Traces of harmonic functions,capacities, and traces of symmetric Markov processes

We derive explicit isomorphism formulas between weighted Dirichlet integrals for harmonic functions and boundary Dirichlet forms. Applications yield results on traces of Markov processes and convergence quasieverywhere of harmonic functions.     

Phase Space Transformations of Gaussian Diffusions

It is shown that for Gaussian diffusions, the transformation back to Brownian motion, usually accomplished via the Girsanov (or Feynman–Kac) formula and time-shift, can be obtained by a classical canonical, i.e. symplectic, transformation in phase space. The method is based on constants of motion, in this case the Wronskian. Similar transformations for general diffusions are briefly discussed.     

Foliated Manifolds and Conformal Heat Morphisms

The objects under study, in this article, are Riemannian manifoldsfoliated by hypersurfaces. Looking at the transverse direction as time,we construct the generalised heat operator and, in the spirit of atime-space extension of harmonic morphisms, we introduce the conformal heat morphisms. Concrete examples of these concepts arepresented, as well as, a characterisation of conformal heat morphisms.Lastly, we calculate the heat Lie algebra of the generalised heatoperator.     

Position dependent non-linear Schrödinger hierarchies: Involutivity, commutation relations, renormalisation and classical invariants

We consider a family of explicitly position dependent hierarchies , containing the NLS (non-linear Schrödinger) hierarchy. All are involutive and fulfill DIn=nIn−1, where D=D⁻¹V₀, V₀ being the Hamiltonian vector field afforded by the common ground state I₀=uv. The construction requires renormalisation of certain function parameters.It is shown that the ‘quantum space’ C[I₀,I₁,…] projects down to its classical counterpart C[p], with p=I₁/I₀, the momentum density. The quotient is the kernel of D. It is identified with classical semi-invariants for forms in two variables.     

Gaussian random fields,infinite dimensional Ornstein-Uhlenbeck processes,and symmetric Markov processes

A general treatment of infinite dimensional Ornstein-Uhlenbeck processes (OUPs) is presented. Emphasis is put on their connection with ordinary Gaussian random fields, and OUPs as symmetric Markov processes. We also discuss the relation to second quantisation and Gaussian Markov random fields.Supported in part by the Swedish Natural Science Research Council, NFR.     

Lagrangian Approach to Evolution Equations: Symmetries and Conservation Laws

We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive conservation laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local conservation laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrödinger and Korteweg—de Vries type systems.     

Random fields as solutions of the inhomogeneous quaternionic Cauchy-Riemann equation. I. Invariance and analytic continuation

We consider random fieldsA satisfying the quaternionic Cauchy-Riemann equationA=F, whereF is white noise. Under appropriate conditions onF, A is invariant under the proper Euclidean group in four dimensions, but in general not under time reflection. The Schwinger functions can be analytically continued to Wightman functions satisfying the relativistic postulates on invariance, specrral property and locality.