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Invariant measures of stochastic partial differential equations and conditioned diffusions. (English) Zbl 1063.60092

Summary: This work establishes and exploits a connection between the invariant measure of stochastic partial differential equations (SPDEs) and the law of bridge processes. Namely, it is shown that the invariant measure of \(u_t =u_{xx}+ f(u)+ \sqrt{2\varepsilon}\eta(x, t)\), where \(\eta(x, t)\) is a space-time white-noise, is identical to the law of the bridge process associated to \(dU= a(U)dx+ \sqrt{\varepsilon}dW(x)\), provided that \(a\) and \(f\) are related by \(\varepsilon a''(u)+ 2a'(u)a(u)= -2f(u)\), \(u\in\mathbb{R}\). Some consequences of this connection are investigated, including the existence and properties of the invariant measure for the SPDE on the line, \(x\in\mathbb{R}\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)

References:

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