Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise

By Robert C. Dalang, Davar Khoshnevisan, Tusheng Zhang. Annals of Probability, 2019

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on R+ × [0,1], with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists ε > 0 such that |b(z)| ≥ |z|(log |z|)1+ε for all sufficiently-large values of |z|. When σ ≡ 0, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [Phys. D 238 (2009) 209–215] have recently shown that there is finite-time blowup when σ is a nonzero constant. In this paper, we prove that the Bonder–Groisman condition is unimprovable by showing that the reaction–diffusion equation with noise is “typically” well posed when |b(z)| = O(|z| log+ |z|) as |z| → ∞. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.

Subfields: Finance, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/3e082189b63babf833df63cec8f7a5533dad4f2c