scholarly journals Fractional stochastic wave equation driven by a Gaussian noise rough in space

Bernoulli ◽  
2020 ◽  
Vol 26 (4) ◽  
pp. 2699-2726
Author(s):  
Jian Song ◽  
Xiaoming Song ◽  
Fangjun Xu
Keyword(s):  
Wave Equation ◽  
Gaussian Noise ◽  
Stochastic Wave Equation

Energy of a string driven by a two-parameter Gaussian noise white in time

2001 ◽  
Vol 38 (04) ◽  
pp. 960-974 ◽  
Author(s):  
Boris P. Belinskiy ◽  
Peter Caithamer
Keyword(s):  
Wave Equation ◽  
Gaussian Noise ◽  
Spatial Dimension ◽  
Spatial Covariance ◽  
Stochastic Wave Equation ◽  
Two Parameter

In this paper we consider the stochastic wave equation in one spatial dimension driven by a two-parameter Gaussian noise which is white in time and has general spatial covariance. We give conditions on the spatial covariance of the driving noise sufficient for the string to have finite expected energy and calculate this energy as a function of time. We show that these same conditions on the spatial covariance of the driving noise are also sufficient to guarantee that the energy of the string has a version which is continuous almost surely.

Energy of a string driven by a two-parameter Gaussian noise white in time

2001 ◽  
Vol 38 (4) ◽  
pp. 960-974 ◽  
Author(s):  
Boris P. Belinskiy ◽  
Peter Caithamer
Keyword(s):  
Wave Equation ◽  
Gaussian Noise ◽  
Spatial Dimension ◽  
Spatial Covariance ◽  
Stochastic Wave Equation ◽  
Two Parameter

In this paper we consider the stochastic wave equation in one spatial dimension driven by a two-parameter Gaussian noise which is white in time and has general spatial covariance. We give conditions on the spatial covariance of the driving noise sufficient for the string to have finite expected energy and calculate this energy as a function of time. We show that these same conditions on the spatial covariance of the driving noise are also sufficient to guarantee that the energy of the string has a version which is continuous almost surely.

A Modulus for the 3-Dimensional Wave Equation With Noise: Dealing With a Singular Kernel

1993 ◽  
Vol 45 (6) ◽  
pp. 1263-1275
Author(s):  
C. Mueller
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Modulus Of Continuity ◽  
Gaussian Noise ◽  
Singular Kernel ◽  
3 Dimensional ◽  
Noise Term ◽  
Dimensional Wave

AbstractWe give a modulus of continuity for solutions of the wave equation with a noise term:utt = Δu + a(u) + b(u)G, x ∈ ℝ3where G is a Gaussian noise. This case is more difficult than in lower dimensions because the fundamental solution of the wave equation is singular.

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