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Coupling and invariant measures for the heat equation with noise. (English) Zbl 0795.60056
The author considers periodic solutions for the heat equation $$u_ t=Du_{xx} -\alpha u+a(u)+b(u) \dot W$$. Here $$\dot W=\dot W(t,x)$$ is a two-parameter white noise and $$D>0$$ and $$\alpha \geq 0$$ are constants, $$t \geq 0$$ and $$x \in S'=R \pmod {2\pi}$$. The equation is interpreted in the weak sense of Walsh. It is shown that for two initial conditions $$u^ 1(0,x)$$, $$u^ 2(0,x)$$ which are continuous in $$x \in S'$$ there are solutions $$u^ 1(t,x)$$, $$u^ 2(t,x)$$ which are equal for $$x \in S'$$ and $$t$$ greater than some stopping time $$t$$.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 35K05 Heat equation
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