Regularity and irregularity of \((1+\beta)\)-stable super-Brownian motion
This paper treats the regularity problem for super-Brownian motions \(X = \{ X_t \}\) with \((1 + \beta)\)-stable branching. Since \(X\) is almost surely discontinuous for the case \(0 < \beta < 1\), in view of the jumps \(X_t\) cannot possess a density for a dense set of times with probability one, and the regularity properties of the densities (that do exist) have remained unsolved. The main result of this paper gives a complete answer to this problem, i.e. Theorem 1.1. (a) There is a continuous version of the density \(\bar{X}(t, \cdot)\) on \({\mathbb R}^d\) if and only if \(d=1\). (b) Moreover, if \(2 \leqslant d \leqslant 2/{\beta}\), then \[ \| \bar{X}(t, \cdot) \|_U = \text{ess.} \sup_{x \in U} | \bar{X}(t,x) | = \infty \] holds whenever \(X_t(U) > 0\) for any open set \(U \subset {\mathbb R}^d\), P-a.s. \noindent The above second assertion says that the density is very badly behaved when \(d > 1\). The next result establishes local unboundedness of the density in time for a fixed spatial parameter \(x \in {\mathbb R}^d\) in any dimension where the density exists. More precisely, Theorem 1.2 asserts that for \(0 < \beta < 1\), \(d < 2 / {\beta}\), and \(s > 0\), \(\| \bar{X}(\cdot, x) \|_{(s, s+ \delta)}\) \(=\) \(\infty\) holds for any \(\delta > 0\), \(X_s\)-a.e. \(x\), P-a.s. The authors' last result deals with the regularity of the local time in the spatial parameter. Theorem 1.3 asserts that (a) for \(d=1\), there exists a jointly continuous version of the local time \(Y_t(x)\) in \({\mathbb R}_+ \times {\mathbb R}\), while (b) if \(0 < \beta < 1\) and \(2 \leqslant d < 2 + 2/{\beta}\), then \(\| Y_t(\cdot) \|_U = \infty\) holds whenever \(\int_U Y_t(x) \text{d} x >0\) for any open \(U \subset {\mathbb R}^d\) and any \(t >0\), P-a.s. This work is due to the first author's previous paper [Probab.\ Theory Relat. Fields 123, No. 2, 157--201 (2002; Zbl 1009.60053)].
stochastic partial differential equation
scientific article; zbMATH DE number 2072325