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arXiv:2605.29508

Quantum Markovian Dynamics from a Double Covariance Stochastic Framework

Andrei Khrennikov, Irina Basieva

quant-phmath.DSmath.PRSubmitted May 28, 2026• Announced May 29, 2026• 0 citations• Revised Aug 8, 2026 (v2)
v1 · May 28, 2026v2 · Aug 8, 2026
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Abstract

We develop an interacting extension of the Double Covariance Model (DCM), a stochastic subquantum framework in which macroscopic quantum dynamics emerge through coarse-graining of correlated microscopic fluctuations. Starting from local stochastic differential equations on subsystem Hilbert spaces, we derive a closed evolution equation for a coarse-grained double covariance operator using multi-scale Itô calculus and sliding-window averaging. The construction explicitly incorporates two separated temporal scales: a fast microscopic fluctuation scale governing subquantum stochastic processes and a slower macroscopic observation scale associated with coarse-grained dynamics. Within the hydrodynamic limit, where the ratio between microscopic correlation time and averaging-window scale vanishes, rapidly fluctuating corrections disappear and the effective dynamics converges to a deterministic macroscopic transport equation. We show that the emergent macroscopic dynamics has the exact Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form: coherent Hamiltonian evolution arises from deterministic subquantum flow, while dissipative channels emerge from quadratic noise correlations. The framework further demonstrates how non-separable interaction Hamiltonians can arise from strictly local, state-dependent stochastic feedback fields. In the fluctuation-free limit, the model reduces naturally to the standard von Neumann equation, providing a unified stochastic foundation for both open and closed quantum dynamics.

Comments
QIP26, Vaxjo, Sweden; this version: heuristic proofs of v1 are developed to the rigorous mathematical proofs based on Ito calculus for the Hilbert space valued stochastic processes

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