The objective of the subsequent notes is to give an overview of the considerations and results contained in the two studies [7, 8], written in collaboration with with X. Chen, J. Song and S. Tindel. The reader is thus referred to these two publications for further details, and in particular for the proof of the assertions below.
[l] R. M. Balan: The stochastic wave equation with multiplicative fractional noise: a Malliavin calculus approach. Potential
Analysis, 36 (2012), no.l, 1-34.
[2] R. M. Balan. Stratonovich solution for the wave equation. Journal of Theoretical Probability (2022).
[3] R. M. Balan, L. Chen and X. Chen: Exact asymptotics of the stochastic wave equation with time-independent noise.
Ann. Inst. Henri Poincare Probab. Stat. 58 (2022), no. 3, 1590-1620.
[4] R. M. Balan and J. Song: Hyperbolic Anderson model with space-time homogeneous Gaussian noise. ALEA 14 (2017),
799-849.
[5] X. Chen, A, Deya, C, Ouyang, and S. Tindel. A K-rough path above the space-time fractional Brownian motion.
Stochastics and Partial Differential Equations: Analysis and Computations (2021) Vol.31, 819-866.
[6] X. Chen, A, Deya, C, Ouyang, and S. Tindel. Moment estimates for some renormalized parabolic Anderson models.
The Annals of Probability (2021) Vol.49, 2599-2636.
[7] X. Chen, A. Deya, J. Song and S. Tindel. Solving the hyperbolic Anderson model 1: Skorohod setting. arXiv:2112. 04954
(2021).
[8] X. Chen, A. Deya, J. Song and S. Tindel. Solving the hyperbolic Anderson model 2:
Stratonovich setting. In revision, arXiv:2205.15773 (2022).
Strichartz estimates and
[9] A. Deya. A nonlinear wave equation with fractional perturbation. Ann. Probab. 47 (2019), no. 3, 1775-1810.
[10] M. Gubinelli, H. Koch, and T. Oh. Renormalization of the two-dimensional stochastic nonlinear wave equations. Trans.
Amer. Math. Soc. 370 (2018), 7335-7359.
[11] M. Gubinelli, H. Koch, and T. Oh. Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation
with quadratic nonlinearity. To appear in J. Eur. Math. Soc.
[12] L. Quer-Sardanyons and S. Tindel. The 1-d stochastic wave equation driven by a fractional Brownian sheet. Stochastic
Processes and their Applications 117 (2007), 1448-1472.
[13] V. S. Rychkov. Littlewood-Paley theory and function spaces with A1° 0 weights. Math. Nachr., 224 (2001), no. 1.
lNSTITUT ELIE CARTAN, UNIVERSITY OF LORRAINE
Email address: Aurelien. Deya©uni v- lorraine. fr
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