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

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.