arXiv subjects

John Herman

Publications and source records attributed to John Herman.

2 recordsLinked to original sources

arXiv2019

Extension Technique for Functions of Diffusion Operators: a stochastic approach

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.

math.PR↗

arXiv2019

Space-time coupled evolution equations and their stochastic solutions

We consider a class of space-time coupled evolution equations (CEEs), obtained by a subordination of the heat operator. Our CEEs reformulate and extend known governing equations of non-Markovian processes arising as scaling limits of continuous time random walks, with widespread applications. In particular we allow for initial conditions imposed on the past, general spatial operators on Euclidean domains and a forcing term. We prove existence, uniqueness and stochastic representation for solutions.

math.AP↗