Global solutions to stochastic wave equations with superlinear coefficients

Annie Millet,Marta Sanz-Solé

doi:10.1016/j.spa.2021.05.002

Abstract

We prove existence and uniqueness of a random field solution (u(t,x);(t,x)∈[0,T]×Rd) to a stochastic wave equation in dimensions d=1,2,3 with diffusion and drift coefficients of the form |z|(ln+(|z|))a for some a>0. The proof relies on a sharp analysis of moment estimates of time and space increments of the corresponding stochastic wave equation with globally Lipschitz coefficients. We give examples of spatially correlated Gaussian driving noises where the results apply.

Highlights

We study the stochastic wave equation in spatial dimension d ∈ {1, 2, 3}, with a multiplicative noise W
We deduce that almost all sample paths of the solution to (1.3) are locally Holder continuous, jointly in (t, x), with exponent θ ∈
We only describe the computations in the case d = 3; the case d = 2 is easier and dealt with in a similar way

Summary

Introduction

We study the stochastic wave equation in spatial dimension d ∈ {1, 2, 3}, with a multiplicative noise W ,. To the best of our knowledge, existence or absence of blow-up has been studied so far in the setting of functional-valued solutions, rather than for random field solutions, and mostly but , with strong conditions on the space covariance Such upper bounds depend on the value at the origin and the Lipschitz constants of the coefficients b and σ (see Propositions 3.4 and 4.12, and the notation (2.6)) These bounds are obtained from Lp-estimates of increments in time and in space of the process (u(t, x))(t,x) (see Propositions 3.3 and 4.11) via a version of Kolmogorov’s theorem ([9, Theorem A.3.1]). Since we are interested in random field solutions, in contrast with the case d = 1, we cannot take a space-time white noise. The results of this paper can be extended to equation (1.3) with coefficients b(t, x; u(t, x)) and σ(t, x; u(t, x))

Preliminaries and notations

The stochastic wave equation in dimension one

The stochastic wave equation in dimensions 2 and 3

Examples of covariance densities

Conclusion

Findings