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The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 43 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Large deviations for many Brownian bridges with symmetrised initialterminal condition
, 2006
"... Consider a large system of N Brownian motions in R d with some nondegenerate initial measure on some fixed time interval [0, β] with symmetrised initialterminal condition. That is, for any i, the terminal location of the ith motion is affixed to the initial point of the σ(i)th motion, where σ is ..."
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Cited by 7 (6 self)
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Consider a large system of N Brownian motions in R d with some nondegenerate initial measure on some fixed time interval [0, β] with symmetrised initialterminal condition. That is, for any i, the terminal location of the ith motion is affixed to the initial point of the σ(i)th motion, where σ is a uniformly distributed random permutation of 1,..., N. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature 1/β. In this paper, we describe the largeN behaviour of the empirical path measure (the mean of the Dirac measures in the N paths) and of the mean of the normalised occupation measures of the N motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and FenchelLegendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the wellknown DonskerVaradhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the largeN asymptotic of the symmetrised trace of e −βHN, where HN is an Nparticle Hamilton operator in a trap.
F.: 2003, ‘The quantum mechanical path integral: toward a realistic interpretation’, Unpublished and copyright. See http://www/eskimo.com/ msharlow
"... In this paper, I explore the feasibility of a realistic interpretation of the quantum mechanical path integral — that is, an interpretation according to which the particle actually follows the paths that contribute to the integral. I argue that an interpretation of this sort requires spacetime to ha ..."
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Cited by 4 (2 self)
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In this paper, I explore the feasibility of a realistic interpretation of the quantum mechanical path integral — that is, an interpretation according to which the particle actually follows the paths that contribute to the integral. I argue that an interpretation of this sort requires spacetime to have a branching structure similar to the structures of the branching spacetimes proposed by previous authors. I point out one possible way to construct branching spacetimes of the required sort, and I ask whether the resulting interpretation of quantum mechanics is empirically testable. 1 1. The Path Integral: A Philosophical Perspective In the path integral formulation of quantum mechanics [Feynman and Hibbs 1965], the transition amplitude between two quantum states of a system is expressed as a sum over contributions from possible classical histories of that system. The mathematical object used to represent this sum is an integral known as a "path integral. " It is important to note that the histories used in this integral are classical histories. For example, if the system is a particle, then the path integral runs over classical trajectories of the particle. These
Stationary quantum Markov process for the Wigner function
, 2007
"... Abstract. Many stochastic models have been investigated for quantum mechanics because of its stochastic nature. In 1988, Cohendet et al. introduced a dichotomic variable to quantum phase space and proposed a background Markov process for the time evolution of the Wigner function. However, in their m ..."
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Abstract. Many stochastic models have been investigated for quantum mechanics because of its stochastic nature. In 1988, Cohendet et al. introduced a dichotomic variable to quantum phase space and proposed a background Markov process for the time evolution of the Wigner function. However, in their method the whole distribution function is required to determine the next step of a constituent particle. In this paper, we discuss a stationary quantum Markov process which enables us to treat each particle independently introducing U(1) extension for the phase space. The process has branching and vanishing points but we can keep a finite time interval between the branchings. The procedure to make the simulation of the process is also discussed. Stationary quantum Markov process for the Wigner function 2 1.
Stationary quantum Markov process for the Wigner function on a lattice phase space
, 2007
"... Abstract. As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space ZN ×ZN with N odd. By introducing a phase factor extension to the phase space, each particle can be treated independently. This ..."
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Abstract. As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space ZN ×ZN with N odd. By introducing a phase factor extension to the phase space, each particle can be treated independently. This is an improvement on earlier methods that require the whole distribution function to determine the evolution of a constituent particle. The process has branching and vanishing points, though a finite time interval can be maintained between the branchings. The procedure to perform a simulation using the process is presented. Stationary quantum Markov process for the Wigner function on a lattice phase space 2 1.
Mimimal Relative Entropy Martingale Measure of Birth and Death Process
"... In this article, we investigate the MEMM (Minimal relative Entropy Martingale Measure) of Birth and Death processes and the MEMM of generalized Birth and Death processes. We see that the existence problem of the MEMM is reduced to the problem of solving the corresponding HamiltonJacobiBellman equa ..."
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In this article, we investigate the MEMM (Minimal relative Entropy Martingale Measure) of Birth and Death processes and the MEMM of generalized Birth and Death processes. We see that the existence problem of the MEMM is reduced to the problem of solving the corresponding HamiltonJacobiBellman equation. 1 Introduction The relative entropy plays very important roles in various fields, for example in the statistical physics, in the information theory, and statistical estimation theory. Recently the ralative entropy has been proved that it is related to the mathematical finance theory. We investigate the MEMM (Minimal relative Entropy Martingale Measure) of Birth and Death processes in this context. In 2 we formulate our problems as a variation problems. In 3 we introduce the HamiltonJacobiBellman equation corresponding to the variation problems. In 4 we see that the existence problem of MEMM is reduced to the problem of solving the HamiltonJacobiBellman equation, and we give a...
ENTROPY, GEOMETRY, AND THE QUANTUM POTENTIAL
, 2005
"... Abstract. We sketch and emphasize here the automatic emergence of a quantum potential Q in e.g. classical WDW type equations upon inserting a (Bohmian) complex wave function ψ = Rexp(iS/�). The interpretation of Q in terms of momentum fluctuations via the Fisher information and entropy ideas is disc ..."
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Abstract. We sketch and emphasize here the automatic emergence of a quantum potential Q in e.g. classical WDW type equations upon inserting a (Bohmian) complex wave function ψ = Rexp(iS/�). The interpretation of Q in terms of momentum fluctuations via the Fisher information and entropy ideas is discussed along with the essentially forced role of R 2 as a probability density. We also review the constructions