White noise in space and time as the time-derivative of a cylindrical Wiener process

<p style='text-indent:20px;'>We consider a class of one-dimensional nonlinear stochastic parabolic problems associated to Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time.</p>

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White noise delta functions and infinite-dimensional Laplacians

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025720500289 ◽

2020 ◽

pp. 2050028

Author(s):

Luigi Accardi ◽

Ai Hasegawa ◽

Un Cig Ji ◽

Kimiaki Saitô

Keyword(s):

Brownian Motion ◽

White Noise ◽

Time Derivative ◽

Delta Function ◽

Itô Formula ◽

Ito Formula ◽

Infinite Dimensional ◽

Delta Functions ◽

Definition Of ◽

White Noise Distribution

In this paper, we introduce a new white noise delta function based on the Kubo–Yokoi delta function and an infinite-dimensional Brownian motion. We also give a white noise differential equation induced by the delta function through the Itô formula introducing a differential operator directed by the time derivative of the infinite-dimensional Brownian motion and an extension of the definition of the Volterra Laplacian. Moreover, we give an extension of the Itô formula for the white noise distribution of the infinite-dimensional Brownian motion.

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LARGE DEVIATION ASYMPTOTICS FOR RANDOM-WALK TYPE PERTURBATIONS

Stochastics and Dynamics ◽

10.1142/s0219493707001950 ◽

2007 ◽

Vol 07 (01) ◽

pp. 75-89

Author(s):

ZHIHUI YANG

Keyword(s):

Random Walk ◽

Random Walks ◽

White Noise ◽

Wiener Process ◽

Stochastic Resonance ◽

Large Deviation ◽

Correction Term ◽

Exit Problem ◽

Wiener Processes ◽

Noise Type

Symmetric random walks can be arranged to converge to a Wiener process in the area of normal deviation. However, random walks and Wiener processes have, in general, different asymptotics of the large deviation probabilities. The action functionals for random-walks and Wiener processes are compared in this paper. The correction term is calculated. Exit problem and stochastic resonance for random-walk-type perturbation are also considered and compared with the white-noise-type perturbation.

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