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Van Vleck-Pauli formula for Wiener integrals and Jacobi fields. (English) Zbl 0865.60067

Ikeda, N. (ed.) et al., Itô’s stochastic calculus and probability theory. Tribute dedicated to Kiyosi Itô on the occasion of his 80th birthday. Tokyo: Springer. 141-156 (1996).
It is well-known that the Fourier-Laplace transform of a quadratic functional of a Gaussian process can be expressed in terms of the regularized determinant of the quadratic form. This formula can be extended to the case of a Fourier-Laplace transform conditioned by a finite-dimensional subspace of the Gaussian space. The authors explore these formulas when the Gaussian process is a two-dimensional Wiener process and the quadratic functional is a linear combination of Lévy’s stochastic area and the integrals of the square of the components. In this case, the Hilbert-Schmidt operator associated with the quadratic functional can be expressed as the sum of a Volterra operator and an operator with finite range. As shown in the first section, in that case the computation of the regularized determinant can be reduced to that of a finite-dimensional determinant. In the case under study, the determinant turns out to be the two-dimensional determinant of an operator which can be defined using a Jacobi field along the critical path for the Lagrangian associated to the quadratic functional. The final result is an analogue of the Van Vleck-Pauli formula in quantum mechanics.
For the entire collection see [Zbl 0852.00016].

MSC:

60J65 Brownian motion
58J65 Diffusion processes and stochastic analysis on manifolds
60H05 Stochastic integrals
70H03 Lagrange’s equations
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