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17
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) 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 L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 43 (1 self)
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We present an introduction (also for nonexperts) 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 L²spaces 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, socalled 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...
Poisson representations of branching Markov and measurevalued branching processes
, 2008
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Multiple SpaceTime Scale Analysis For Interacting Branching Models
 JOURN. OF PROBAB
, 1996
"... We study a class of systems of countably many linearly interacting diffusions whose components take values in [0; 1) and which in particular includes the case of interacting (via migration) systems of Feller's continuous state branching diffusions. The components are labelled by a hierarchical ..."
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Cited by 9 (2 self)
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We study a class of systems of countably many linearly interacting diffusions whose components take values in [0; 1) and which in particular includes the case of interacting (via migration) systems of Feller's continuous state branching diffusions. The components are labelled by a hierarchical group. The longterm behaviour of this system is analysed by considering spacetime renormalised systems in a combination of slow and fast time scales and in the limit as an interaction parameter goes to infinity. This leads to a new perspective on the large scale behaviour (in space and time) of critical branching systems in both the persistent and nonpersistent cases and including that of the associated historical process. Furthermore we obtain an example for a rigorous renormalization analysis. The qualitative behaviour of the system is characterised by the socalled interaction chain, a discrete time Markov chain on [0; 1) which we construct. The transition mechanism of this chain is given in...
Asymptotic behavior of continuous time and state branching processes
 J. Austral. Math. Soc. (Ser. A
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Limit theorems for conditioned multitype DawsonWatanabe processes and Feller diffusions, Electron
 J. Probab
"... A multitype DawsonWatanabe process is conditioned, in subcritical and critical cases, on nonextinction in the remote future. On every finite time interval, its distribution law is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is ..."
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Cited by 7 (0 self)
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A multitype DawsonWatanabe process is conditioned, in subcritical and critical cases, on nonextinction in the remote future. On every finite time interval, its distribution law is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. The explicit form of the Laplace functional of the conditioned process is used to obtain several results on the long time behaviour of the mass of the conditioned and unconditioned processes. The general case is considered first, where the mutation matrix which modelizes the interaction between the types, is irreducible. Several twotype models with decomposable mutation matrices are also analysed. AMS 2000 Subject Classifications: 60J80, 60G57. KEYWORDS: multitype measurevalued branching processes, conditioned DawsonWatanabe process, critical and subcritical DawsonWatanabe process, remote survival, long time behavior. 1
Convergence Of Branching Processes To The Local Time Of A Bessel Process
 In Proceedings of the Eighth International Conference “Random Structures and Algorithms
, 1997
"... We study GaltonWatson branching processes conditioned on the total progeny to be n which are scaled by a sequence cn tending to infinity as o( p n). It is shown that this process weakly converges to the totallocal time of a twosided threedimensional Bessel process. This is done by means of char ..."
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Cited by 7 (4 self)
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We study GaltonWatson branching processes conditioned on the total progeny to be n which are scaled by a sequence cn tending to infinity as o( p n). It is shown that this process weakly converges to the totallocal time of a twosided threedimensional Bessel process. This is done by means of characteristic functions and a generating function approach. 1.
Infinite canonical superBrownian motion and scaling limits
, 2008
"... We construct a measure valued Markov process which we call infinite canonical superBrownian motion, and which corresponds to the canonical measure of superBrownian motion conditioned on nonextinction. Infinite canonical superBrownian motion is a natural candidate for the scaling limit of various ..."
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Cited by 6 (0 self)
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We construct a measure valued Markov process which we call infinite canonical superBrownian motion, and which corresponds to the canonical measure of superBrownian motion conditioned on nonextinction. Infinite canonical superBrownian motion is a natural candidate for the scaling limit of various random branching objects on Z d when these objects are (a) critical; (b) meanfield and (c) infinite. We prove that ICSBM is the scaling limit of the spreadout oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the uppercritical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions. This paper also serves as a survey of recent results linking superBrownian to scaling limits in statistical mechanics.
A Williams decomposition for spatially dependent superprocesses ∗
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A conditional limit theorem for generalized diffusion processes
 J. Math. Kyoto Univ
, 2003
"... Let {X(t) : t ≥ 0} be a onedimensional generalized diffusion process with initial state X(0)> 0, hitting time τX(0) at the origin and speed measure m which is regularly varying at infinity with exponent 1/α − 1> 0. It is proved that, for a suitable function u(c), the probability law of {u(c)− ..."
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Cited by 2 (0 self)
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Let {X(t) : t ≥ 0} be a onedimensional generalized diffusion process with initial state X(0)> 0, hitting time τX(0) at the origin and speed measure m which is regularly varying at infinity with exponent 1/α − 1> 0. It is proved that, for a suitable function u(c), the probability law of {u(c)−1X(ct) : 0 < t ≤ 1} converges as c→ ∞ to the conditioned 2(1−α)dimensional Bessel excursion on natural scale and that the latter is equivalent to the 2(1 − α)dimensional Bessel meander up to a scale transformation. In particular, the distribution of u(c)−1X(c) converges to the Weibull distribution. From the conditional limit theorem we also derive a limit theorem for some of regenerative process associated with {X(t) : t ≥ 0}. Key words: generalized diffusion, hitting time, conditional limit theorem, Bessel diffusion, excursion, meander.