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Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups. (English) Zbl 1063.60098

Consider the problem of computation of expectations \(E \langle X_t, A X_t\rangle\), where \(A\) is a linear operator on a separable complex Hilbert space \((H, \langle \cdot, \cdot \rangle)\) and \((X_t)_{t \in [0,T]}\) is an \(H\)-valued diffusion process satisfying the Itô-type stochastic evolution equation \[ dX_t = G X_t dt + \sum_{k=1}^m L_k X_t dW^k_t, \qquad X_0 = x_0 \in H, \] driven by independent standard Brownian motions \(W^k\), \(L_1, \dots, L_m\) operators on \(H\) and \(G = - i B - \frac{1}{2} \sum^m_{j=1} L^*_j L_j\) with self-adjoint operator \(B\) on \(H\). The author introduces approximations \(X_{t,n}\) of \(X_t\) via orthogonal projection solving an approximating (finite-dimensional) equation of the above one with truncated operators \(L_{1,n}, \dots, L_{m,n}\) and \( G_{n}\). These approximating equations preserve the mean square conservative relationship \(E | | X_t| | ^2 = | | x_0| | ^2\) in a wide class of applications. As the main result, he verifies (naturally very complex) worst-case estimates on the rate of convergence of \(E \langle \tilde{X}_{t,n}, A \tilde{X}_{t,n} \rangle\) to \(E \langle X_t, A X_t \rangle\) where \(\tilde{X}_{t,n}\) is obtained by the explicit Euler approximation. He also considers an extrapolation method based on the explicit Euler scheme. Eventually, a quantum master equation for a boson system (quantum harmonic oscillator) is studied as a first application.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
37H10 Generation, random and stochastic difference and differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
65C05 Monte Carlo methods
65C20 Probabilistic models, generic numerical methods in probability and statistics
65C30 Numerical solutions to stochastic differential and integral equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
81S25 Quantum stochastic calculus
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
93E15 Stochastic stability in control theory
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