## The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics (1996)

Citations: | 26 - 1 self |

### BibTeX

@TECHREPORT{Stannat96thetheory,

author = {Wilhelm Stannat},

title = {The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics},

institution = {},

year = {1996}

}

### OpenURL

### Abstract

We present an introduction (also for non-experts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²-spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, so-called generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 --semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p --conditions, singular and fractional diffusion operators. Subsequently, we analyz...

### Citations

1106 | Semigroups of linear operators and applications to partial differential equations - Pazy - 1983 |

1057 | Brownian Motion and Stochastic Calculus - KARATZAS, SHREVE - 1991 |

806 | Theory of Ordinary Differential Equations - Coddington, Levisson - 1955 |

737 |
Quelque méthodes de résolution des problémes aux limites non linéaires, Gauthier–Villars
- Lions
- 1969
(Show Context)
Citation Context ...ded satisfying no L p --condition. Concerning the L 2 -analysis, e.g. whether a properly chosen extension generates a C 0 --semigroup, we modify the approach by J.L. Lions and E. Magenes in [LMa] and =-=[L]-=- and adapt their methods so that we can handle operators of particular interest in probability theory. The main difference to the known analysis is that the measure of the underlying L 2 --space is no... |

540 |
Methods of modern mathematical physics IV: Analysis of operators
- Reed, Simon
- 1978
(Show Context)
Citation Context ...orm E : the associated resolvent (G ff ) ff?0 is sub--Markovian if and only if D2 u 2 F ) u +s1 2 V and E(u; u \Gamma u +s1)s0 : D2 is an extension of the second Beurling--Deny criterion (cf. [BeDe], =-=[ReSi3]-=-) to the framework of generalized Dirichlet forms. This characterization allows to deduce a sufficient condition for the sub--Markov property of (G ff ) ff?0 which is easy to check in applications (cf... |

365 | Methods of modern mathematical physics. I. Functional analysis - Reed, Simon - 1980 |

321 | Heat Kernels and Spectral Theory - Davies - 1989 |

265 | Methods of modern mathematical physics. II. Fourier analysis, self-adjointness - Reed, Simon - 1975 |

234 |
Introduction to Theory of (Non-Symmetric) Dirichlet Forms
- Ma, Röckner
- 1992
(Show Context)
Citation Context ...are in general neither symmetric nor coercive, but still generate associated C0 --semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], =-=[MR1]-=-) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators ... |

151 | Operators in Hilbert Spaces - Linear - 1980 |

125 | One-parameter semigroups - Davies - 1980 |

106 |
General Theory of Markov Processes
- SHARPE
- 1988
(Show Context)
Citation Context ...t]), is called the natural filtration. A right process w.r.t. some filtration (M t ) t0 is always a right process w.r.t. the natural filtration. (ii) If E is a Radon topological space in the sense of =-=[Sh]-=- (i.e. homeomorphic to a universally measurable subset of a compact metric space, e.g. a locally 75 compact separable metric space) then any right process M with state space E w.r.t. some filtration (... |

105 | Equations of evolution - Tanabe - 1979 |

75 | Probability and potentials - MEYER - 1966 |

71 |
Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups
- Gross
- 1993
(Show Context)
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63 | Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures - Schwartz - 1973 |

60 |
Fractional diffusion and wave equations
- Schneider, Wyss
- 1989
(Show Context)
Citation Context ...) = R hru; rvi dx for all t 2 R, the generator (L ff ; D (L ff )) associated with E ff is an extension of the fractional diffusion operator ( @ @t ) ff u + \Deltau; u 2 C 1 0 \Gamma R d+1 \Delta (cf. =-=[SWy]-=-). (b) Transformation of generalized positivity preserving forms via excessive functions In this subsection we suppose that the bilinear form associated with (A; V) and (; D(; H)) is a generalized pos... |

49 |
Branching processes with immigration and related limit theorems, Theory Probab
- Kawazu, Watanabe
- 1971
(Show Context)
Citation Context ...tudied in the 1--dimensional case both from the analytic and the probabilistic point of view by W. Feller (cf. [Fe1] and [Fe2]), J. Lamperti and P. Ney (cf. [LaNe]) and K. Kawazu and S. Watanabe (cf. =-=[KaWa]-=-). Note that the vector field Q T x satisfies (3.2)--(3.4). Hence there exists a generalized Dirichlet form E associated with (A; V) and the closure of hQ T x; rui \Gamma d(L=2)u, u 2 C 1 0 (R d + ), ... |

