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  • 1
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    Chaos - The Interplay Between Stochastic and Deterministic Behaviour : Proceedings of the XXXIst Winter School of Theoretical Physics Held in Karpacz, Poland 13-24 February 1995 (1995)
    Berlin ; Heidelberg : Springer
    Details
    ISBN: 9783540601883
    URL: http://www.springerlink.com/openurl.asp?genre=book&isbn=978-3-540-60188-3
    Language: English
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  • 2
    Electronic Resource
    Electronic Resource
    Ornstein–Uhlenbeck–Cauchy process (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 6843-6860 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We combine earlier investigations of linear systems subject to Lévy fluctuations with recent attempts to give meaning to so-called Lévy flights in external force fields. We give a complete construction of the Ornstein–Uhlenbeck–Cauchy process as a fully computable paradigm example of Doob's stable noise-supported Ornstein–Uhlenbeck process. Despite the nonexistence of all moments, we determine local characteristics (forward drift) of the process, generators of forward and backward dynamics, and relevant (pseudodifferential) evolution equations. The induced nonstationary spatial process is proved to be Markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1290054
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  • 3
    Electronic Resource
    Electronic Resource
    Non-negative Feynman–Kac kernels in Schrödinger's interpolation problem (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 1-15 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The local formulations of the Markovian interpolating dynamics, which is constrained by the prescribed input-output statistics data, usually utilize strictly positive Feynman–Kac kernels. This implies that the related Markov diffusion processes admit vanishing probability densities only at the boundaries of the spatial volume confining the process. We discuss an extension of the framework to encompass singular potentials and associated non-negative Feynman–Kac-type kernels. It allows us to deal with a class of continuous interpolations admitted by general non-negative solutions of the Schrödinger boundary data problem. The resulting nonstationary stochastic processes are capable of both developing and destroying nodes (zeros) of probability densities in the course of their evolution, also away from the spatial boundaries. This observation conforms with the general mathematical theory (due to M. Nagasawa and R. Aebi) that is based on the notion of multiplicative functionals, extending in turn the well known Doob's h-transformation technique. In view of emphasizing the role of the theory of non-negative solutions of parabolic partial differential equations and the link with "Wiener exclusion" techniques used to evaluate certain Wiener functionals, we give an alternative insight into the issue, that opens a transparent route towards applications.© 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532004
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  • 4
    Electronic Resource
    Electronic Resource
    Rotational diffusions as seen by relativistic observers (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 3393-3401 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The major unsolved problem in the framework of Nelson's stochastic mechanics is addressed and an attempt is made to provide a description of relativistic spin-1/2 particles in terms of Markovian diffusions on S3. Random rotations are here labeled by the proper time of a particle in relativistic motion and are continuously distributed along a space-time trajectory followed by the particle in Minkowski space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529887
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  • 5
    Electronic Resource
    Electronic Resource
    Fermion excitations of the nonlinear Schrödinger field in the attractive case (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 612-614 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The recent inverse scattering method analysis by L. Martínez Alonso [J. Math. Phys. 25, 1935 (1984)] is extended to demonstrate that the Bose quantized (attractive) nonlinear Schrödinger field in 1+1 dimensions, admits fermion excitations in its (quantum soliton) spectrum.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527214
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  • 6
    Electronic Resource
    Electronic Resource
    Boson approximants for lattice Fermi systems (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 2039-2044 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We demonstrate that Bose systems can be used to simulate properties of their Fermi partners in thermal bath, provided the spectral property HB =PHBP +(1−P)HB(1−P), HF =PHBP can be proved in the state space of the Bose system. The approximation accuracy can be made arbitrarily good by varying the (free) coupling parameter λ∈(0,∞), and permits studying of fermionic partition or correlation functions in a finite volume by means of the standard techniques (Bose:path integral, Monte Carlo, etc).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526876
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  • 7
    Electronic Resource
    Electronic Resource
    Fermi states of Bose systems in three space dimensions (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 490-494 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Recently an exact spectral solution was constructed by Sudarshan and Tata for the (Naitch-theta) Fermi version of the Lee model. We demonstrate that it provides a partial solution for the related pure Bose spectral problems. Moreover, the (Naitch-theta) Bose (Bolsterli–Nelson) version of the Lee model is shown to possess Fermi partners, both exhibiting the partial solubility interplay: finding solutions in the Fermi case would presumably be easier than in the original Bose model. Fermi states of the underlying Bose systems in three space dimensions are explicitly identified.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526636
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  • 8
    Electronic Resource
    Electronic Resource
    Fermi states of Bose systems in three space dimensions. II. The solvable version of the Hubbard model (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 2684-2692 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An exact spectral solution of the Hubbard problem on the tetrahedron is given. It is demonstrated that it provides at the same time a partial spectral solution for the Bose version of this (Fermi) model. A remarkable relationship of the Hamiltonians HB=HF+(1−P)HB(1−P) is established, where P is a projection in the state space of the interacting Bose system. This partial observation is then extended to the spatial lattice composed of the four-site Hubbard cells, which interact due to the superimposed perturbation of the hopping type, affecting all sites of the lattice; H=∑(i j) ∑σ=± bi j c@B|iσ cjσ+U ∑i ni+ni− being the corresponding Hamiltonian. It is thus demonstrated that the joint Bose–Fermi spectral problem (previously established in 1+1 dimensions) makes sense in 1+3 dimensions as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526738
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  • 9
    Electronic Resource
    Electronic Resource
    Feynman–Kac kernels in Markovian representations of the Schrödinger interpolating dynamics (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 732-751 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Probabilistic solutions of the so-called Schrödinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input–output statistics data for the process taking place in a finite-time interval. The key issue is to select the jointly continuous in all variables positive Feynman–Kac kernel, appropriate for the phenomenological (physical) situation. We extend the existing formulations of the problem to cases when the kernel is not a fundamental solution of a parabolic equation, and prove the existence of a continuous Markovian interpolation in this case. Next, we analyze the compatibility of this stochastic evolution with the original parabolic dynamics, which is assumed to be governed by the temporally adjoint pair of (parabolic) partial differential equations, and prove that the pertinent random motion is a diffusion process. In particular, in conjunction with Born's statistical interpretation postulate in quantum theory, we consider stochastic processes which are compatible with the Schrödinger picture quantum evolution. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531412
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  • 10
    Electronic Resource
    Electronic Resource
    Cauchy noise and affiliated stochastic processes (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 1057-1073 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: By departing from the previous attempt [Phys. Rev. E 51, 4114 (1995)] we give a detailed construction of conditional and perturbed Markov processes, under the assumption that the Cauchy law of probability replaces the Gaussian law (appropriate for the Wiener process) as the model of primordial noise. All considered processes are regarded as probabilistic solutions of the so-called Schrödinger interpolation problem, whose validity is thus extended to the jump-type processes and their step process approximants. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532706
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