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Validity of magnus formula5/31/2023 ![]() Where is the exponential mapping on the Lie group. Elements of can be seen as vector fields on. At this point, we can observe that the Chen's expansion formula is a purely algebraic statement, thus expanding the exponentialĪnother framework, close to this linear case, in which the Chen's expansion makes sense are Lie groups. Is absolutely convergent on the interval. We deduce that if is such that, then the series ![]() First, we observe that a combinatorial argument shows thatįor the matrix norm we have the estimate so we conclude that for some constant , Details can be found in this paper by Strichartz. Proof: We only give the sketch of the proof. Proposition: There exists such that for , The formal analogy between this expansion and the signature leads to the following result: The solution admits a representation as an absolutely convergent Volterra series Let us consider matrices and let be the solution of the differential equation The first case of study are linear equations. This expansion is of course formal but analytically makes sense in a number of situations that we now describe. In the previous lecture, we proved the Chen’s expansion formula which establishes the fact that the signature of a path is the exponential of a Lie series.
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