Intermittency and Chaos for a Nonlinear Stochastic Wave Equation in Dimension 1

Abstract

We consider a non-linear stochastic wave equation driven by space-time white noise in dimension 1. First of all, we state some results about the intermittency of the solution, which had only been carefully studied in some particular cases so far. Then, we establish a comparison principle for the solution, following the ideas of Mueller. We think it is of particular interest to obtain such a result for an hyperbolic equation. Finally, using the results mentioned above, we aim to show that the solution exhibits a chaotic behavior, in a similar way as was established in [9] for the heat equation. We study the two cases where 1. the initial conditions have compact support, where the global maximum of the solution remains bounded and 2. the initial conditions are bounded away from 0, where the global maximum is almost surely infinite. Interesting estimates are also provided on the behavior of the global maximum of the solution.Keywords and phrases. Stochastic partial differential equations, wave equation, intermittency.where σ : R → R is a globally Lipschitz function with Lipschitz constant Lip σ , the noise (Ẇ (t, x), t 0, x ∈ R) is space-time white noise and κ > 0. We consider non-random, bounded and measurable initial condition u 0 : R → R + and initial derivative v 0 : R → R.Equation (1.1) has been studied by Carmona and Nualart [6] and Walsh [23]. There are also results available in the more delicate setting where x ∈ R d for d > 1; see Conus and Dalang [8], Dalang [13], Dalang and Frangos [14], and Dalang and Mueller [15].The parabolic equivalent to equation (1.1) is a well-studied family of Stochastic PDEs. In particular, it contains the well-known Parabolic Anderson Model. For more about this parabolic family, we refer the reader to Foondun and Khoshnevisan [18].

Moment bounds and asymptotics for the stochastic wave equation

“…As an immediate consequence of Theorem 3.1, we obtain the result m p ≤ Cp 3/2 for p ≥ 2 of [17] (see Theorem 3.11). We extend their lower bound on the upper Lyapunov exponent m 2 to the lower Lyapounov exponent, by showing that m 2 ≥ c(κ/2) 1/2 .…”

“…If the initial conditions are constants, then m p (x) =: m p and m p (x) =: m p do not depend on x. Intermittency is the property that m p = m p =: m p and m 1 < m 2 /2 < · · · < m p /p < · · · . It is implied by the property m 1 = 0 and m 2 > 0 (see [7, Definition III.1.1, on p. 55]), which is called full intermittency, while weak intermittency, defined in [29] and [17,Theorem 2.3] is the property m 2 > 0 and m p < +∞, for all p ≥ 2. Another property of the parabolic Anderson model is described by the behavior of exponential growth indices, initiated by Conus and Khoshnevisan in [17].…”

“…It is implied by the property m 1 = 0 and m 2 > 0 (see [7, Definition III.1.1, on p. 55]), which is called full intermittency, while weak intermittency, defined in [29] and [17,Theorem 2.3] is the property m 2 > 0 and m p < +∞, for all p ≥ 2. Another property of the parabolic Anderson model is described by the behavior of exponential growth indices, initiated by Conus and Khoshnevisan in [17]. They defined This is again a property of moments of the solution u(t, x).…”

“…Recently Conus and Balan [1] studied the problem when the noise is Gaussian, spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index H > 1/2.Regarding exponential growth indices, Conus and Khoshnevisan [18, Theorem 5.1] show that for initial data with exponential decay at ±∞, 0 < λ(p) ≤ λ(p) < +∞, for all p ≥ 2. They also show that if the initial data consists of functions with compact support, then λ(p) = λ(p) = κ, for all p ≥ 2.One objective of our study is to understand how irregular (and possibly unbounded) initial data affects the random field solutions to (1.6); another is to continue the study of moment Lyapounov exponents and exponential growth indices of [17,18]. We will only assume that the initial position g belongs to L 2 loc (R), the set of locally square integrable Borel functions, and the initial velocity µ belongs to M (R), the set of locally finite Borel measures.…”

“…Secondary 60G60, 35R60.This equation has been intensively studied during last two decades by many authors: see e.g., [6,8,9,41,46] for some early work, [20,46] for an introduction, [23,24] for the intermittency problems, [15,21,22,25,35,42,43] for the stochastic wave equation in the spatial domain R d , d > 1, [26,45] for regularity of the solution, [4,5] for the stochastic wave equation with values in Riemannian manifolds, [13,39,40] for wave equations with polynomial nonlinearities, and [36,37,44] for smoothness of the law.Concerning intermittency properties, Dalang and Mueller showed in [23] that for the wave equation in spatial domain R 3 with spatially homogeneous colored noise, with ρ(u) = u and constant initial position and velocity, the Lyapunov exponents m p and m p are both bounded, from above and below respectively, by some constant times p 4/3 . For the stochastic wave equation in spatial dimension 1, Conus et al [17] show that if the initial position and velocity are bounded and measurable functions, then the moment Lyapunov exponents satisfy m p ≤ Cp 3/2 for p ≥ 2, and m 2 ≥ c(κ/2) 1/2 for positive initial data. The difference in the exponents-3/2 versus 4/3 in the three dimensional wave equation-reflects the distinct nature of the driving noises.…”