Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

• Published in: Probability theory and related fields Vol. 167; no. 3-4; pp. 817 - 915
• Main Authors: Güneysu, B., Matte, O., Møller, J. S.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2017; Springer Nature B.V
• Subjects: Byproducts; Calculus; Differential equations; Economics; Finance; Hilbert space; Hypotheses; Insurance; Integrals; Invariants; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Probability Theory and Stochastic Processes; Quantitative Finance; Radiation; Scalars; Statistics for Business; Stochastic models; Stochasticity; Studies; Theoretical; Translations; 81V10; 60H10; 60H30; Quantized radiation; Feynman–Kac; Stochastic differential equation in Hilbert space; 47D08
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-016-0694-4

We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman–Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman–Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson’s model.