On the boundary of the zero set of super-Brownian motion and its local time
| Autor: | Edwin Perkins, Thomas Hughes |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
Super-Brownian motion Boundary (topology) Hausdorff dimension 01 natural sciences Combinatorics 010104 statistics & probability Mathematics::Probability FOS: Mathematics 60J68 0101 mathematics Super brownian motion Stochastic pde Mathematics Zero–one law Zero set 010102 general mathematics Probability (math.PR) Mathematics::Spectral Theory Zero-one law Local time 60H15 60J55 28A78 Statistics Probability and Uncertainty Mathematics - Probability |
| Zdroj: | Ann. Inst. H. Poincaré Probab. Statist. 55, no. 4 (2019), 2395-2422 |
| Popis: | If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that $\text{dim}(\partial\{x:X(t,x)>0\})=2-2\lambda_0\in(0,1)$ a.s. on $\{X_t\neq 0\}$, where $-\lambda_0\in(-1,-1/2)$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck process. This confirms a conjecture of Mueller, Mytnik and Perkins who proved the above with positive probability. To establish this result we derive some new basic properties of a recently introduced boundary local time and analyze the behaviour of $X(t,\cdot)$ near the upper edge of its support. Numerical estimates of $\lambda_0$ suggest that the above Hausdorff dimension is approximately $.224$. Comment: 23 pages |
| Databáze: | OpenAIRE |
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