论文标题
在带顶点噪声的量子图的抛物线库奇问题上
On the parabolic Cauchy problem for quantum graphs with vertex noise
论文作者
论文摘要
我们研究了与量子图或多项式类型非线性以及加性高斯噪声扰动的顶点条件相关的抛物线库奇问题。顶点条件是每个顶点中的标准连续性和Kirchhoff假设。在只有Kirchhoff条件受到干扰的情况下，我们可以证明具有连续路径在标准状态空间$ \ MATHCAL $ \ MATHCAL {H} $中的平方集成函数中具有连续路径的唯一性。我们还表明，解决方案是马尔可夫和候车者。此外，假设统一的自动化操作员的归一化量函数的顶点值统一是有限的，我们表明，温和的解决方案在与汉密尔顿操作员相关的分数域空间中具有连续的路径，$ \ mathcal {h}_α$ for $ $ a <\ frac <\ frac}当汉密尔顿运营商是受电位扰动的标准拉普拉斯时期，就是这种情况。我们还表明，如果在两种类型的顶点条件下都存在噪声，那么该问题将在分数域空间中具有连续路径的温和解决方案$ \ MATHCAL {h}_α$，$α< - \ frac {1} {1} {4} {4} $。这些规律性结果是在单个间隔和经典边界Dirichlet或Neumann噪声的情况下，Da Prato和Zabczyk [9]获得的量子图类似物。
We investigate the parabolic Cauchy problem associated with quantum graphs including Lipschitz or polynomial type nonlinearities and additive Gaussian noise perturbed vertex conditions. The vertex conditions are the standard continuity and Kirchhoff assumptions in each vertex. In the case when only Kirchhoff conditions are perturbed, we can prove existence and uniqueness of a mild solution with continuous paths in the standard state space $\mathcal{H}$ of square integrable functions on the edges. We also show that the solution is Markov and Feller. Furthermore, assuming that the vertex values of the normalized eigenfunctions of the self-adjoint operator governing the problem are uniformly bounded, we show that the mild solution has continuous paths in the fractional domain space associated with the Hamiltonian operator, $\mathcal{H}_α$ for $α<\frac{1}{4}$. This is the case when the Hamiltonian operator is the standard Laplacian perturbed by a potential. We also show that if noise is present in both type of vertex conditions, then the problem admits a mild solution with continuous paths in the fractional domain space $\mathcal{H}_α$ with $α<-\frac{1}{4}$ only. These regularity results are the quantum graph analogues obtained by da Prato and Zabczyk [9] in case of a single interval and classical boundary Dirichlet or Neumann noise.
