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The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. (English) Zbl 0623.34058
The Campbell-Baker-Hausdorff-Dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems. Specifically, suppose u(t) takes values in a $$C^{\infty}$$ manifold and satisfies the initial value problem $$u'(t)=A(t)(u(t))$$, $$u(0)=a$$, where A(t) is a vector field on the manifold depending continuously on t. Then $$u(t)=\exp z(t)(a)$$ (here exp z(a) means the solution at $$s=1$$ to $$v'(s)=z(v(s))$$, $$v(0)=a)$$ for $z(t)\sim \sum^{\infty}_{r=1}\sum_{\sigma \in P_ r}((- 1)^{e(\sigma)}/r^ 2\left[ \begin{matrix} r-1\\ e(\sigma)\end{matrix} \right])$ $\int_{T_ r(t)}[\cdot \cdot \cdot [A(s_{\sigma (1)})A(s_{\sigma (2)})]...]A(s_{\sigma (r)})]ds$ as $$t\to 0$$, where $$T_ r(t)=\{s\in {\mathbb{R}}^ r: 0<s_ 1<s_ 2<...<s_ r<t\}$$, $$P_ r$$ is the set of permutations on $$\{$$ 1,...,r$$\}$$, e($$\sigma)$$ is the number of errors in ordering consecutive terms in $$\{\sigma$$ (1),...,$$\sigma$$ (r)$$\}$$, and [ ] is the usual commutator of vector fields. Under appropriate analyticity assumptions the series for z(t) is convergent for small t. This formula gives an explicit formulation of results of K.-T. Chen published in 1957. Applications are given to problems in sub- Riemannian geometry, and to improving convergence estimates for the Campbell-Baker-Hausdorff-Dynkin formula in the context of Banach algebras. Our formula can be thought of as a noncommutative generalization of the familiar formula $$u(t)=a \exp (\int^{t}_{0}A(s)ds)$$ in the scalar linear case in the same way that the Campbell-Baker-Hausdorff-Dynkin formula is a noncommutative generalization of the familiar formula $$e^ xe^ y=e^{x+y}$$.

##### MSC:
 34E05 Asymptotic expansions of solutions to ordinary differential equations 22E60 Lie algebras of Lie groups 53B99 Local differential geometry
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##### References:
  Bourbaki, N, Elements of mathematics, Lie groups and Lie algebras, (1975), Hermann Paris, English translation · Zbl 0329.17002  Brockett, R.W, Volterra series and geometric control theory, Automatica, 12, 167-176, (1976) · Zbl 0342.93027  Brockett, R.W, Control theory and singular Riemannian geometry, (), 11-27 · Zbl 0483.49035  Chen, K.-T, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. math., 65, 163-178, (1957) · Zbl 0077.25301  Chen, K.-T, Formal differential equations, Ann. math., 73, 110-133, (1961) · Zbl 0098.05702  Chen, K.-T, An expansion formula for differential equations, Bull. amer. math. soc., 68, 341-344, (1962) · Zbl 0166.08003  Chen, K.-T, Expansions of solutions of differential systems, Arch. rational mech. anal., 13, 348-363, (1963) · Zbl 0117.04802  Chen, K.-T, On a generalization of Picard’s approximation, J. differential equations, 2, 438-448, (1966) · Zbl 0148.05803  Dynkin, E, Evaluation of the coefficients of the Campbell-Hausdorff formula, Dokl. akad. nauk, 57, 323-326, (1947), [Russian] · Zbl 0029.24507  Fliess, M, Fonctionelle causales non lineares et indeterminees non commutatives, Bull. soc. math. France, 109, 3-40, (1981) · Zbl 0476.93021  Grobner, W, Die Lie-reihen und ihre anwendungen, (1967), VEB Deutscher Verlag der Wissenschaften Berlin · Zbl 0157.40301  Jacobson, N, Lie algebras, (1962), “Interscience,” Wiley New York · JFM 61.1044.02  Koranyi, A, Geometric aspects of analysis on the Heisenberg group, (), 209-258  Mitchell, J, On Carnot-caratheodory metrics, J. differential geometry, 21, 35-45, (1985) · Zbl 0554.53023  Nachman, A, The wave equation on the Heisenberg group, Comm. partial differential equations, 7, 675-714, (1982) · Zbl 0524.35065  Nagel, A; Stein, E.M; Wainger, S, Balls and metrics defined by vector fields I, Acta. math., 155, 103-147, (1985) · Zbl 0578.32044  Nelson, E, Analytic vectors, Ann. math., 70, 572-615, (1954) · Zbl 0091.10704  Strichartz, R.S, Sub-Riemannian geometry, J. differential. geometry, 24, 221-263, (1986) · Zbl 0609.53021  \scT. J. S. Taylor, Some aspects of differential geometry associated with hypoelliptic second order operators, preprint. · Zbl 0698.35041
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