Bismut’s Way of the Malliavin Calculus for Non-Markovian Semi-groups: An Introduction

2020

Symmetry

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We give a new approximation with respect of the traditional parametrix method of the solution of a parabolic equation whose generator is of big order and under the Hoermander form. This generalizes to a higher order generator the traditional approximation of Stratonovitch diffusion which put in relation random ordinary differential equation (the leading process is random and of finite energy. When a trajectory of it is chosen, the solution of the equation is defined) and stochastic differential equation (the leading process is random and only continuous and we cannot choose a path in the solution which is only almost surely defined). We consider simple operators where the computations can be fully performed. This approximation fits with the semi-group only and not for the full path measure in the case of a stochastic differential equation.

Section: Discussionmentioning

confidence: 93%

“…We refer to the reviews [3,6] for the study of stochastic analysis without probability for non-markovian semi-groups.…”

Section: Introductionmentioning

confidence: 99%

Toward a Wong–Zakai Approximation for Big Order Generators

2020

Symmetry

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“…It is the theory of Stratonovitch differential equations in Itô Calculus [2]. But equation (7) has a solution in this case only almost surely defined. Blagoveschchenkii-Freidlin [3] have done pioneering works in the derivation in the starting point of the solution almost surely defined, which were later considerably extended.…”

Section: Dedicated To Professor Volovich For His Birthdaymentioning

confidence: 99%

“…See the expository paper Meyer [4] and the book of Kunita [5] for that. We refer to the review papers [6] and [7] for stochastic analysis for non-markovian semi-groups.…”

Section: Dedicated To Professor Volovich For His Birthdaymentioning

confidence: 99%

The flow theorem for big order generators

2022

Int. J. Mod. Phys. A

“…This theorem is classical in analysis , but it enters in our general program to implement stochastic analysis tools in the theory of Non-Markovian semi-group. See the review [7] and [13] for that. See [10], [11] for another presentation.…”

Section: Introductionmentioning

confidence: 99%

Some Relations Between Bounded Below Elliptic Operators and Stochastic Analysis

Springer Proceedings in Mathematics &Amp; Statistics

2019

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We apply Malliavin Calculus tools to the case of a bounded below elliptic rightinvariant Pseudodifferential operators on a Lie group. We give examples of bounded below pseudodifferential elliptic operators on R d by using the theory of Poisson process and the Garding inequality. In the two cases, there is no stochastic processes besides because the considered semi-groups do not preserve positivity.