## The Annals of Probability

- Ann. Probab.
- Volume 23, Number 1 (1995), 178-222.

### Approximation and Support Theorem in Holder Norm for Parabolic Stochastic Partial Differential Equations

Vlad Bally, Annie Millet, and Marta Sanz-Sole

#### Abstract

The solution $u(t, x)$ of a parabolic stochastic partial differential equation is a random element of the space $\mathscr{E}_{\alpha,\beta}$ of Holder continuous functions on $\lbrack 0, T \rbrack \times \lbrack 0, 1 \rbrack$ of order $\alpha = \frac{1}{4} - \varepsilon$ in the time variable and $\beta = \frac{1}{2} - \varepsilon$ in the space variable, for any $\varepsilon > 0$. We prove a support theorem in $\mathscr{E}_{\alpha,\beta}$ for the law of $u$. The proof is based on an approximation procedure in Holder norm (which should have its own interest) using a space-time polygonal interpolation for the Brownian sheet driving the SPDE, and a sequence of absolutely continuous transformations of the Wiener space.

#### Article information

**Source**

Ann. Probab., Volume 23, Number 1 (1995), 178-222.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988383

**Digital Object Identifier**

doi:10.1214/aop/1176988383

**Mathematical Reviews number (MathSciNet)**

MR1330767

**Zentralblatt MATH identifier**

0835.60053

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Secondary: 60H05: Stochastic integrals

**Keywords**

Brownian sheet parabolic stochastic partial differential equations polygonal approximation support theorem Holder norm

#### Citation

Bally, Vlad; Millet, Annie; Sanz-Sole, Marta. Approximation and Support Theorem in Holder Norm for Parabolic Stochastic Partial Differential Equations. Ann. Probab. 23 (1995), no. 1, 178--222. doi:10.1214/aop/1176988383. https://projecteuclid.org/euclid.aop/1176988383