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2021 Extension technique for functions of diffusion operators: a stochastic approach
Sigurd Assing, John Herman
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Electron. J. Probab. 26: 1-32 (2021). DOI: 10.1214/21-EJP624


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.

Funding Statement

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.


Download Citation

Sigurd Assing. John Herman. "Extension technique for functions of diffusion operators: a stochastic approach." Electron. J. Probab. 26 1 - 32, 2021.


Received: 25 November 2019; Accepted: 3 April 2021; Published: 2021
First available in Project Euclid: 12 May 2021

Digital Object Identifier: 10.1214/21-EJP624

Primary: 60J45 , 60J55 , 60J60
Secondary: 35J25 , 35J70 , 47G20

Keywords: Dirichlet-to-Neumann map , elliptic equation , Krein strings , trace process


Vol.26 • 2021
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