46 | Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab - Albeverio, Röckner - 1991 |

40 | Ordinary Differential Equations: An Introduction to Nonlinear Analysis, Walter de Gruyter - Amann - 1990 |

39 | One-Parameter Semigroups of Positive Operators - NAGEL |

15 |
Espaces de Dirichlet
- Beurling, Deny
- 1958
(Show Context)
Citation Context ...linear form E : the associated resolvent (G ff ) ff?0 is sub--Markovian if and only if D2 u 2 F ) u +s1 2 V and E(u; u \Gamma u +s1)s0 : D2 is an extension of the second Beurling--Deny criterion (cf. =-=[BeDe]-=-, [ReSi3]) to the framework of generalized Dirichlet forms. This characterization allows to deduce a sufficient condition for the sub--Markov property of (G ff ) ff?0 which is easy to check in applica... |

14 |
Conditioned branching processes and their limiting diffusions
- Lamperti, Ney
- 1968
(Show Context)
Citation Context ... corresponding to the operator A has been studied in the 1--dimensional case both from the analytic and the probabilistic point of view by W. Feller (cf. [Fe1] and [Fe2]), J. Lamperti and P. Ney (cf. =-=[LaNe]-=-) and K. Kawazu and S. Watanabe (cf. [KaWa]). Note that the vector field Q T x satisfies (3.2)--(3.4). Hence there exists a generalized Dirichlet form E associated with (A; V) and the closure of hQ T ... |

14 |
Markov processes associated with semi-Dirichlet forms
- Ma, Overbeck, et al.
- 1995
(Show Context)
Citation Context ...ter we will discuss several examples for generalized Dirichlet forms. We will concentrate on examples which are not symmetric nor coercive Dirichlet forms nor semi--Dirichlet forms (cf. [FOT], [MR1], =-=[MOvR]-=-) but also not time dependent Dirichlet forms as treated in [O1]. In the first section we give some examples of Dirichlet forms with time dependent potentials, in the next section we consider first or... |

11 |
Dissipative operators and hyperbolic systems of partial differential equations
- Phillips
- 1959
(Show Context)
Citation Context ...the operator ( \Gamma c; C 1 0 (R d )) is negative definite on H. Hence there exists at least one closed extension ( \Gamma c; D()) of ( \Gamma c; C 1 0 (R d )) on H generating a C 0 --semigroup (cf. =-=[P]-=-, [Da1, Section 6.1]). Then (; D()) is an extension of (; C 1 0 (R d )) generating a C 0 --semigroup. Uniqueness: To prove uniqueness let (; D(; H)) be an arbitrary extension of (; C 1 0 (R d )) gener... |

9 | Nonsymmetric) Dirichlet operators on L : existence, uniqueness and associated Markov processes
- Stannat
- 1997
(Show Context)
Citation Context ...ess, and Theorem IV.2.2 can be viewed as the analogue of the well--known existence result on Hunt processes associated with Feller semigroups (cf. [BlG]). For example, Theorem IV.2.2 has been used in =-=[St4]-=- to construct diffusion processes properly associated in the resolvent sense with (particular extensions of non--symmetric) differential operators of type Lu(x) = P d i;j=1 @ @x j (a ij (x) @u @x i (x... |

8 | Markov processes associated with positivity preserving coercive forms - Ma, ockner, et al. - 1995 |

8 |
Schrödinger Equations and Diffusion Theory, Birkhäuser
- Nagasawa
- 1993
(Show Context)
Citation Context ...e diffusions with singular drifts. The existence of such processes is of interest in the theory of stochastic mechanics, where they are often called Nelson processes (cf. the monograph by M. Nagasawa =-=[Nag]-=- and references therein). Summarizing, the framework of generalized Dirichlet forms contains many new examples for differential operators that can only be treated by an L 2 --approach for a properly c... |

8 |
On a construction of Markov processes associated with time dependent Dirichlet spaces. Forum Mathematicum 4
- OSHIMA
- 1992
(Show Context)
Citation Context ...ll generate associated C0 --semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. =-=[O1]-=-). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p --conditions,... |

6 | Generalized Dirichlet forms and associated Markov processes
- Stannat
- 1994
(Show Context)
Citation Context ...he semigroup (U t ) t0 corresponding to the closure of (; C 1 0 (R d )) cannot be restricted to a C 0 --semigroup on H 1;2 (R d ). Hence our framework is more general than the framework considered in =-=[St2]-=- and [St3]. The reason why ( b U t ) t0 cannot be restricted to H 1;2 (R d ) in general is given as follows. Assume that B satisfies (2.5) - (2.7) and is continuously differentiable. Denote bysthe cor... |

