On the boundary of the zero set of super-Brownian motion and its local\n time

2018/02/10 by Thomas J.R. Hughes, Hughes, Thomas, Edwin Perkins +1
Economics, Econometrics and Finance · Mathematics · #60J68 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1802.03681

openalex publication_date 2018/02/10 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

If X(t,x) is the density of one-dimensional super-Brownian motion, we prove\nthat dim(\∂ x:X(t,x)>0 )=2-2\λ₀\∈(0,1) a.s. on\n X_t\≠ 0 , where -\λ₀\∈(-1,-1/2) is the lead eigenvalue of a\nkilled Ornstein-Uhlenbeck process. This confirms a conjecture of Mueller,\nMytnik and Perkins who proved the above with positive probability. To establish\nthis result we derive some new basic properties of a recently introduced\nboundary local time and analyze the behaviour of X(t,\⋅) near the upper\nedge of its support. Numerical estimates of \λ₀ suggest that the above\nHausdorff dimension is approximately .224.\n

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