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Kolmogorov’s test for the Brownian snake. (English) Zbl 1017.60039

Let \(W=(W_{s},s\geq 0)\) be a Brownian snake started at \(0\in {\mathbb R}^{d},\) let \(h\) be a nonincreasing nonnegative function defined on \((0,\infty),\) and let \(Q=\{(t,x)\in {\mathbb R}_{+}\times {\mathbb R}^{d}:\left|x\right|<\sqrt{t}h(t)\}.\) Let \(T=\inf \{t>0:\rho _{t}\geq \sqrt{t}h(t)\}\) be the exit time of \(W\) from the domain \(Q,\) where \(\rho _{t}\) is the radius of the smallest ball centered at the origin that contains the snake at level \(t,\) and let \(\mathbb{N}_{0}\) be the excursion measure. The authors show that \(\mathbb{N}_{0}(T=0)=0\) or \(1\) according as \(I_{r}=\int_{0}^{r}(h(t)^{d+2}/t) e^{-h(t)^{2}/2}dt<\infty \) for some \(r>0\) or \(I_{r}=\infty \) for all \(r>0.\) This implies the following result.
Corollary. Let \(u(t,x)\) be the maximal nonnegative solution to \(\partial _{t}u+(\Delta /2)u=2u^{2}\) in \(Q,\) and let \((\gamma _{t},t\geq 0)\) be a Brownian motion started at \(0\in \mathbb{R}^{d}\) under \(\mathbb{P}.\) Then \(\mathbb{P}\)-a.s. there exists \(s>0\) such that \(\int_{0}^{s}u(t,\gamma _{t})dt<\infty \) if and only if \(I_{r}<\infty \) for some \(r>0.\)

MSC:

60F20 Zero-one laws
60J65 Brownian motion
60G40 Stopping times; optimal stopping problems; gambling theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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