Results 171 to 180 of about 8,689 (223)
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1981
We have presented the equations of quantum mechanics in Chapter 1. In only a handful of cases, however, do these equations have solutions equal to well-known special functions, or can the spectra be written in closed form. Thus calculations in quantum mechanics are made by some approximate method, such as computing the first few terms in a formal power
James Glimm, Arthur Jaffe
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We have presented the equations of quantum mechanics in Chapter 1. In only a handful of cases, however, do these equations have solutions equal to well-known special functions, or can the spectra be written in closed form. Thus calculations in quantum mechanics are made by some approximate method, such as computing the first few terms in a formal power
James Glimm, Arthur Jaffe
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Quantum white noise Feynman–Kac formula
Random Operators and Stochastic Equations, 2018Abstract In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and ...
Ettaieb, Aymen +2 more
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Filtering the Feynman--KAC Formula
SIAM Journal on Numerical Analysis, 2002Numerical approximations of the pointwise solution of the linear partial differential equation (PDE) (called generalized heat equation here) \[ u_t = \frac{1}{2} \Delta u + c u,\;u(0,x)=f(x) \] are discussed. The main emphasis is on Monte-Carlo type approximations based on the well-known Feynman-Kac formula which leads to the problem of approximating ...
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Reports on Mathematical Physics, 1993
The authors construct a process \(Z^ a\), somehow in the sense of nonstandard analysis (called 4-asymptotic-stable motion): \[ (\Omega,A,P) \times (\Omega, A, P) \to (\mathbb{C}^ \infty, {\mathcal B}^ \infty) \] (where \((\mathbb{C}, {\mathcal B})\) is the Borel measurable space of the complex plane), \[ \mathbb{Z}^ a = (Z^ a_ t : t \geq 0, Z^ a_ t ...
Mądrecki, A., Rybaczuk, M.
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The authors construct a process \(Z^ a\), somehow in the sense of nonstandard analysis (called 4-asymptotic-stable motion): \[ (\Omega,A,P) \times (\Omega, A, P) \to (\mathbb{C}^ \infty, {\mathcal B}^ \infty) \] (where \((\mathbb{C}, {\mathcal B})\) is the Borel measurable space of the complex plane), \[ \mathbb{Z}^ a = (Z^ a_ t : t \geq 0, Z^ a_ t ...
Mądrecki, A., Rybaczuk, M.
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GENERALIZED FEYNMAN–KAC FORMULA WITH STOCHASTIC POTENTIAL
Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2002In this paper we study the solution of the stochastic heat equation where the potential V and the initial condition f are generalized stochastic processes. We construct explicitly the solution and we prove that it belongs to the generalized function space [Formula: see text].
Ouerdiane, Habib, Silva, José Luis
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Stochastic Feynman-Kac formula
Journal d'Analyse Mathématique, 1982The stochastic partial differential equation called stochastic Feynman- Kac formula is investigated. For any distribution \(g\in {\mathcal D}'(R^ d)\) the author proves that the equation has a unique solution g exp- \(\int^{t}_{0}V(\cdot +w_ s)ds\) defining a semimartingale with the strong Markov property with continuous trajectories in \({\mathcal D}'(
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2015
In this chapter, we establish the connection between the deterministic EIT forward problem and the class of reflecting diffusion processes. We proceed along the lines of the recent paper [137] by Piiroinen and the author: We derive Feynman-Kac formulae in terms of these processes for the solutions to the forward problems corresponding to the continuum ...
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In this chapter, we establish the connection between the deterministic EIT forward problem and the class of reflecting diffusion processes. We proceed along the lines of the recent paper [137] by Piiroinen and the author: We derive Feynman-Kac formulae in terms of these processes for the solutions to the forward problems corresponding to the continuum ...
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A discrete Feynman-Kac formula
Journal of Statistical Planning and Inference, 1993The author is concerned with an analogue of the Feynman-Kac formula for a simple symmetric random walk. For a clear understanding of the study several examples are discussed in detail. Particularly the eigenvalue problem and the random number of steps are also illustrated by examples.
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1996
Throughout this chapter, we suppose that E is a Banach space with norm ‖ • ‖, and (Σ, e) is a measurable space. Further, we suppose that S is a C0-semigroup of continuous linear operators acting on E and that Q: e → ℒ(E) is a spectral measure, so that $$ X = \left( {\Omega ,{{\left\langle {{S_t}} \right\rangle }_{t \geqslant 0}},{{\left\langle ...
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Throughout this chapter, we suppose that E is a Banach space with norm ‖ • ‖, and (Σ, e) is a measurable space. Further, we suppose that S is a C0-semigroup of continuous linear operators acting on E and that Q: e → ℒ(E) is a spectral measure, so that $$ X = \left( {\Omega ,{{\left\langle {{S_t}} \right\rangle }_{t \geqslant 0}},{{\left\langle ...
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