关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Diffusions, Markov processes, and martingales

1. Stochastic processes and random fields 2. Continuous semimartingales and stochastic integrals 3. Semimartingales with spatial parameter and stochastic integrals 4. Stochastic flows 5. Convergence of stochastic flows 6. Stochastic partial differential equations.

Stochastic flows and stochastic differential equations

What are quantum fluctuations

(1994). The Quantum Theory of Motion. Journal of Modern Optics: Vol. 41, No. 1, pp. 168-169.

The Quantum Theory of Motion

Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrodinger equations by the Hamiltonian Feynman path integrals (=Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrodinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.

Hamiltonian Feynman path integrals via the Chernoff formula

A Cameron-Martin formula is established for the Feynman map definition of the Feynman integral in non-relativistic quantum mechanics. This formula involves the Morse or Maslov indices and shows the connection with the Feynman-Hibbs definition of the path integral and the Fresnel integrals of Albeverio and Hoegh-Krohn. Applications are given to the Feynman-Kac-Ito formula for anharmonic oscillator potentials On etablit une formule de Cameron-Martin pour l'integrale de Feynman en mecanique quantique non relativiste. Cette formule fait intervenir les indices de Morse et de Maslov et montre la relation avec la definition de Feynman-Hibbs de l'integrale de chemins et les integrales de Fresnel considerees par Albeverio et Hoegh-Krohn. On donne des applications a la formule de Feynman-Kac-Ito pour des potentiels d'oscillateurs anharmoniques

Feynman maps, Cameron-Martin formulae and anharmonic oscillators

We consider the limiting case λ→0 of the Cauchy problem, ∂gλ(x,t)/∂t = (1/2) λΔxgλ(x,t)+(V(x)/λ) gλ( x,t), with gλ (x,0) = exp{−S0(x)/λ}T0(x), V, S0 being real‐valued functions on N, T0 a complex‐valued function on N; V, S0, T0 being independent of λ, Δx being the Laplace–Beltrami operator on N, some complete Riemannian manifold. We prove some new results relating the limiting behavior of the solution to the above Cauchy problem to the solution of the corresponding classical mechanical problem  D2Z(s)/∂s2 = −∇ZV[Z(s)], s∈[0,t], with Z(t) = x and Z(0) = ∇S0(Z(0)).One of our results is equivalent to the fact that for short times Schrodinger quantum mechanics on the Riemannian manifold N tends to classical Newtonian mechanics on N as h/ tends to zero.

Classical mechanics, the diffusion (heat) equation and the Schrödinger equation on a Riemannian manifold

Existence and uniqueness of strong solutions for a class of stochastic functional differential equations in Hilbert spaces are established. Sufficient conditions which guarantee the transference of mean–square and pathwise exponential stability from stochastic partial differential equations to stochastic functional partial differential equations are studied. The stability results derived are also applied to stochastic ordinary differential equations with hereditary characteristics. In particular, as a direct consequence our main results improve some of those by Mao & Shah in which it was proved that under certain conditions pathwise exponential stability is transferred from non–delay equations to delay equations if the constant time–lag appearing in the problem is sufficiently small, while in our treatment the transference actually holds for arbitrary bounded delay variables not only in finite but in infinite dimensions.

Aubrey Truman

Papers

Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

Hamiltonian Feynman path integrals via the Chernoff formula

Feynman maps, Cameron-Martin formulae and anharmonic oscillators

Classical mechanics, the diffusion (heat) equation and the Schrödinger equation on a Riemannian manifold

Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property