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Finite width for a random stationary interface. (English) Zbl 0890.60056

Summary: We study the asymptotic shape of the solution \(u(t,x) \in [0,1]\) to a one-dimensional heat equation with a multiplicative white noise term. At time zero the solution is an interface, that is \(u(0,x)\) is 0 for all large positive \(x\) and \(u(0,x)\) is 1 for all large negative \(x\). The special form of the noise term preserves this property at all times \(t \geq 0\). The main result is that, in contrast to the deterministic heat equation, the width of the interface remains stochastically bounded.

MSC:

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