The fractional stochastic heat equation driven by time-space white noise
نویسندگان
چکیده
Abstract We study the stochastic time-fractional heat equation $$\begin{aligned} \frac{\partial ^{\alpha }}{\partial t^{\alpha }}Y(t,x)=\lambda \varDelta Y(t,x)+\sigma W(t,x);\; (t,x)\in (0,\infty )\times \mathbb {R}^{d}, \end{aligned}$$ ∂ α t Y ( , x ) = λ Δ + σ W ; ∈ 0 ∞ × R d where $$d\in {N}=\{1,2,...\}$$ N { 1 2 . } and $$\frac{\partial }}$$ is Caputo derivative of order $$\alpha \in (0,2)$$ , $$\lambda >0$$ > $$\sigma {R}$$ are given constants. Here $$\varDelta $$ denotes Laplacian operator, W ( t x ) time-space white noise, defined by W(t,x)=\frac{\partial }{\partial t}\frac{\partial ^{d}B(t,x)}{\partial x_{1}...\partial x_{d}}, B $$B(t,x)=B(t,x,\omega ); t\ge 0, {R}^d, \omega \varOmega ω ≥ Ω being Brownian motion with probability law $$\mathbb {P}$$ P . consider (0.1) in sense distribution, we find an explicit expression for $$\mathcal {S}'$$ S ′ -valued solution Y ), space tempered distributions. Following terminology Y. Hu [11], say that mild if $$Y(t,x) L^2(\mathbb {P})$$ L all It well-known classical case = 1$$ only dimension $$d=1$$ prove (1,2)$$ or $$d=2$$ If < < not any d
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ژورنال
عنوان ژورنال: Fractional Calculus and Applied Analysis
سال: 2023
ISSN: ['1311-0454', '1314-2224']
DOI: https://doi.org/10.1007/s13540-023-00134-7