Large Deviation Principle for Stochastic Reaction–Diffusion Equations with Superlinear Drift on $$\mathbb {R}$$ R Driven by Space–Time White Noise

Yue Li (), Shijie Shang () and Jianliang Zhai ()
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Yue Li: University of Science and Technology of China
Shijie Shang: University of Science and Technology of China
Jianliang Zhai: University of Science and Technology of China

Journal of Theoretical Probability, 2024, vol. 37, issue 4, 3496-3539

Abstract: Abstract In this paper, we consider stochastic reaction–diffusion equations with superlinear drift on the real line $$\mathbb {R}$$ R driven by space–time white noise. A Freidlin–Wentzell large deviation principle is established by a modified weak convergence method on the space $$C([0,T], C_\textrm{tem}(\mathbb {R}))$$ C ( [ 0 , T ] , C tem ( R ) ) , where $$C_\textrm{tem}(\mathbb {R}):=\{f\in C(\mathbb {R}): \sup _{x\in \mathbb {R}} \left( |f(x)|e^{-\lambda |x|}\right) 0\}$$ C tem ( R ) : = { f ∈ C ( R ) : sup x ∈ R | f ( x ) | e - λ | x | 0 } . Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space–time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the specially designed family of norms on the Fréchet space $$C([0,T], C_\textrm{tem}(\mathbb {R}))$$ C ( [ 0 , T ] , C tem ( R ) ) , one-order moment estimates of the stochastic convolution, and two nonlinear Gronwall-type inequalities play an important role.

Keywords: Stochastic reaction–diffusion equation; Large deviation principle; Unbounded domain; Space–time white noise; Weak convergence method; Superlinear drift term; Primary 60H15; Secondary 60F10; 35R60 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-024-01345-1

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