Sergio Albeverio

Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms

SummaryUsing the theory of Dirichlet forms on topological vector spaces we construct solutions to stochastic differential equations in infinite dimensions of the type

A random walk on p-adics - the generator and its spectrum

dX_t = dW_t + \beta (X_t )dt

Extreme events in nature and society

for possibly very singular drifts β. Here (Xt)t≧0 takes values in some topological vector spaceE and (Wt)t≧0 is anE-valued Brownian motion. We give applications in detail to (infinite volume) quantum fields where β is e.g. a renormalized power of a Schwartz distribution. In addition, we present a new approach to the case of linear β which is based on our general results and second quantization. We also prove new results on general diffusion Dirichlet forms in infinite dimensions, in particular that the Fukushima decomposition holds in this case.

Concurrence of Arbitrary Dimensional Bipartite Quantum States

We study the Hamiltonians for nonrelativistic quantum mechanics—and for the heat equation—in terms of energy forms ∫∇f∇fd’gm, where dμ is a positive, not necessarily finite measure on Rn. We cover the cases of very singular interactions (e.g., N particles in R3 interacting by two‐body ’’δ potentials’’). We also exhibit, on the other hand, regularity conditions for μ in order that H be realized as a perturbation of the Laplacian by a measurable or generalized functon. The Hamiltonians defined by energy forms alwasy generate Markov semigroups, and the associated processes are symmetric homogeneous strong Markov diffusion Hunt processes with continuous paths realizations. Ergodicity, transiency, and recurrency are also discussed. The associated stochastic differential equaiton is discussed in the situaion were μ is finite but the drift coefficient is only restricted to be l (Rr,dμ). These results provide a large class of examples where solutions of the heat equaion can be expressed by averages with respect t...

Parabolic SPDEs driven by Poisson white noise

We construct a random walk with continuous time taking values in the p-adic numbers. We compute its transition semigroup and infinitesimal generator and exhibits its spectrum. We also give the associated Dirichlet form, which is of the jump type.

The Wightman Axioms and the Mass Gap for Strong Interactions of Exponential Type in Two-Dimensional Space-Time

Abstract We prove a sufficient condition for the closability of classical Dirichlet forms on L 2 ( E ; μ ) which is also necessary if all components of the Dirichlet form are closable. Here E is a locally convex topological vector space and μ, a (not necessarily quasi-invariant) probability measure on E . The same condition is shown to imply the existence of a closed extension whose domain can be described explicitly. In the special case where μ is quasi-invariant with respect to certain vectors in E our result generalises previous theorems on closability. In addition, we prove a Cameron-Martin-type formula for a large class of measures μ. If E is finite dimensional our characterisation of closability is the analogue of the corresponding one-dimensional result. Applications to quantum fields and the connection with the well-studied case of abstract Wiener spaces are discussed.

Classical Dirichlet forms on topological vector spaces : the construction of the associated diffusion process

Extreme Events: Magic, Mysteries, and Challenges.- Extreme Events: Magic, Mysteries, and Challenges.- General Considerations.- Anticipating Extreme Events.- Mathematical Methods and Concepts for the Analysis of Extreme Events.- Dynamical Interpretation of Extreme Events: Predictability and Predictions.- Endogenous versus Exogenous Origins of Crises.- Scenarios.- Epilepsy: Extreme Events in the Human Brain.- Extreme Events in the Geological Past.- Wind and Precipitation Extremes in the Earths Atmosphere.- Freak Ocean Waves and Refraction of Gaussian Seas.- Predicting the Lifetime of Steel.- Computer Simulations of Opinions and their Reactions to Extreme Events.- Networks of the Extreme: A Search for the Exceptional.- Prevention, Precaution, and Avoidance.- Risk Management and Physical Modelling for Mountainous Natural Hazards.- Prevention of Surprise.- Disasters as Extreme Events and the Importance of Network Interactions for Disaster Response Management.

Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics I.*)

We derive an analytical lower bound for the concurrence of a bipartite quantum state in arbitrary dimension. A functional relation is established relating concurrence, the Peres-Horodecki criterion, and the realignment criterion. We demonstrate that our bound is exact for some mixed quantum states. The significance of our method is illustrated by giving a quantitative evaluation of entanglement for many bound entangled states, some of which fail to be identified by the usual concurrence estimation method.