Study of a fractional stochastic heat equation

Abstract

In this article, we study a d-dimensional stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional space-time white noise: ∂ t u − ∆u = ρ 2 u 2 +Ḃ , t ∈ [0, T ] , x ∈ R d , u 0 = φ. Two types of regimes are exhibited, depending on the ranges of the Hurst index H = (H 0 , ..., H d) ∈ (0, 1) d+1. In particular, we show that the local well-posedness of (SNLH) resulting from the Da Prato-Debussche trick, is easily obtained when 2H 0 + d i=1 H i > d. On the contrary, (SNLH) is much more difficult to handle when 2H 0 + d i=1 H i ≤ d. In this case, the model has to be interpreted in the Wick sense, thanks to a time-dependent renormalization. Helped with the regularising effect of the heat semigroup, we establish local well-posedness results for (SNLH) for all dimension d ≥ 1. Contents