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A boundary local time for super-Brownian motion and new path properties for superprocess densities

University of British Columbia

We establish new fractal properties for superprocess densities. Consider the density of super-Brownian motion in one dimension at a fixed time. We construct a random measure supported on the boundary of the zero set of the density, which we call the boundary local time, and prove that it is locally non-degenerate, in the sense that it almost surely has positive mass on every neighbourhood of the boundary of the support. We then use the boundary local time to give an almost sure lower bound on the Hausdorff dimension of the boundary of the zero set, confirming a conjecture of Mueller et al. [69], whose work on this problem motivated ours. The Hausdorff dimension is characterized in terms of the lead eigenvalue of a certain killed Ornstein-Uhlenbeck operator which also appears in the construction of the boundary local time. The proof of local non-degeneracy includes a novel analysis of the density near the right endpoint of its support, including a new bound on the expected mass in a neighbourhood of the endpoint. We then consider the density of (α, β)-superprocess in dimension d with α ∈ (0, 2), β ∈ (0, 1], and βd < α. A classical result called instantaneous propagation asserts that, at a fixed time, the density has positive mass on any open set almost surely when conditioned on survival. We give general conditions under which the integral the density against a measure μ is positive almost surely when conditioned on survival. In particular, we show that this is determined by the relationship between the dimension of the measure, which we define in two ways, and a critical parameter determined by α, β and d. When the density does not charge μ almost surely, we obtain upper and lower bounds on the probability that it does so. In both cases our results correspond to new existence and non-existence results for solutions to the partial differential equation which is dual to the process. In the regime where the density is continuous and positive at a fixed point almost surely, we prove that the density function is strictly positive.

Text

2020-08-20

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75612

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/