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 \(t\).


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