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A quick probability-oriented introduction to operator splitting methods

M. B. Vovchanskyi

TL;DR

This article delivers a comprehensive, accessible overview of operator splitting methods within probability and stochastic analysis, emphasizing multiplicative and BCH-based perspectives while also introducing additive splittings. Grounded in the theory of $C_0$-semigroups, Trotter–Kato and Chernoff product formulas, it surveys Lie–Trotter and Strang schemes, their extensions to SPDEs/SDEs, and how boundary conditions and discretization interact with convergence. It also discusses rate-of-convergence results, order reduction phenomena, and higher-order exponential splitting via the Baker–Campbell–Hausdorff framework, connecting algebraic and variational viewpoints to practical numerical schemes. The paper aggregates probabilistic, variational, and numerical insights to map current knowledge, highlight limitations, and point to directions for future research and applications in stochastic analysis and beyond.

Abstract

This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic processes'', devoted to publishing lecture notes from the aforementioned workshop. The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are given. The bibliography contains illustrative deterministic examples and a selection of probability-related works.

A quick probability-oriented introduction to operator splitting methods

TL;DR

This article delivers a comprehensive, accessible overview of operator splitting methods within probability and stochastic analysis, emphasizing multiplicative and BCH-based perspectives while also introducing additive splittings. Grounded in the theory of -semigroups, Trotter–Kato and Chernoff product formulas, it surveys Lie–Trotter and Strang schemes, their extensions to SPDEs/SDEs, and how boundary conditions and discretization interact with convergence. It also discusses rate-of-convergence results, order reduction phenomena, and higher-order exponential splitting via the Baker–Campbell–Hausdorff framework, connecting algebraic and variational viewpoints to practical numerical schemes. The paper aggregates probabilistic, variational, and numerical insights to map current knowledge, highlight limitations, and point to directions for future research and applications in stochastic analysis and beyond.

Abstract

This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic processes'', devoted to publishing lecture notes from the aforementioned workshop. The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are given. The bibliography contains illustrative deterministic examples and a selection of probability-related works.
Paper Structure (13 sections, 7 theorems, 214 equations)

This paper contains 13 sections, 7 theorems, 214 equations.

Table of Contents

  1. The scope of the paper and first examples
  2. Matrices and the Lie product formula
  3. Semigroups and the Chernoff product formula for abstract Cauchy problems
  4. The Trotter-Kato formula
  5. A general formulation of a splitting scheme. Multiplicative and additive splitting methods
  6. A naive probabilistic example. An error of a splitting scheme and the sources of it
  7. Some remarks on the relation between ACPs and practice
  8. The rate of convergence. Some remarks about error representations in the deterministic case
  9. Boundary conditions, inhomogeneous ACPs and the phenomenon of order reduction
  10. General exponential splitting methods. The Baker-Campbell-Hausdorff formula
  11. The Itô-Taylor expansion, the Kunita expansion and exponential methods for SDEs
  12. General results for SPDEs and SDEs
  13. A few examples: diffeomorphic flows of SDEs, Brownian web and non-homeomorphic Harris flows

Key Result

Theorem 1

For $A_1,A_2\in M_n$ where $\| \cdot\|$ is an arbitrary matrix norm and $C=C(A,B).$

Theorems & Definitions (120)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 1: e.g. ReedSai72methodsHall13quantum
  • Remark 1
  • Remark 2
  • Remark 3
  • Example 5
  • Remark 4
  • ...and 110 more