Helge Holden

Ideas and Methods in Mathematical Analysis, Stochastics and Applications

Preface Picture of Raphael Hoegh-Krohn Bibliography of Raphael Hoegh-Krohn 1. On the scientific work of Raphael Hoegh-Krohn Part I. Stochastic Analysis 2. Some Euclidean integer-valued random fields with Markov properties 3. A Beurling-Deny type structure theorem for Dirichlet forms on general state spaces 4. Log-concavity of radial Schrodinger wave functions and convergence of planetesimal diffusions 5. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals 6. Large deviations for weak solutions of stochastic differential equations 7. On a class of probabilistic integrodifferential equations 8. Operator integrals and martingale integrals with general parameter 9. Wick multiplication and Ito-Skorohod stochastic differential equations 10. Multiscale analysis in random dynamics 11. A limiting distribution connected with fractional parts of linear forms 12. Rapport sur les representations chaotiques 13. Markov properties of solutions of stochastic partial differential equations in a finite volume 14. On stochastic evolution equations with non-homogeneous boundary conditions 15. Grey noise 16. Existence of invariant measures for diffusion process with infinite dimensional state space Part II. Infinite Dimensional Groups 17. On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds 18. Fields of left-invariant standard Brownian motion processes on a smooth bundle of compact-simple Lie groups 19. Unitary highest weight representations of gauge groups Part III. Operator Algebras 20. Some non-commutative orbifolds 21. Ergodic actions of non-abelian compact groups 22. Positive projections onto Jordan algebras and their enveloping von Neumann algebras Part IV. Nonlinear Analysis and Applications 23. How many singularities can there be in an energy minimizing map from the ball to the sphere? 24. Front tracking for petroleum reservoirs 25. Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equations 26. A new representation of soliton solutions of the Kadomtsev-Petviashvili equation 27. On scalar conservation laws in one dimension.

Stochastic multiplicative measures, generalized Markov semigroups, and group-valued stochastic processes and fields

Abstract We extend the well-known relation between Markov semigroups and processes with independent increments to the case of generalized group-valued semigroups indexed by some measurable space and stochastic multiplicative measures. We give a Kolmogorov-type theorem and a classification theorem, connecting the stochastic multiplicative measures with vector-valued Levy-Khinchine fields. The results have applications to the study of stochastic connections and differential equations for forms on Lie groups.

Markov Cosurfaces and Gauge Fields

We give a general construction of homogeneous Markov fields associated with d-1 hypersurfaces on a d-dimensional Riemannian manifold, with values in a Lie group G. In the case d = 2 these “Markov cosurfaces” coincide with pure continuum gauge fields with values in G. The constructed Markov cosurfaces have properties of the type of those postulated for Wilson loops Schwinger functions in the case of gauge fields and lead to relativistic quantum fields. The Markov cosurfaces provide an extension of Markov processes for the case where time points are replaced by d-1-dimensional hypersurfaces,and a corresponding extension of stochastic differential equations if given. The Markov cosurfaces are also shown to be obtainable from lattice models.

Construction of quantized Higgs-like fields in two dimensions

Abstract A mathematical construction of gauge invariant Higgs-like fields in two-dimensional space-time is presented, including passage to the continuum and infinite volume limits. In the limit, a quantum field theory obeying the Osterwalder-Schrader axioms is obtained. The method is based on representing Schwinger functions in terms of stochastic multiplicative curve integrals and Brownian bridges.

Random fields with values in Lie groups and Higgs fields

We review recent work based on extending the known relations between Markov processes and Markov semigroups to the case where time is replaced by hypersurfaces of codimension 1 and the state space is a group. Relations with gauge fields, Higgs fields and relativistic fields are also discussed.