Journal of the Mathematical Society of Japan

Smoothness of the joint density for spatially homogeneous SPDEs

Yaozhong HU, Jingyu HUANG, David NUALART, and Xiaobin SUN

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In this paper we consider a general class of second order stochastic partial differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and has a homogeneous spatial covariance. Using the techniques of Malliavin calculus we derive the smoothness of the density of the solution at a fixed number of points $(t,x_1), \dots, (t,x_n)$, $t$ > 0, with some suitable regularity and nondegeneracy assumptions. We also prove that the density is strictly positive in the interior of the support of the law.

Article information

J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1605-1630.

First available in Project Euclid: 27 October 2015

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Stochastic partial differential equations Malliavin calculus spatially homogeneous covariances smoothness of joint density strict positivity


HU, Yaozhong; HUANG, Jingyu; NUALART, David; SUN, Xiaobin. Smoothness of the joint density for spatially homogeneous SPDEs. J. Math. Soc. Japan 67 (2015), no. 4, 1605--1630. doi:10.2969/jmsj/06741605.

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