2020/11/26 by Siva Athreya, Oleg Butkovsky, Athreya, Siva +5 · 4 citations
Economics, Econometrics and Finance · Mathematics · Computer Science · #Stochastic processes and financial applications #Mathematical Biology Tumor Growth #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2011.13498
We study stochastic reaction--diffusion equation ∂tut(x)=\frac12 ∂2xxut(x)+b(ut(x))+Wt(x), tgt;0, x∈ D where b is a generalized function in the Besov space Bβq,∞(\mathbb R), D⊂\mathbb R and W is a space-time white noise on \mathbb R+× D. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever β-1/q≥-1, β>-1 and q∈[1,∞]. This class includes equations with b being measures, in particular, b=δ0 which corresponds to the skewed stochastic heat equation. For β-1/q > -3/2, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê~(2020).