A Survey of the Matrix Exponential Formulae with Some Applications

Zheng, Baoying

doi:10.4208/jms.v49n4.16.04

Cited by 1 publication

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“…For unbounded operators, it is also related to Chernoff and Trotter's results on the approximation of semi-groups [18,29]. The product formula also justifies the construction of some splitting schemes [8].…”

Section: Motivationsmentioning

confidence: 80%

The non-linear sewing lemma I: weak formulation

Brault¹,

Lejay²

2019

Electron. J. Probab.

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We give an unified framework to solve rough differential equations. Based on flows, our approach unifies the former ones developed by Davie, Friz-Victoir and Bailleul. The main idea is to build a flow from the iterated product of an almost flow which can be viewed as a good approximation of the solution at small time. In this second article, we give some tractable conditions under which the limit flow is Lipschitz continuous and its links with uniqueness of solutions of rough differential equations. We also give perturbation formulas on almost flows which link the former constructions.This theory was very fruitful to study stochastic equations driven by Gaussian process which is not covered by the Itô framework, like the fractional Brownian motion [13,26]. More generally, the rough path framework allows one to separate the probabilistic from the deterministic part in such equation and to overcome some probabilistic conditions such as using adapted or non-anticipative processes.Recently, the ideas of the rough path theory were extended to stochastic partial differential equations (SPDE) with the works of [21,22] which have led to significant progress in the study of some SPDE. This theory also found applications in machine learning and the recognizing of the Chinese ideograms [10,24].Since the seminal article [25] by T. Lyons in 1998, several approaches emerged to solve (1). They are based on two main technical arguments: fixed point theorems [20,25] and flow approximations [2,12,14,16,19]. In particular, the rough flow theory allows one to extend work about stochastic flows, which has been developed in '80s by Le Jan-Watanabe-Kunita and others, to a non-semimartinagle setting [4].The main goal of this article is to give a framework which unifies the approaches by flow and pursue further investigations on their properties and their relations with families of solutions to (1).A flow is a family of maps t , u from a Banach space to itself such that ,˝, " , for any ď ď . Typically, the map which associates the initial condition to the solution of (1) has a flow property. The existence of a such flow heavily depends on the existence and uniqueness of the solution. However, it was proved in [7,8] and extended to the rough path case in [6] that when non-uniqueness holds, it is possible to build a measurable flow by a selection technique. In this article we are interested by the construction of a Lipschitz flows.The main idea to build the flow associated is to find a good approximation , of , when |´| is small enough. We iterate this approximation on a subdivision " t ď ď¨¨¨ď ď u of r , s by setting , :" ,˝¨¨¨˝, . Ifdoes converges when the mesh of goes to zero, , the limit is necessarily a flow This computation is similar to the ones of numerical schemes as Euler's methods of different order [11]. Moreover, this idea is found among the Trotter's formulas for bounded or unbounded linear operators which allows to compute the semi-group of the sum of two non-commutative operators only knowing the semi-groups associated to e...

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Section: Motivationsmentioning

confidence: 80%