Jiang-Lun Wu

Parabolic SPDEs driven by Poisson white noise

Stochastic partial differential equations (SPDEs) of parabolic type driven by (pure) Poisson white noise are investigated in this paper. These equations are interpreted as stochastic integral equations of the jump type involving evolution kernels. Existence and uniqueness of the solution is established.

CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS AND THEIR ANALYTIC CONTINUATION TO WIGHTMAN FUNCTIONS

We construct Euclidean random fields X over by convoluting generalized white noise F with some integral kernels G, as X=G*F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise. Furthermore, by representing the kernels Gα of the pseudo-differential operators for α∈(0, 1) and m0>0 as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of X=Gα*F, obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property. Finally we give some remarks on scattering theory for these models.

Invariant Measures and Symmetry Property of Lévy Type Operators

Second order elliptic integro-differential operators (Lévy type operators) are investigated. The notion of regular (infinitesimal) invariant probability measures for such operators is posed. Sufficient conditions for the existence of such regular infinitesimal invariant probability measures are obtained and the symmetrization problem is discussed.

Models of Local Relativistic Quantum Fields with Indefinite Metric (in All Dimensions)

Abstract.A condition on a set of truncated Wightman functions is formulated and shown to permit the construction of the Hilbert space structure included in the Morchio--Strocchi modified Wightman axioms. The truncated Wightman functions which are obtained by analytic continuation of the (truncated) Schwinger functions of Euclidean scalar random fields and covariant vector (quaternionic) random fields constructed via convoluted generalized white noise, are then shown to satisfy this condition. As a consequence such random fields provide relativistic models for indefinite metric quantum field theory, in dimension 4 (vector case), respectively in all dimensions (scalar case).

Euclidean random fields obtained by convolution from generalized white noise

We study Euclidean random fields X over Rd of the form X=G*F, where F is a generalized white noise over Rd and G is an integral kernel. We give conditions for the existence of the characteristic functional and moment functions and we construct a convergent lattice approximation of X. Finally, we perform the analytic continuation of the moment functions and the characteristic functional of X, obtaining the corresponding relativistic functions on Minkowski space.

Molecular Versus Electromagnetic Wave Propagation Loss in Macro-Scale Environments

Molecular communications (MC) has been studied as a bio-inspired information carrier for micro-scale and nano-scale environments. On the macro-scale, it can also be considered as an alternative to electromagnetic (EM) wave based systems, especially in environments where there is significant attenuation to EM wave power. This paper goes beyond the unbounded free space propagation to examine three macro-scale environments: the pipe, the knife edge, and the mesh channel. Approximate analytical expressions shown in this paper demonstrate that MC has an advantage over EM wave communications when: 1) the EM frequency is below the cut-off frequency for the pipe channel, 2) the EM wavelength is considerably larger than the mesh period, and 3) when the receiver is in the high diffraction loss region of an obstacle.

Analytic and Probabilistic Aspects of Lévy Processes and Fields in Quantum Theory

A review of work on the description of generators and processes associated with stochastic (partial or pseudo-) differential equations driven by general white noises (including jump as well as diffusion parts) is given. Processes with finite- or infinite-dimensional state space are described in a unified way using the theory of Dirichlet forms, combined with the technique of subordination of processes. In particular the analytic problems arising from subordinating sub-Markov semigroups are described. As examples the subordination of stochastic quantization processes is presented. It is also described how stochastic partial differential or pseudodifferential equations are used to construct relativistic quantum fields in indefinite metric with nontrivial scattering in four space- time dimensions.

Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric

Abstract We study models of self-interacting massless spin 1 local relativistic quantum fields with indefinite metric in space-time dimension four. We prove that these models for large times converge to free fields and we derive explicit formulae for their (nontrivial, gauge invariant) scattering amplitudes. These scattering amplitudes have properties expected for S-matrix theory.

Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter H ∈ (1 2, 1). We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals.

Scattering behaviour of quantum vector fields obtained from Euclidean covariant SPDEs

We discuss Euclidean covariant vector random fields as the solution of stochastic partial differential equations of the form DA = η, where D is a covariant (w.r.t. a representation τ of SO(d)) differential operator with “positive mass spectrum” and η is a non-Gaussian white noise. We obtain explicit formulae for the Fourier transformed truncated Wightman functions, using the analytic continuation of Schwinger functions discussed by Becker, Gielerak and Lugewicz. Based on these formulae we give necessary and sufficient conditions on the mass spectrum of D which imply nontrivial scattering behaviour of relativistic quantum vector fields associated to the given sequence of Wightman functions. We compute the scattering amplitudes explicitly and we find that the masses of particles in the obtained theory are determined by the mass spectrum of D.