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The WKB Method, Path-Integrals, and Large Deviations

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Stochastic Processes in Cell Biology

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 41))

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Abstract

In Sects. 3.4 and 3.5 we highlighted a limitation of the diffusion approximation of jump Markov processes for large system size N, namely, that it can lead to exponentially large errors in solutions to FPT problems.

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Notes

  1. 1.

    The Perron–Frobenius theorem asserts that a real square matrix with positive entries has a unique largest real eigenvalue (the Perron eigenvalue) and that the corresponding eigenvector has strictly positive components. The theorem can also be extended to matrices with non-negative entries, provided that the matrix is irreducible. However, there can now be complex eigenvalues with the same absolute value as the Perron eigenvalue. In the case of a transition matrix, the Perron eigenvalue is zero. Strictly speaking, the Perron–Frobenius theorem applies to finite-dimensional matrices, so we will assume that it still holds in cases where the number of discrete states is infinite.

  2. 2.

    Technically speaking, one first has to normal-order the operator \(\mathcal{H}\) by moving all operators A † to the right of all operators A using repeated application of the commutation rule. However, given the dependence of the transition rates on the system size N, this normal ordering introduces O(1∕N) corrections, which can be ignored to leading order.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, USA

    Paul C. Bressloff

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Bressloff, P.C. (2014). The WKB Method, Path-Integrals, and Large Deviations. In: Stochastic Processes in Cell Biology. Interdisciplinary Applied Mathematics, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-08488-6_10

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