A counterexample to maximal L_p-regularity of the stochastic heat equation in polygons: the case p>4

2015/08/14 by Kyeong-Hun Kim, Kim, Kyeong-Hun
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories #math.PR

paper · pdf · doi:10.48550/arxiv.1508.03402

There is an error in the proof of main theorem

openalex publication_date 2015/08/14 · arxiv created 2016/05/06 · arxiv updated 2016/05/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let D be a domain in R^d and u be the solution to the stochastic heat equation du=Δu dt+ g dW_t, t>0, x∈ D, with zero initial and boundary data. Here W_t is a one-dimensional Wiener process on a probability space Ω. It has been proved (see below for references) that for any p≥ 2 the inequality ‖∇ u‖_{Lp(Ω× [0,T]× D)} ≤ c ‖g‖_{Lp(Ω× [0,T]× D)} holds if ∂ D∈ C¹. In this note we prove that if p>4 then this inequality fails in any polygon in R² having an angle greater than or equal to (pπ)/(2(p-2)). We also show that a similar statement holds in higher dimensional polygons. The counterexample introduced here is based on personal communication with N.V. Krylov.

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