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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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