scholarly journals Generalized Mehler Semigroup on White Noise Functionals and White Noise Evolution Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1025
Author(s):  
Un Cig Ji ◽  
Mi Ra Lee ◽  
Peng Cheng Ma
Keyword(s):  
Evolution Equation ◽  
White Noise ◽  
Explicit Form ◽  
Infinitesimal Generator ◽  
Evolution Equations ◽  
Time Derivative ◽  
White Noise Functionals

In this paper, we study a representation of generalized Mehler semigroup in terms of Fourier–Gauss transforms on white noise functionals and then we have an explicit form of the infinitesimal generator of the generalized Mehler semigroup in terms of the conservation operator and the generalized Gross Laplacian. Then we investigate a characterization of the unitarity of the generalized Mehler semigroup. As an application, we study an evolution equation for white noise distributions with n-th time-derivative of white noise as an additive singular noise.

scholarly journals Infinite dimensional rotations and Laplacians in terms of white noise calculus

1992 ◽  
Vol 128 ◽  
pp. 65-93 ◽  
Author(s):  
Takeyuki Hida ◽  
Nobuaki Obata ◽  
Kimiaki Saitô
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Quantum Physics ◽  
Variational Calculus ◽  
Rotation Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian ◽  
White Noise Functionals ◽  
White Noise Calculus

The theory of generalized white noise functionals (white noise calculus) initiated in [2] has been considerably developed in recent years, in particular, toward applications to quantum physics, see e.g. [5], [7] and references cited therein. On the other hand, since H. Yoshizawa [4], [23] discussed an infinite dimensional rotation group to broaden the scope of an investigation of Brownian motion, there have been some attempts to introduce an idea of group theory into the white noise calculus. For example, conformal invariance of Brownian motion with multidimensional parameter space [6], variational calculus of white noise functionals [14], characterization of the Levy Laplacian [17] and so on.

Analytic characterizations of infinite dimensional distributions

2017 ◽  
Vol 20 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Luigi Accardi ◽  
Un Cig Ji ◽  
Kimiaki Saitô
Keyword(s):  
White Noise ◽  
Characterization Theorem ◽  
Growth Conditions ◽  
Necessary Condition ◽  
Infinite Dimensional ◽  
S Transform ◽  
White Noise Functionals

We revisit the analytic characterization theorem for S-transform of infinite dimensional distributions. Then we prove that the nuclearity of the space of test white noise functionals is a necessary condition for the characterization of the S-transform in terms of analytic and growth conditions.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

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