On the parabolic Cauchy problem for quantum graphs with vertex noise

Abstract

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, ${\mathcal{H}}_{\mathit{\alpha }}$ for $\mathit{\alpha }<\frac{1}{4}$. 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 ${\mathcal{H}}_{\mathit{\alpha }}$ with $\mathit{\alpha }<-\frac{1}{4}$ 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.