Nonlocal operators with Neumann conditions
Abstract
We construct a strong Markov process $X$ corresponding to the Dirichlet form of Servadei and Valdinoci and use the process to solve the corresponding Neumann boundary problem for the fractional Laplacian and the half-line.
When started in $(0,\infty)$, the process behaves like the isotropic $\alpha$-stable process. At the first exit time from $(0,\infty)$, $X$ jumps to a point $y$ on the negative half-line, spends an exponential time at $y$, then jumps back to the positive half-line and starts afresh. The asymptotic behavior of $X$ strongly depends on the parameter $\alpha$. In particular, its lifetime is finite if $\alpha \in (1,2)$, and infinite if $\alpha \in (0,1]$. Our construction is based on concatenation of Markov processes.
We identify the Dirichlet form corresponding to $X$ with the form of Servadei and Valdinoci, prove a Hardy inequality, and propose various characterizations of the domain for the form. Under suitable assumptions, the solution of the Neumann boundary problem is given by the Green operator corresponding to the process $X$.
