Stochastic processes and their spectral representations over non-archimedean fields

In this paper, we study stochastic processes with values in finite- and infinite-dimensional vector spaces over infinite fields K of zero characteristic with nontrivial non-Archimedean norms. For different types of stochastic processes controlled by measures with values in K and in complete topological vector spaces over K, we study stochastic integrals, vector-valued measures, and integrals in spaces over K. We also prove theorems on spectral decompositions of non-Archimedean stochastic processes.     

On some definable sets over fields with analytic structure

We discover geometric properties of certain definable sets over non-Archimedean valued fields with analytic structures. Results include a parameterized smooth stratification theorem and the existence of a bound on the piece number of fibers for these sets. In addition, we develop a dimension theory for these sets and also for the formulas which define them.     

On infinite products of non-Archimedean measure spaces

We study generalized ‘probabilistic measures’ taking values in non-Archimedean fields (in particular, fields of p-adic numbers). We prove the theorem on the existence of probability on a product of non-Archimedean probabilistic spaces.     

Quasi-Invariant and Pseudo-Differentiable Measures with Values in Non-Archimedean Fields on a Non-Archimedean Banach Space

Quasi-invariant and pseudo-differentiable measures on a Banach space X over a non-Archimedean locally compact infinite field with a non-trivial valuation are defined and constructed. Measures are considered with values in non-Archimedean fields, for example, the field Q _p of p-adic numbers. Theorems and criteria are formulated and proved about quasi-invariance and pseudo-differentiability of measures relative to linear and non-linear operators on X. Characteristic functionals of measures are studied. Moreover, the non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov theorems are investigated. Infinite products of measures are considered and the analog of the Kakutani theorem is proved. Convergence of quasi-invariant and pseudo-differentiable measures in the corresponding spaces of measures is investigated.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 149–199, 2003.     

Topological simplicity,commensurator super-rigidity and nonlinearities of Kac-Moody groups (appendix by P. Bonvin)

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator super-rigidity theorem proved in the Appendix by P. Bonvin.     

Constrained second order optimization on non-archimedean fields

Constrained optimization on non-Archimedean fields is presented. We formalize the notion of a tangent plane to the surface defined by the constraints making use of an implicit function Theorem similar to its real counterpart. Then we derive necessary and sufficient conditions of second order for the existence of a local minimizer of a function subject to a set of equality and inequality constraints, based on a concept of continuity and differentiability that is stronger than the conventional one.     

Involutions of Type G<Subscript>2</Subscript> Over Fields of Characteristic Two

We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G₂ over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G₂ over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two.     

On Nilpotent Orbits of SL<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> and Sp<Subscript>2<Emphasis Type="Italic">n</Emphasis></Subscript> over a Local Non-Archimedean Field

We relate the partition-type parametrization of rational (arithmetic) nilpotent adjoint orbits of the classical groups SL _n and Sp_2n over local non-Archimedean fields with a parametrization, introduced by DeBacker in 2002, which uses the associated Bruhat-Tits building to relate the question to one over the residue field.     

Infinitely divisible distributions over locally compact non-archimedean fields

This paper is devoted to stochastic processes with values in finite-dimensional vector spaces over infinite, locally compact fields of zero and positive characteristics with nontrivial non-Archimedean norms. Infinitely divisible distributions are studied. Theorems about their characteristic functionals are proved. Particular cases are demonstrated as applications to non-Archimedean analogs of Gaussian and Poisson processes and their generalizations.     

Mumford Curves with Maximal Automorphism Group II: Lamé Type Groups in Genus 5-8

A Mumford curve of genus g=5,6,7 or 8 over a non-Archimedean field ofcharacteristic p (such that if p=0, the residue field characteristic exceeds 5) has at most 12(g–1) automorphisms. In this paper, all curves that attain this bound and their automorphism groups (called of Lamé type) are explicitly determined.     

Topological transformation groups of manifolds over non-Archimedean fields,their representations,and quasi-invariant measures,II

Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure, the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied. We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous, unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism group and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological groups. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 18, Functional Analysis, 2006.     

Stein domains in Banach algebraic geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.     

Contractible Lie groups over local fields

Let G be a Lie group over a local field of characteristic p > 0 which admits a contractive automorphism α : G → G (i.e., α ⁿ (x) → 1 as n → ∞, for each x ∈ G). We show that G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results extend to Lie groups over arbitrary complete ultrametric fields. Supported by the German Research Foundation (DFG), grants GL 357/2-1 and GL 357/6-1.     

Measures on diffeomorphism groups for non-archimedean manifolds: Group representations and their applications

Nondegenerate σ-additive measures with ranges in ℝ and ℚ_q (q≠p are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups G′ are constructed on diffeomorphism and homeomorphism groups G for separable non-Archimedean Banach manifolds M over a local fieldK,K ⊃ ℚ_q, where ℚ_q is the field of p-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 381–396, June, 1999.     

Topological transformation groups of manifolds over non-Archimedean fields,their representations,and quasi-invariant measures. I

Diffeomorphism groups and loop groups of manifolds on Banach spaces over non-Archimedean fields are defined. Moreover, for these groups, finite-and infinite-dimensional manifolds over the corresponding fields are considered. The group structure, the differential-geometric structure, and also the topological structure of diffeomorphism groups and loops groups are studied. We prove that these groups do not locally satisfy the Campbell-Hausdorff formula. The principal distinctions in the structure for the Archimedean and classical cases are found. The quasi-invariant measures on these groups with respect to dense subgroups are constructed. Stochastic processes on topological transformation groups of manifolds and, in particular, on diffeomorphism groups and on loop groups and also the corresponding transition probabilities are constructed. Regular, strongly continuous, unitary representations of dense subgroups of topological transformation groups of manifolds, in particular, those of diffeomorphism groups and loop groups associated with quasi-invariant measures on groups and also on the corresponding configurational spaces are constructed. The conditions imposed on the measure and groups under which these unitary representations are irreducible are found. The induced representations of topological groups are studied by using quasi-invariant measures on topological groups. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 39, Functional Analysis, 2006.     

DISTINGUISHED LINE BUNDLES FOR COMPLEX SURFACE AUTOMORPHISMS

We equate dynamical properties (e.g., positive entropy, existence of a periodic curve) of complex projective surface automorphisms with properties of the pull-back actions of such automorphisms on line bundles. We use the properties of the cohomological actions to describe the measures of maximal entropy for automorphisms with positive entropy.     

On multi-orthogonal bases in finite-dimensional non-Archimedean normed spaces

The paper deals with the problem of the existence multi-orthogonal bases in finite-dimensional normed spaces over K, where K is a non-Archimedean complete valued field.