## 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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