# Preprint: INITIAL MEASURES FOR THE STOCHASTIC HEAT EQUATION

## Daniel Conus, Mathew Joseph, Davar Khoshnevisan, and Shang-Yuan Shiu

Abstract. 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 $$\bf R$$, 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 $$\mathcal{L}f =cf''$$ 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 $$t>0$$.

Keywords. The stochastic heat equation, singular initial data.

AMS Classification (2000) Primary: 60H15; Secondary: 35R60.

Support. Research supported in part by the NSF grants DMS-0747758 (M.J.) and DMS-1006903 (D.K.).

Pre/E-Prints. This paper is available in

Daniel Conus Lehigh University, Department of Mathematics, Christmas--Saucon Hall, 14 East Packer Avenue, Bethlehem, PA 18015, U.S.A. (daniel [dot sign] conus [at sign] lehigh [dot sign] edu>) Department of Mathematics University of Utah, 155 S, 1400 E JWB 233, Salt Lake City, UT 84112-0090, U.S.A. (joseph [at sign] math [dot sign] utah [dot sign] edu & davar[at sign] math [dot sign] utah [dot sign] edu) Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, TAIWAN (shiu [at sign] math [dot sign] sinica [dot sign] edu [dot sign] tw)

Last Update: October 18, 2011
© 2011 - Daniel Conus, Mathew Joseph, Davar Khoshnevisan, and Shang-Yuan Shiu