Intermittence and nonlinear parabolic stochastic partial differential equations

Mahboubi

2015

Stoch PDE: Anal Comp

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Consider the stochastic partial differential equation ∂tu = Lu + σ(u)ξ, where ξ denotes space-time white noise and L := −(−∆) α/2 denotes the fractional Laplace operator of index α/2 ∈ ( 1 /2 , 1]. We study the detailed behavior of the approximate spatial gradient ut(x) − ut(x − ε) at fixed times t > 0, as ε ↓ 0. We discuss a few applications of this work to the study of the sample functions of the solution to the KPZ equation as well.

“…Throughout, we write 9) as substitute for the approximate spatial gradient of any real function f . We also adopt the following notation…”

Section: )mentioning

confidence: 99%

“…This bound follows essentially from the Appendix of [9]; see also [7]. Therefore, we will not describe a proof.…”

Section: )mentioning

confidence: 99%

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Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion

Foondun

Mahboubi

2015

Stoch PDE: Anal Comp

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“…Foondun and Khoshnevisan [17] have shown that: provided that: (a) inf x |σ(x)/x| > 0; and (b) inf x u 0 (x) > 0. 2 Together these results show that if u 0 is bounded away from 0 and σ is sublinear, then the solution to (1.1) is "weakly intermittent" [that is, highly peaked for large t].…”

Section: Introductionmentioning

confidence: 99%

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Conus

Probab. Theory Relat. Fields

2010

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We study the stochastic heat equation ∂ t u = Lu + σ(u)Ẇ in (1 + 1) dimensions, whereẆ is space-time white noise, σ : R → R is Lipschitz continuous, and L is the generator of a symmetric Lévy process that has finite exponential moments, and u 0 has exponential decay at ±∞. We prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting. [See, however,[19,20] for the analysis of the location of the peaks in a different model.]Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.

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

Conus

Joseph

Springer Proceedings in Mathematics &Amp; Statistics

et al. 2013

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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].