scholarly journals Wick product in the stochastic Burgers equation: A curse or a cure?

2011 ◽  
Vol 75 (3-4) ◽  
pp. 145-168 ◽  
Author(s):  
S. Kaligotla ◽  
S.V. Lototsky
Keyword(s):  
Burgers Equation ◽  
Wick Product ◽  
Stochastic Burgers Equation

Wick Product and Backward Heat Equation

2005 ◽  
Vol 2 (4) ◽  
pp. 367-379 ◽  
Author(s):  
Alberto Lanconelli
Keyword(s):  
Heat Equation ◽  
Wick Product

scholarly journals Hölder-Type Inequalities for Norms of Wick Products

2008 ◽  
Vol 2008 ◽  
pp. 1-22 ◽  
Author(s):  
Alberto Lanconelli ◽  
Aurel I. Stan
Keyword(s):  
Random Variable ◽  
Upper Bounds ◽  
Minimal Condition ◽  
Wick Product ◽  
Identity Operator ◽  
Second Quantization ◽  
Measurable Functions ◽  
Quantization Operator ◽  
Wick Products ◽  
Finite Moments

Various upper bounds for the L2-norm of the Wick product of two measurable functions of a random variable X, having finite moments of any order, together with a universal minimal condition, are proven. The inequalities involve the second quantization operator of a constant times the identity operator. Some conditions ensuring that the constants involved in the second quantization operators are optimal, and interesting examples satisfying these conditions are also included.

AN EXTENDED STOCHASTIC INTEGRAL AND A WICK CALCULUS ON PARAMETRIZED KONDRATIEV-TYPE SPACES OF MEIXNER WHITE NOISE

2008 ◽  
Vol 11 (04) ◽  
pp. 541-564 ◽  
Author(s):  
N. A. KACHANOVSKY
Keyword(s):  
White Noise ◽  
Stochastic Equation ◽  
Stochastic Integral ◽  
Holomorphic Functions ◽  
Generalized Functions ◽  
Wick Product ◽  
Stochastic Integration ◽  
Type Spaces ◽  
Wick Calculus

Using a general approach that covers the cases of Gaussian, Poissonian, Gamma, Pascal and Meixner measures, we consider an extended stochastic integral and construct elements of a Wick calculus on parametrized Kondratiev-type spaces of generalized functions; consider the interconnection between the extended stochastic integration and the Wick calculus; and give an example of a stochastic equation with a Wick-type nonlinearity. The main results consist of studying the properties of the extended (Skorohod) stichastic integral subject to the particular spaces under consideration; and of studying the properties of a Wick product and Wick versions of holomorphic functions on the parametrized Kondratiev-type spaces. These results are necessary, in particular, in order to describe properties of solutions of normally ordered white noise equations in the "Meixner analysis".

WICK CALCULUS FOR THE SQUARE OF A GAUSSIAN RANDOM VARIABLE WITH APPLICATION TO YOUNG AND HYPERCONTRACTIVE INEQUALITIES

2012 ◽  
Vol 15 (03) ◽  
pp. 1250018 ◽  
Author(s):  
ALBERTO LANCONELLI ◽  
LUIGI SPORTELLI
Keyword(s):  
Gaussian Measure ◽  
Type Inequality ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Wick Product ◽  
Probabilistic Interpretation ◽  
Chi Square ◽  
Wick Calculus

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian7 and Poissonian12 cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.

scholarly journals Interconnection between Wick multiplication and integration on spaces of nonregular generalized functions in the Lévy white noise analysis

2019 ◽  
Vol 11 (1) ◽  
pp. 70-88
Author(s):  
N.A. Kachanovsky ◽  
T.O. Kachanovska
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Stochastic Equations ◽  
Generalized Functions ◽  
Wick Product ◽  
Pettis Integral ◽  
White Noise Analysis ◽  
Lévy White Noise

We deal with spaces of nonregular generalized functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to describe a relationship between Wick multiplication and integration on these spaces. More exactly, we show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); and prove a theorem about a representation of the extended stochastic integral via the Pettis integral from the Wick product of the original integrand by a Lévy white noise. As examples of an application of our results, we consider some stochastic equations with Wick type nonlinearities.

Generalized Bernoulli Wick differential equation

2021 ◽  
Vol 24 (01) ◽  
pp. 2150008
Author(s):  
Sami H. Altoum ◽  
Aymen Ettaieb ◽  
Hafedh Rguigui
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exponential Growth ◽  
Present Method ◽  
Dual Space ◽  
Entire Functions ◽  
Wick Product ◽  
Derivation Operator ◽  
New Type ◽  
Minimal Type

Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in [Formula: see text]. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.

A note on Gamma Wick products

2018 ◽  
Vol 21 (01) ◽  
pp. 1850004 ◽  
Author(s):  
Florin Catrina ◽  
Aurel I. Stan
Keyword(s):  
Integral Representation ◽  
Random Variables ◽  
Hölder Inequality ◽  
Wick Product ◽  
Wick Products

An integral representation of the Wick product for Gamma distributed random variables, with mean greater than [Formula: see text], is presented first. We use this integral representation to prove a Hölder inequality for norms of Gamma Wick products.

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