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Long-time behavior for discretization schemes of Fokker-Planck equations via couplings

Ansgar Jüngel, Katharina Schuh

TL;DR

This work analyzes the long-time behavior of finite-volume discretizations for Fokker–Planck equations by integrating Conforti’s coupling method with a discrete Bakry–Émery framework. It proves exponential decay in relative $φ$-entropy and contractivity in Wasserstein distances for both continuous- and discrete-time Markov chains, with rates governed by a diagonal-dominance gap of the Hessian of the potential and independent of the domain size. The results accommodate non-additive potentials and rely on coupling-based proofs rather than discrete Bochner identities, and extend to explicit Euler discretizations with quantified discretization errors. Overall, the paper provides robust, quantitative long-time guarantees for discretized Fokker–Planck dynamics on finite-volume grids, including discrete-time analogs and explicit convergence to the underlying SDE.

Abstract

Continuous-time Markov chains associated to finite-volume discretization schemes of Fokker-Planck equations are constructed. Sufficient conditions under which quantitative exponential decay in the $φ$-entropy and Wasserstein distance are established, implying modified logarithmic Sobolev, Poincaré, and discrete Beckner inequalities. The results are not restricted to additive potentials and do not make use of discrete Bochner-type identities. The proof for the $φ$-decay relies on a coupling technique due to Conforti, while the proof for the Wasserstein distance uses the path coupling method. Furthermore, exponential equilibration for discrete-time Markov chains is proved, based on an abstract discrete Bakry-Emery method and a path coupling.

Long-time behavior for discretization schemes of Fokker-Planck equations via couplings

TL;DR

This work analyzes the long-time behavior of finite-volume discretizations for Fokker–Planck equations by integrating Conforti’s coupling method with a discrete Bakry–Émery framework. It proves exponential decay in relative -entropy and contractivity in Wasserstein distances for both continuous- and discrete-time Markov chains, with rates governed by a diagonal-dominance gap of the Hessian of the potential and independent of the domain size. The results accommodate non-additive potentials and rely on coupling-based proofs rather than discrete Bochner identities, and extend to explicit Euler discretizations with quantified discretization errors. Overall, the paper provides robust, quantitative long-time guarantees for discretized Fokker–Planck dynamics on finite-volume grids, including discrete-time analogs and explicit convergence to the underlying SDE.

Abstract

Continuous-time Markov chains associated to finite-volume discretization schemes of Fokker-Planck equations are constructed. Sufficient conditions under which quantitative exponential decay in the -entropy and Wasserstein distance are established, implying modified logarithmic Sobolev, Poincaré, and discrete Beckner inequalities. The results are not restricted to additive potentials and do not make use of discrete Bochner-type identities. The proof for the -decay relies on a coupling technique due to Conforti, while the proof for the Wasserstein distance uses the path coupling method. Furthermore, exponential equilibration for discrete-time Markov chains is proved, based on an abstract discrete Bakry-Emery method and a path coupling.
Paper Structure (19 sections, 12 theorems, 132 equations, 1 figure)

This paper contains 19 sections, 12 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The setting
  3. Main results and key ideas
  4. Comparison to the literature
  5. Definition of the continuous-time Markov chain
  6. Main results for continuous-time Markov chains
  7. Convergence of the Markov chain approximation to the SDE
  8. Exponential decay in $\phi$-entropy
  9. Exponential decay in the Wasserstein distance
  10. Main results for discrete-time Markov chains
  11. Exponential decay in $\phi$-entropy
  12. Exponential decay in the Wasserstein distance
  13. Coupling method
  14. Proofs for continuous-time Markov chains
  15. Exponential decay in $\phi$-entropy
  16. ...and 4 more sections

Key Result

Lemma 1

The unique invariant measure $m_h$ of the continuous-time Markov chain $(Y_t^h)_{t\ge 0}$ with the transition rates 1.c is given by with the normalization constant $Z=\sum_{i\in{\mathcal{K}}_h}\exp(-V^h(i)/\sigma^2)$.

Figures (1)

  • Figure 1: Coupling for the jump rates of the states $i$ and $i+he_j$. For the five possible jumps for state $i$ illustrated by red dashed lines, the corresponding coupled jumps for state $i+he_j$ are given by blue solid lines.

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Remark 2: Choice of $c(i,\gamma)$
  • Theorem 3: Convergence to SDE
  • Theorem 4: Exponential decay
  • Remark 5: Additive potentials
  • Remark 6: Strictly diagonally dominance of $\mathrm{D}^2 V$
  • Theorem 7: Convergence in Wasserstein distance
  • Theorem 8: Convergence in $L^1$ Wasserstein distance
  • Remark 9: Discussion of Theorem \ref{['thm.W1']}
  • ...and 12 more