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Regularity and irregularity of superprocesses with \((1 + \beta)\)-stable branching mechanism. (English) Zbl 1385.60001

SpringerBriefs in Probability and Mathematical Statistics. Cham: Springer (ISBN 978-3-319-50084-3/pbk; 978-3-319-50085-0/ebook). viii, 77 p. (2016).
Let \(X= \{X_t; t\geq 0\}\) be \(d\)-dimensional super-Brownian motion with \((1+\beta)\)-stable branching, i.e., a measure-valued stochastic process in \(\mathbb{R}^d\), obtained as scaling limit of a \(d\)-dimensional Brownian motion whose critical branching mechanism is in the domain of attraction of \((1+\beta)\)-stable laws, \(\beta\in(0,1)\). The authors give an introductory, largely self-contained, up-to-date review of the main results on irregularity, respectively regularity properties of \(X_t\) for fixed \(t>0\). After analyzing the jumps, they prove the basic dichotomy respective the dimension \(d\): Denote by \({\mathcal M}_f\) the space of finite measures on \(\mathbb{R}^d\) with the weak topology. Let \(d< 2/\beta\), \(t>0\) fixed, and \(X_0\in{\mathcal M}_f\). If \(d=1\), then almost surely there exists a continuous version of the density function \(X_t(x)\) of the measure \(X_t(dx)\), but if \(d>1\), then almost surely for all open \(U\subseteq\mathbb{R}^d\), \(\text{ess\,sup}_{x\in Y}|X_t(x)|=\infty\) whenever \(X_t(U)>0\). For \(d=1\), the focus is on the following results concerning local Hölder continuity, the local Hölder exponent at fixed points, and the multi-fractal spectrum, again for fixed \(t>0\) and \(X_0\in{\mathcal M}_f\): cm
(1)
Define \(\eta_c:= 2/(1+\beta)\). Then, for each \(\eta<\eta_c\), \(X_t(\cdot)\) is locally Hölder continuous of index \(\eta\). For every \(\eta\geq\eta_c\), with probability 1, for any open \(U\subseteq\mathbb{R}\) with \(X_t(U)>0\), \(\sup_{x,y\in U, x\neq y} (|X_t(x)- X_t(y)|/|x-y|^\eta)=\infty\).
(2)
Denote by \(H_X(x)\) the pointwise Hölder exponent of the density function \(X_t(\cdot)\) at \(x\in\mathbb{R}\), and define \(\eta_c:= 3/(1+\beta)-1\). If \(\eta_c\leq 1\), then for every fixed \(x\in\mathbb{R}\), \(H_X(x)= \eta_c\) almost surely on \(\{X_t(x)>0\}\).
(3)
Denote by \(D_U(\eta)\) the Hausdorff dimension of \(\{x\in U: H_X(x)=\eta\}\), for \(U\subseteq\mathbb{R}\) open and \(\eta\in(\eta_c,\eta_c]\). Then, \(D_U(\eta)=(\beta+1)(\eta-\eta_c)\) almost surely whenever \(X_t(U)>0\).
The authors explain and discuss the methods and results. The essential parts of the proofs are provided, for others references are given. Familiarity with classical branching or super-processes is not assumed.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G57 Random measures
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