1,020 research outputs found
Quantum Feynman-Kac perturbations
We develop fully noncommutative Feynman-Kac formulae by employing quantum
stochastic processes. To this end we establish some theory for perturbing
quantum stochastic flows on von Neumann algebras by multiplier cocycles.
Multiplier cocycles are constructed via quantum stochastic differential
equations whose coefficients are driven by the flow. The resulting class of
cocycles is characterised under alternative assumptions of separability or
Markov regularity. Our results generalise those obtained using classical
Brownian motion on the one hand, and results for unitarily implemented flows on
the other.Comment: 27 pages. Minor corrections to version 2. To appear in the Journal of
the London Mathematical Societ
Solution of the Fokker-Planck equation with boundary conditions by Feynman-Kac integration.
In this paper, we apply the results about d and d-function perturbations in order to formulate within the Feynman-Kac integration the solution of the forward Fokker-Planck equation subject to Dirichlet or Neumann boundary conditions. We introduce the concept of convex order to derive upper and lower bounds for path integrals with d and d- functions in the integrand. We suggest the use of bounds as an approximation for the solution.Feynman-Kac integration; Functions; Integration; Path integral; Perturbations theory; SDE;
Cauchy Noise and Affiliated Stochastic Processes
By departing from the previous attempt (Phys. Rev. {\bf 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
Schr\"{o}dinger interpolation problem, whose validity is thus extended to the
jump-type processes and their step process approximants.Comment: Latex fil
Construction of Self-Adjoint Berezin-Toeplitz Operators on Kahler Manifolds and a Probabilistic Representation of the Associated Semigroups
We investigate a class of operators resulting from a quantization scheme
attributed to Berezin. These so-called Berezin-Toeplitz operators are defined
on a Hilbert space of square-integrable holomorphic sections in a line bundle
over the classical phase space. As a first goal we develop self-adjointness
criteria for Berezin-Toeplitz operators defined via quadratic forms. Then,
following a concept of Daubechies and Klauder, the semigroups generated by
these operators may under certain conditions be represented in the form of
Wiener-regularized path integrals. More explicitly, the integration is taken
over Brownian-motion paths in phase space in the ultra-diffusive limit. All
results are the consequence of a relation between Berezin-Toeplitz operators
and Schrodinger operators defined via certain quadratic forms. The
probabilistic representation is derived in conjunction with a version of the
Feynman-Kac formula.Comment: AMS-LaTeX, 30 pages, no figure
Feynman-Kac formula for Levy processes and semiclassical (Euclidean) momentum representation
We prove a version of the Feynman-Kac formula for Levy processes and
integro-differential operators, with application to the momentum representation
of suitable quantum (Euclidean) systems whose Hamiltonians involve
L\'{e}vy-type potentials. Large deviation techniques are used to obtain the
limiting behavior of the systems as the Planck constant approaches zero. It
turns out that the limiting behavior coincides with fresh aspects of the
semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of
Levy processes are considered as illustrations and precise asymptotics are
given for the terms in both configuration and momentum representations
Stochastic modelling of nonlinear dynamical systems
We develop a general theory dealing with stochastic models for dynamical
systems that are governed by various nonlinear, ordinary or partial
differential, equations. In particular, we address the problem how flows in the
random medium (related to driving velocity fields which are generically bound
to obey suitable local conservation laws) can be reconciled with the notion of
dispersion due to a Markovian diffusion process.Comment: in D. S. Broomhead, E. A. Luchinskaya, P. V. E. McClintock and T.
Mullin, ed., "Stochaos: Stochastic and Chaotic Dynamics in the Lakes",
American Institute of Physics, Woodbury, Ny, in pres
Linear response theory and transient fluctuation theorems for diffusion processes: a backward point of view
On the basis of perturbed Kolmogorov backward equations and path integral
representation, we unify the derivations of the linear response theory and
transient fluctuation theorems for continuous diffusion processes from a
backward point of view. We find that a variety of transient fluctuation
theorems could be interpreted as a consequence of a generalized
Chapman-Kolmogorov equation, which intrinsically arises from the Markovian
characteristic of diffusion processes
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