5 | A note on quasicontinuous kernels representing quasi-linear positive maps - Albeverio, Ma - 1991 |

5 | Markov processes: Ray processes and right processes - Getoor - 1970 |

5 | Dirichlet forms and Markov processes: a generalized framework including both elliptic and parabolic cases, to appear in: Potential Analysis
- Stannat
- 1995
(Show Context)
Citation Context ...up (U t ) t0 corresponding to the closure of (; C 1 0 (R d )) cannot be restricted to a C 0 --semigroup on H 1;2 (R d ). Hence our framework is more general than the framework considered in [St2] and =-=[St3]-=-. The reason why ( b U t ) t0 cannot be restricted to H 1;2 (R d ) in general is given as follows. Assume that B satisfies (2.5) - (2.7) and is continuously differentiable. Denote bysthe corresponding... |

4 |
Représentant Précis d’Un Potentiel Parabolique. Séminaire de Théorie du Potentiel
- Pierre
- 1980
(Show Context)
Citation Context ... the class of generalized Dirichlet forms includes in particular symmetric and coercive Dirichlet forms (cf. [MR1],[FOT]) but also time dependent Dirichlet forms as considered in [O1] (cf. also [Pi1],=-=[Pi2]-=-). The latter are given as follows: Let (E (t) ; V ) t2R be a family of coercive Dirichlet forms with uniform sector constant K and common domain V in some L 2 --space such that t 7! E (t) (u; v) is B... |

4 |
Parabolic capacity and Sobolev spaces
- Pierre
- 1983
(Show Context)
Citation Context ...spaces including the examples mentioned above. The third chapter is a generalization of the classical work on parabolic potential theory in an L 2 --setting as developed by M. Pierre in [Pi1], [Pi2], =-=[Pi3]-=- and its extensions to the case of time dependent Dirichlet spaces by Y. Oshima in [O1] to the framework of generalized Dirichlet forms. Suppose from now on that E is a Hausdorff topological space and... |

3 |
Two singular diffusion problems, The Annals of Mathematics 54
- Feller
- 1951
(Show Context)
Citation Context ...Theorem X.49]. The singular diffusion equation corresponding to the operator A has been studied in the 1--dimensional case both from the analytic and the probabilistic point of view by W. Feller (cf. =-=[Fe1]-=- and [Fe2]), J. Lamperti and P. Ney (cf. [LaNe]) and K. Kawazu and S. Watanabe (cf. [KaWa]). Note that the vector field Q T x satisfies (3.2)--(3.4). Hence there exists a generalized Dirichlet form E ... |

2 | Probl`emes d"evolution avec constraintes et potentiels paraboliques - Pierre - 1979 |

2 | Verallgemeinerte zeitabh angige Dirichletformen, Diploma thesis. Universit at Bonn - Stannat - 1993 |

1 | Measure - valued Markov processes, in: Hennequin, P.L. (Editor), Ecole d'Et'e de Probabilit'es de - Dawson - 1991 |

1 |
2nd Berkely Symp. Univ. of Calif
- Feller, Proc
- 1951
(Show Context)
Citation Context ...49]. The singular diffusion equation corresponding to the operator A has been studied in the 1--dimensional case both from the analytic and the probabilistic point of view by W. Feller (cf. [Fe1] and =-=[Fe2]-=-), J. Lamperti and P. Ney (cf. [LaNe]) and K. Kawazu and S. Watanabe (cf. [KaWa]). Note that the vector field Q T x satisfies (3.2)--(3.4). Hence there exists a generalized Dirichlet form E associated... |

1 |
In'equations d"evolution paraboliques avec convexes d'ependant du temps, Applications aux in'equations quasi--variationelles d'evolution
- Mignot, Puel
- 1977
(Show Context)
Citation Context ...riational problem: Find u 2 V, ush, such that A 1 (u; v) \Gamma h b v; uisA 1 (u; u) for all v 2 b F , vsh. The proof is based on methods used in the theory of parabolic variational inequalities (cf. =-=[MiPu]-=-). In Section III.2 we define E--nests and E--exceptional sets with the help of 1-- reduced functions, hence only in terms of the bilinear form E , and construct Choquet capacities whose zero sets are... |

1 | Functional Analysis, 5 - Yosida - 1978 |