Electronic Communications in Probability

Stability of the stochastic heat equation in $L^1([0,1])$

Nicolas Fournier and Jacques Printems

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We consider the white-noise driven stochastic heat equation on $[0,1]$ with Lipschitz-continuous drift and diffusion coefficients. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some estimates which allow us to study the stability of the solution with respect the initial condition, the uniqueness of the possible invariant distribution and the asymptotic confluence of solutions.

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Electron. Commun. Probab. Volume 16 (2011), paper no. 32, 337-352.

Accepted: 30 May 2011
First available in Project Euclid: 7 June 2016

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Fournier, Nicolas; Printems, Jacques. Stability of the stochastic heat equation in $L^1([0,1])$. Electron. Commun. Probab. 16 (2011), paper no. 32, 337--352. doi:10.1214/ECP.v16-1636. https://projecteuclid.org/euclid.ecp/1465261988.

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