18] | M.Hairer and C. Labbé, Multiplicative stochastic heat equations on the whole space, Journal of the European Mathematical Society 20 (2018), no. 4, 1005-1054. · Zbl 1447.60102 |

[19] | F. Harang and A. Mayorcas, Pathwise regularisation of singular interacting particle systems and their mean field limits, 2020, 2010.15517. |

[20] | F. Harang and N. Perkowski, \(C\)∞-regularization of ODEs perturbed by noise, Stochastics and Dynamics (2021), 2140010. · Zbl 1490.60135 |

[21] | Y. Hu and K. Lê, Nonlinear young integrals and differential systems in Hölder media, Transactions of the American Mathematical Society 369 (2016), no. 3, 1935-2002. · Zbl 1356.60086 |

[22] | Y. Hu, F. Lu, and D. Nualart, Feynman-Kac formula for the heat equation driven by fractional noise with Hurst parameter \[H\textless 1/ 2\], The Annals of Probability 40 (2012), no. 3, 1041-1068. · Zbl 1253.60074 |

[23] | Y. Hu, D. Nualart, and J. Song, Feynman-Kac formula for heat equation driven by fractional white noise, The Annals of Probability 39 (2011), no. 1, 291-326. · Zbl 1210.60056 |

[24] | T. P. Hytönen and M. C. Veraar, On Besov regularity of Brownian motions in infinite dimensions, Probability and Mathematical Statistics 28 (2008), Fasc. 1, 143-162. · Zbl 1136.60358 |

[25] | I. Karatzas and I.K.S. Shreve, Brownian motion and stochastic calculus, Graduate Texts in Mathematics (113) (Book 113), Springer New York, 1991. · Zbl 0734.60060 |

[26] | W. König, The parabolic Anderson model, Springer International Publishing, 2016. · Zbl 1378.60007 |

[27] | F. Kühn and R. Schilling, Convolution inequalities for Besov and Triebel-Lizorkin spaces, and applications to convolution semigroups, 2021, 2101.03886. |

[28] | C. Labbé, The continuous Anderson hamiltonian in \[d\le 3\], Journal of Functional Analysis 277 (2019), no. 9, 3187-3235. · Zbl 1432.35064 |

[29] | J. Martin, Refinements of the solution theory for singular SPDEs, Ph.D. thesis, Humboldt-Universität zu Berlin, 2018. |

[30] | O. Mocioalca and F. Viens, Skorohod integration and stochastic calculus beyond the fractional brownian scale, Journal of Functional Analysis 222 (2005), no. 2, 385-434. · Zbl 1068.60078 |

[31] | T. Nilssen, Rough linear PDE’s with discontinuous coefficients – existence of solutions via regularization by fractional Brownian motion, Electronic Journal of Probability 25 (2020), no. none, 1-33. · Zbl 1441.60048 |

[32] | R. Schilling and L. Partzsch, Brownian motion: An introduction to stochastic processes, De Gruyter, 2012. · Zbl 1258.60002 |

[33] | W. van Zuijlen, Theory of function spaces, Lecture notes (2020), http://www.wias-berlin.de/people/vanzuijlen/LN_theory_of_function_spaces.pdf. |

[34] | M. Veraar, Regularity of gaussian white noise on the d-dimensional torus, Banach Center Publications 95 (2011), 385-398. · Zbl 1252.60035 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.