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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Abstract

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

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

Dates
First available in Project Euclid: 27 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1445951159

Digital Object Identifier
doi:10.2969/jmsj/06741605

Mathematical Reviews number (MathSciNet)
MR3417506

Zentralblatt MATH identifier
1334.60111

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

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

Citation

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. https://projecteuclid.org/euclid.jmsj/1445951159


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