ABSTRACT

In this paper, we characterize the topological support in Hölder norm of the law of the solution to a stochastic wave equation with three-dimensional space variable. This note is an extension of Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597]. The result presented here characterize a more general type of stochastic wave equations in 3-d space variable than those considered in Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597]. Here we extend these two previous results in the folowing sense. The first extension is that we cover the case of multiplicative noise and non-zero initial conditions. The second extension is related to the covariance function associated with the noise, here we follow the approach of Hu, Huang and Nualart and ask conditions in terms of the mean Hölder continuity of such covariance function. As in Delgado-Vences and Sanz-Solé [Approximation of a stochastic wave equation in dimension three, with applications to a support theorem in Hölder norm, Bernoulli 20(4) (2014), pp. 2169–2216; Approximation of a stochastic wave equation in dimension three, with application to a support theorem in Hölder norm: The non-stationary case, Bernoulli 22(3) (2016), pp. 1572–1597] the result is a consequence of an approximation theorem, in the convergence of probability, for a sequence of evolution equations driven by a family of regularizations of the driving noise.

抽象的

在本文中，我们描述了具有三维空间变量的随机波动方程解的定律在Hölder范式中的拓扑支持。本注释是Delgado-Vences和Sanz-Solé的扩展[三维中的随机波动方程的逼近，并适用于Hölder范数中的支持定理，Bernoulli 20（4）（2014），第2169-2216页；维数为3的随机波动方程的逼近，并应用于Hölder范式的支持定理：非平稳情况，Bernoulli 22（3）（2016），第1572-1597页]。与在Delgado-Vences和Sanz-Solé中所考虑的相比，此处呈现的结果在3-d空间变量中表征了更为通用的随机波动方程。维数为3的随机波动方程的逼近，并应用到Hölder规范中的支持定理，Bernoulli 20（4）（2014），pp。2169-2216；维数为3的随机波动方程的逼近，并应用于Hölder范式的支持定理：非平稳情况，Bernoulli 22（3）（2016），第1572-1597页]。在这里，我们从以下意义上扩展了这两个先前的结果。第一个扩展是，我们讨论了乘法噪声和非零初始条件的情况。第二个扩展与与噪声相关的协方差函数有关，这里我们遵循Hu，Huang和Nualart的方法，并根据此类协方差函数的平均Hölder连续性询问条件。如在德尔加多-旺斯和桑兹-索莱[维数为3的随机波动方程的逼近，并应用到Hölder规范中的支持定理，Bernoulli 20（4）（2014），pp。2169-2216；维数为3的随机波动方程的逼近，并应用到Hölder范式的支持定理中：非平稳情况，Bernoulli 22（3）（2016），第1572-1597页]，结果是逼近定理的结果在概率收敛中，对于由一系列驱动噪声的正则化驱动的一系列演化方程。