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  • 1
    Electronic Resource
    Electronic Resource
    On the stochastic mechanics of the free relativistic particle (2001)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 42 (2001), S. 4846-4856 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation. © 2001 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1401135
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  • 2
    Electronic Resource
    Electronic Resource
    Stochastic mechanics and the Feynman integral (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 6060-6078 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Feynman integral is given a stochastic interpretation in the framework of Nelson's stochastic mechanics employing a time-symmetric variant of Nelson's kinematics recently developed by the author. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1286880
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  • 3
    Electronic Resource
    Electronic Resource
    Derivation of the wave function collapse in the context of Nelson's stochastic mechanics (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 5565-5577 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The von Neumann collapse of the quantum mechanical wave function after a position measurement is derived by a purely probabilistic mechanism in the context of Nelson's stochastic mechanics. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533046
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  • 4
    Electronic Resource
    Electronic Resource
    Hamilton's principle in stochastic mechanics (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 6774-6800 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper we establish three variational principles that provide new foundations for Nelson's stochastic mechanics in the case of nonrelativistic particles without spin. The resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature. We first develop two stochastic variational principles whose Hamilton–Jacobi-like equations are precisely the two coupled partial differential equations that are obtained from the Schrödinger equation (Madelung equations). The two problems are zero-sum, noncooperative, stochastic differential games that are familiar in the control theory literature. They are solved here by means of a new, absolutely elementary method based on Lagrange functionals. For both games the saddle-point equilibrium solution is given by the Nelson's process and the optimal controls for the two competing players are precisely Nelson's current velocity v and osmotic velocity u, respectively. The first variational principle includes as special cases both the Guerra–Morato variational principle [Phys. Rev. D 27, 1774 (1983)] and Schrödinger original variational derivation of the time-independent equation.It also reduces to the classical least action principle when the intensity of the underlying noise tends to zero. It appears as a saddle-point action principle. In the second variational principle the action is simply the difference between the initial and final configurational entropy. It is therefore a saddle-point entropy production principle. From the variational principles it follows, in particular, that both v(x,t) and u(x,t) are gradients of appropriate principal functions. In the variational principles, the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second. Combining the two variational principles, we get the quantum Hamilton principle, i.e., a variational characterization of the logarithm of the wave function ψ. The Lagrangian is the Lagrangian of classical mechanics with the complex-valued velocity v−iu replacing the classical velocity. The dynamics is given by a stochastic differential equation for real-valued diffusions with complex-valued drift and driving noise processes. From the variational principle we derive a Newton-type law. We finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531187
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  • 5
    Electronic Resource
    Electronic Resource
    Lagrangian dynamics for classical, Brownian, and quantum mechanical particles (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 3375-3388 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In the framework of Nelson's stochastic mechanics [E. Nelson, Dynamical Theories of Brownian Motion (Princeton University, Princeton, 1967); F. Guerra, Phys. Rep. 77, 263 (1981); E. Nelson, Quantum Fluctuations (Princeton University, Princeton, 1985)] we seek to develop the particle counterpart of the hydrodynamic results of M. Pavon [J. Math. Phys. 36, 6774 (1995); Phys. Lett. A 209, 143 (1995)]. In particular, a first form of Hamilton's principle is established. We show that this variational principle leads to the correct equations of motion for the classical particle, the Brownian particle in thermodynamical equilibrium, and the quantum particle. In the latter case, the critical process q satisfies a stochastic Newton law. We then introduce the momentum process p, and show that the pair (q,p) satisfies canonical-like equations. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531570
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  • 6
    Electronic Resource
    Electronic Resource
    Critical Ornstein-Uhlenbeck processes (1986)
    Springer
    Applied mathematics & optimization 14 (1986), S. 265-276 
    Details
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Ornstein-Uhlenbeck position process with the invariant measure is shown to satisfy a variational principle quite analogous to Hamilton's least action principle of classical mechanics. To prove this, a stochastic calculus of variations is developed for processes with differentiable sample paths, and which form a diffusion together with their derivative. The key tool in the derivation of stochastic Euler-Lagrange-type equations is a symmetric variant of Nelson's integration by parts formula for semimartingales simultaneously adapted to an increasing and a decreasing family ofσ-algebras. An energy conservation theorem is also proved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01442240
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  • 7
    Electronic Resource
    Electronic Resource
    Stochastic control and nonequilibrium thermodynamical systems (1989)
    Springer
    Applied mathematics & optimization 19 (1989), S. 187-202 
    Details
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Minus the logarithm of the density of a diffusion process is shown to be the value function of a stochastic control problem, where the controlled equation evolves backward in time. For nonequilibrium thermodynamical systems, this provides a Hamilton-Jacobi-like theory, where the action is a local entropy function. This variational principle may also be seen as a rigorous version of the formal Onsager-Machlup principle. For the Ornstein-Uhlenbeck model of physical Brownian motion, the principle is related to a pathwise Newton law. For the latter model, several other pathwise results are derived, which strengthen the classical thermodynamical results on the averages. In particular, the (stochastic) Helmholtz free energy is shown to be a backward submartingale with respect to the natural filtration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01448198
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  • 8
    Electronic Resource
    Electronic Resource
    Lagrange approach to the optimal control of diffusions (1993)
    Springer
    Acta applicandae mathematicae 32 (1993), S. 101-122 
    Details
    ISSN: 1572-9036
    Keywords: 93E20 ; Stochastic control ; diffusion process ; Lagrange functional ; minimum principle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new approach to the optimal control of diffusion processes based on Lagrange functionals is presented. The method is conceptually and technically simpler than existing ones. A first class of functionals allows to obtain optimality conditions without any resort to stochastic calculus and functional analysis. A second class, which requires Ito's rule, allows to establish optimality in a larger class of problems. Calculations in these two methods are sometimes akin to those in minimum principles and in dynamic programming, but the thinking behind them is new. A few examples are worked out to illustrate the power and simplicity of this approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00998149
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  • 9
    Electronic Resource
    Electronic Resource
    Equilibrium description of a particle system in a heat bath (1989)
    Springer
    Acta applicandae mathematicae 14 (1989), S. 239-258 
    Details
    ISSN: 1572-9036
    Keywords: 82A30 ; 82A05 ; 93B05 ; 93E15 ; 60J65 ; Statistical physics ; invariant measure ; controllability ; stochastic stability ; Langevin equation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract AnN particle system evolving in 3-space under the action of a friction force, an external potential and a fluctuating force is studied, using the Langevin equation. Necessary and sufficient conditions are given for both (a) the existence and (b) the Maxwell-Boltzmann character of an invariant measure. Both probabilistic and system theoretical tools play an important part in this study. In addition, the general results mentioned above are applied to the study of a system ofn oscillators coupled through velocity, with a first-neighbour-only type of interaction.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01307215
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  • 10
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    AsymmetricalgebraicRiccatiequation:Ahomeomorphicparametrizationofthesetof solutions (2001)
    Elsevier
    Details
    Publication Date: 2001-05-01
    Print ISSN: 0024-3795
    Electronic ISSN: 1873-1856
    Topics: Mathematics
    Published by Elsevier
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