It has recently been shown that complete Bernstein functions of the Laplace operator map the Dirichlet boundary condition of a related elliptic PDE to the Neumann boundary condition. The importance of this mapping consists in being able to convert problems involving non-local operators, like fractional Laplacians, into ones that only involve differential operators. We generalise this result to diffusion operators associated with stochastic differential equations, using a method which is entirely based on stochastic analysis.
J. Herman has been supported by EPSRC funding as a part of the MASDOC DTC, Grant reference number EP/HO23364/1.
We are grateful to the anonymous referee for their detailed report which helped to clarify the presentation. Moreover, the fact, quoted in Remark 2.13(a), that uniform convergence can even follow from pointwise convergence, under certain conditions, was kindly pointed out to us by the referee.
"Extension technique for functions of diffusion operators: a stochastic approach." Electron. J. Probab. 26 1 - 32, 2021. https://doi.org/10.1214/21-EJP624