Lévy-driven Volterra Equations in Space and Time
Lévy-driven Volterra Equations in Space and Time
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空间和时间中的 Lévy 驱动 Volterra 方程
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Carsten Chong
中科院分区：
文献类型：
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作者：
Carsten Chong
我们研究由 Lévy 基驱动的时空非线性随机 Volterra 方程。在非线性项的 Lipschitz 条件下，我们给出了加权函数空间中的存在性和唯一性准则，该准则取决于核的可积性质和 Lévy 基的特征。特别注意具有平稳解的方程，或更一般地，具有无限记忆的方程，即积分的时域从负无穷大开始的方程。在这里，与时间为正的情况相反，核上通常的可积条件不再足以证明解的存在性和唯一性，但我们必须对核和 Lévy 特性施加额外的大小条件。此外，一旦保证解的存在，我们就分析它的渐近稳定性，即当时间趋于无穷大时，它的矩是否仍然有界。只要核和特征足够小，或者方程的非线性表现出严格小于一阶的分数增长，就可以证明稳定性。将结果应用于随机热方程进行说明。
We investigate nonlinear stochastic Volterra equations in space and time that are driven by Lévy bases. Under a Lipschitz condition on the nonlinear term, we give existence and uniqueness criteria in weighted function spaces that depend on integrability properties of the kernel and the characteristics of the Lévy basis. Particular attention is devoted to equations with stationary solutions, or more generally, to equations with infinite memory, that is, where the time domain of integration starts at minus infinity. Here, in contrast to the case where time is positive, the usual integrability conditions on the kernel are no longer sufficient for the existence and uniqueness of solutions, but we have to impose additional size conditions on the kernel and the Lévy characteristics. Furthermore, once the existence of a solution is guaranteed, we analyze its asymptotic stability, that is, whether its moments remain bounded when time goes to infinity. Stability is proved whenever kernel and characteristics are small enough, or the nonlinearity of the equation exhibits a fractional growth of order strictly smaller than one. The results are applied to the stochastic heat equation for illustration.