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Error estimates on ergodic properties of discretized Feynman–Kac semigroups

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Abstract

We consider the numerical analysis of the time discretization of Feynman–Kac semigroups associated with diffusion processes. These semigroups naturally appear in several fields, such as large deviation theory, Diffusion Monte Carlo or non-linear filtering. We present error estimates à la Talay–Tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized; as well as on the leading eigenvalue of the generator of the dynamics, which corresponds to the rate of creation of probability. This provides criteria to construct efficient integration schemes of Feynman–Kac dynamics, as well as a mathematical justification of numerical results already observed in the Diffusion Monte Carlo community. Our analysis is illustrated by numerical simulations.

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Acknowledgements

The authors are grateful to Mathias Rousset for his help in understanding Feynman–Kac models. We also thank Frédéric Cérou, Jonathan Mattingly, Julien Roussel, Hugo Touchette and Jonathan Weare for fruitful discussions, and the anonymous referees for their useful comments. The Ph.D. fellowship of Grégoire Ferré is partly funded by the Bézout Labex, funded by ANR, reference ANR-10-LABX-58. The work of Gabriel Stoltz was funded in part by the Agence Nationale de la Recherche, under Grant ANR-14-CE23-0012 (COSMOS) and by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 614492. We also benefited from the scientific environment of the Laboratoire International Associé between the Centre National de la Recherche Scientifique and the University of Illinois at Urbana-Champaign.

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    Grégoire Ferré & Gabriel Stoltz

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Appendix A: Markov contractions and Dobrushin coefficients

Appendix A: Markov contractions and Dobrushin coefficients

Denoting by \(\mathcal {M}(\mathcal {D})\) is the set of measures over \(\mathcal {D}\), we define \(\mathcal {M}_0(\mathcal {D})=\{\eta \in \mathcal {M}(\mathcal {D})\, |\, \eta (\mathcal {D})=0\}\) the set of (unsigned) measures with zero mass. The contraction norm of a Markov operator \(Q:\mathcal {P}(\mathcal {D})\rightarrow \mathcal {P}(\mathcal {D})\) is

the second equality coming from the fact that all elements in \(\mathcal {M}_0(\mathcal {D})\) are proportional to the difference of two probability measures. In particular,

A fundamental tool [9,10,11] for the study of Feynman–Kac type semigroups (15) and introduced by Dobrushin [15, 16] is the so-called Dobrushin ergodic coefficient, which can be defined for a Markov operator Q as:

$$\begin{aligned} \alpha (Q) = \inf _{\begin{array}{c} q,q'\in \mathcal {D}\\ \{A_i\}_{1 \leqslant i \leqslant m}\subset \mathcal {D} \end{array}} \left\{ \sum _{i=1}^m \min \left( Q(q,A_i), Q(q',A_i)\right) \right\} , \end{aligned}$$
(88)

where the infimum in the last equality runs over points \(q,q'\in \mathcal {D}\) and all partitions \((A_i)_{i=1}^m\) of \(\mathcal {D}\). If we interpret \(Q(q,A_i)\) as the probability of going from q into the set \(A_i\), we see that this coefficient provides information on the mixing properties of the operator Q. The link between this coefficient and the contraction properties of Q is made precise by the following relationship [15, 16]:

(89)

As a result, a minorization condition on Q translates into a contraction of the operator through its ergodic coefficient \(\alpha (Q)\). Relation (89) is essentially obtained by a Hahn decomposition of measures of zero mass, as made precise in [15, 16].

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Ferré, G., Stoltz, G. Error estimates on ergodic properties of discretized Feynman–Kac semigroups. Numer. Math. 143, 261–313 (2019). https://doi.org/10.1007/s00211-019-01059-1

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