Pathwise Uniqueness of the Solutions of Stochastic Heat Equation with Square-root Coefficient

Abstract

White-noise case stochastic heat equation was derived from Dawson-Watanabe superprocess. The pathwise uniqueness of their solutions with non-Lipschitz coefficients has attracted wide attention and in particular, the square-root coefficient case was listed as a long standing open problem in the famous literature (i.e. Perkins \cite{Perkins02} p217.) This short note gives an affirmative answer to this open problem. Our idea is using the $\varphi_k(x)$ function constructed by Yamada-Watanable to prove that \beqlb \lab{key} \E_{\mu}| X_{t}(z)-Y_{t}(z) | = \lim_{k \ra \infty} \E_{\mu}|\int_{\r}\frac{1}{\sqrt{\epsilon_k}} \rho_{\epsilon_k}(z-x)[ X_{t}(x)-Y_{t}(x)]dx |=0 . \eeqlb However, $\frac{1}{\sqrt{\epsilon_k}} \rho_{\epsilon_k}(x)=p_{\epsilon_k}(x)$, the stable kernel, is not square-integrable uniformly in $k$ and we cannot directly use the heat kernel or stable kernel convolution transformation. This forces us to go a new way in which we have used the linear and reciprocal linear properties of $\varphi_k$ and $\varphi^{\prime \prime}_k$, respectively, making a $\rho_{\epsilon_k}$ convolution transformation first, then, taking expectation and multiplying singularity factor $\frac{1}{\sqrt{\epsilon_k}}$ and finally letting $k \ra \infty$ to estimate and derive (\ref{key}). In this way, we have avoided the non-uniform square integrability of the stochastic integral terms which involve the singularity. The same idea also can be used to prove the pathwise uniqueness of the nonnegative solutions of stochastic heat equation with $\alpha$ coefficients ($1/2 \leq \alpha$) since by a same idea we can construct the $\varphi_k(x)$ functions with $1/2 \leq \alpha$.