## Hitting properties of a random string.(English)Zbl 1010.60059

The authors study hitting problems, double points and recurrence questions for a Funaki’s model of a random string taking values in $$\mathbb{R}^d$$, that is ${\partial u(x)\over\partial t}= {\partial^2 u(x)\over\partial x^2}+ \dot W,$ where $$\dot W$$ is a two-parameter white noise. Among others they prove the following results: the random string hits points if and only if $$d< 6$$, there exist points $$(t,x)$$ and $$(t,y)$$ such that $$u_t(x)= u_t(y)$$ if and only if $$d< 8$$, there exist points $$(t,x)$$ and $$(s,y)$$ such that $$u_t(x)= u_s(y)$$ if and only if $$d<12$$, and the stationary pinned string is almost surely recurrent if $$d\leq 6$$ and almost surely non-recurrent if $$d\geq 7$$.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 35L05 Wave equation
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