Speaker: Rahma Yasmina Moulay Hachemi (University of Saida Dr. Moulay Tahar)
Title: The time-fractional stochastic heat equation driven by time-space white noise
Abstract: We study the stochastic time-fractional stochastic heat equation:
∂α/∂tα Y(t, x) = λΔY (t, x) + σ W(t, x); (t, x) ∈ (0,∞) × Rd , (0.1), where d ∈ N = {1, 2, ...} and ∂α/∂tα is the Caputo derivative of order α ∈ (0, 2), and λ > 0 and σ ∈ R are given constants.
We consider the equation in the sense of distribution, and we find an explicit expression for the S-valued solution Y (t, x), where S is the space of tempered distributions.
Following the terminology of Y. Hu [1], we say that the solution is mild if Y (t, x) ∈ L2(P) for all t, x. It is well-known that in the classical case with α = 1, the solution is mild if and only if the space dimension d = 1. We prove that if α ∈ (1, 2) the solution is mild if d = 1 or d = 2. If α < 1 we prove that the solution is not mild for any d.
[1] Hu, Yaozhong (2019): Some recent progress on stochastic heat equations. Acta Mathematica Scientia 39B(3); 874-914
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