## 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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