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Functional Analysis and Evolution Equations The Günter Lumer Volume

:	Herbert Amann, Wolfgang Arendt, Matthias Hieber, Frank Neubrander, Serge Nicaise, Joachim von Below
:	Herbert Amann, Wolfgang Arendt, Frank Neubrander, Serge Nicaise, Joachim von Below
:	Functional Analysis and Evolution Equations The Günter Lumer Volume
:	Birkhäuser Basel
:	9783764377946
:	1
:	CHF 48.40
:

:	Analysis
:	English

:	637
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

Gunter Lumer was an outstanding mathematician whose works have great influence on the research community in mathematical analysis and evolution equations. He was at the origin of the breath-taking development the theory of semigroups saw after the pioneering book of Hille and Phillips from 1957. This volume contains invited contributions presenting the state of the art of these topics and reflecting the broad interests of Gunter Lumer.

Feynman-Kac Formulas, Backward Stochastic Differential Equations and Markov Processes (p. 81-82)

Jan A. Van Casteren

This article is written in honor of G. Lumer whom I consider as my semi-group teacher

Abstract. In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic Differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial Differential equations of parabolic type. Some emphasis is put on the fact that the whole theory generalizes Feynman-Kac formulas. A new method of proof of the existence of solutions is given. All the existence arguments are based on rather precise quantitative estimates.

1. Introduction

Backward stochastic Differential equations, in short BSDEs, have been well studied during the last ten years or so. They were introduced by Pardoux and Peng [20], who proved existence and uniqueness of adapted solutions, under suitable squareintegrability assumptions on the coeffcients and on the terminal condition. They provide probabilistic formulas for solution of systems of semi-linear partial Differential equations, both of parabolic and elliptic type. The interest for this kind of stochastic equations has increased steadily, this is due to the strong connections of these equations with mathematical finance and the fact that they provide a generalization of the well-known Feynman-Kac formula to semi-linear partial differential equations. In the present paper we will concentrate on the relationship between time-dependent strong Markov processes and abstract backward stochastic Differential equations. The equations are phrased in terms of a martingale type problem, rather than a strong stochastic Differential equation. They could be called weak backward stochastic Differential equations. Emphasis is put on existence and uniqueness of solutions. The paper in [27] deals with the same subject, but it concentrates on comparison theorems and viscosity solutions.

The notion of squared gradient operator is implicitly used by Bally at al in [4]. The latter paper was one of the motivations to write the present paper with an emphasis on the squared gradient operator. In addition, our results are presented in such a way that the state space of the underlying Markov process, which in most of the other papers on BSDEs is supposed to be Rn, can be any diffusion with an abstract state space, which throughout our text is denoted by E. In fact in the existing literature the underlying Markov process is a (strong) solution of a (forward) stochastic Differential equation: see, e.g., [4], [8] and [7] and [19]. For more on this see Remark 2.9 below. In particular our results are applicable in case the Markov process under consideration is Brownian motion on a Riemannian manifold. Our condition on the generator (or coefficient) of the BSDE f in terms of the squared gradient is very natural. In the Lipschitz context it is more or less optimal. Moreover, our proof of existence is not based on standard regularization methods by using convolution products with smooth functions, but on a homotopy argument due to Crouzeix [11], which seems more direct than the classical approach. We also obtain rather precise quantitative estimates. Only very rudimentary sketches of proofs are given, details will appear elsewhere.

	Contents	5
	Life and Work of Günter Lumer	9
	In Remembrance of Günter Lumer	18
	Expansions in Generalized Eigenfunctions of the Weighted Laplacian on Star-shaped Networks	20
	1. Introduction	20
	2. Data and functional analytic framework	24
	3. Expansion in generalized eigenfunctions	24
	4. Application of Stone’s formula and limiting absorption principle	27
	5. A Plancherel-type formula and a functional calculus for the operator	30
	References	34
	Diffusion Equations with Finite Speed of Propagation	36
	1. Introduction	36
	2. The Cauchy problem for a strongly degenerate quasi-linear equation	39
	3. The evolution of the support of the solutions of the relativistic heat equation	49
	References	51
	Subordinated Multiparameter Groups of Linear Operators: Properties via the Transference Principle	54
	1. Introduction	54
	2. The transference principle and generator formulas	56
	3. Examples	66
	References	68
	An Integral Equation in AeroElasticity	70
	1. Introduction	70
	2. The Possio Equation	71
	3. Possio Integral: Time domain version	73
	4. Special case	76
	5. The general case	79
	6. Generalization	83
	References	84
	Eigenvalue Asymptotics Under a Nondissipative Eigenvalue Dependent Boundary Condition for Second- order Elliptic Operators	85
	1. Introduction	85
	2. Eigenvalue asymptotics for constant dynamical coeffcient	88
	3. Dynamical coeffcient of constant sign	92
	4. Dynamical coeffcient changing sign	93
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