2016/12/19 by Hao Wang, Wang, Hao
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.48550/arxiv.1612.06055
This paper has been withdrawn by the author due to a gap in the proof of the main theorem
openalex publication_date 2016/12/19 · arxiv created 2017/02/09 · arxiv updated 2017/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citePerkins02 p217.) This short note gives an affirmative answer to this open problem. Our idea is using the φk(x) function constructed by Yamada-Watanable to prove that \beqlb \labkey \Eμ| Xt(z)-Yt(z) | = limk \ra ∞ \Eμ|∫\r(1)/(√(εk)) ρεk(z-x)[ Xt(x)-Yt(x)]dx |=0 . \eeqlb However, (1)/(√(εk)) ρεk(x)=pεk(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 φk and φ′ ′k, respectively, making a ρεk convolution transformation first, then, taking expectation and multiplying singularity factor (1)/(√(εk)) and finally letting k \ra ∞ to estimate and derive (\refkey). 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 α coefficients (1/2 ≤ α) since by a same idea we can construct the φk(x) functions with 1/2 ≤ α.