Initial measures for the stochastic heat equation. (English. French summary) Zbl 1288.60077

Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar; Shiu, Shang-Yuan

Ann. Inst. Henri Poincaré, Probab. Stat. 50, No. 1, 136-153 (2014).

Summary: We consider a family of nonlinear stochastic heat equations of the form \(\partial_{t}u=\mathcal{L}u+\sigma(u)\dot{W}\), where \(\dot{W}\) denotes space-time white noise, \(\mathcal{L}\) the generator of a symmetric Lévy process on \(\mathbb{R} \), and \(\sigma\) is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure \(u_{0}\). Tight a priori bounds on the moments of the solution are also obtained. {
} In the particular case that \(\mathcal{L}f=cf^{\prime\prime}\) for some \(c>0\), we prove that if \(u_{0}\) is a finite measure of compact support, then the solution is with probability one a bounded function for all times \(t>0\).

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

stochastic heat equation; singular initial data

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