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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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