scholarly journals Properties of $\varphi$-sub-Gaussian stochastic processes related to the heat equation with random initial conditions

2020 ◽  
pp. 17-24
Author(s):  
O. Hopkalo ◽  
L. Sakhno ◽  
O. Vasylyk
Keyword(s):  
Heat Equation ◽  
Stochastic Processes ◽  
Initial Conditions ◽  
Unbounded Domains ◽  
Stationary Processes ◽  
Sample Paths ◽  
Random Initial Conditions ◽  
Gaussian Stochastic Processes

In this paper, there are studied sample paths properties of stochastic processes representing solutions (in $L_2(\Omega)$ sense) of the heat equation with random initial conditions given by $\varphi$-sub-Gaussian stationary processes. The main results are the bounds for the distributions of the suprema for such stochastic processes considered over bounded and unbounded domains.

scholarly journals Sample Paths Properties of Stochastic Processes from Orlicz Spaces, with Applications to Partial Differential Equations

2020 ◽  
Vol 8 (3) ◽  
pp. 722-739
Author(s):  
Lyudmyla Sakhno ◽  
Yuriy Kozachenko ◽  
Enzo Orsingher ◽  
Olha Hopkalo
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Processes ◽  
Orlicz Spaces ◽  
Unbounded Domains ◽  
Partial Differential ◽  
Sample Paths

In the present paper we obtain conditions for stochastic processes from Orlicz spaces to have almost sure bounded and continuous sample paths, the study is concerned with the processes defined on unbounded domains. Estimates for the distributions of suprema of the processes are also presented. Conditions are given in terms of entropy integrals and majorant characteristics of Orlicz spaces. Possible applications to solutions of partial differential equations are discussed. Examples of processes are given for which conditions of the main results are satisfied.

scholarly journals The Cauchy problem for the heat equation on the plane with a random right part from the Orlicz space

2020 ◽  
pp. 103-109
Author(s):  
A. I. Slyvka-Tylyshchak ◽  
M. M. Mykhasiuk ◽  
O. O. Pohoriliak
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Orlicz Space ◽  
Initial Conditions ◽  
Classical Problem ◽  
Mathematical Physics ◽  
Heat Equations ◽  
Random Initial Conditions ◽  
The Cauchy Problem ◽  
The Right

The heat equation with random conditions is a classical problem of mathematical physics. Recently, a number of works appeared, which in many ways investigated this equation according to the type of random initial conditions. We consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on the plane with a random right part. We consider the right part as a random function of the Orlicz space. The conditions of existence with probability one classical solution of the problem are investigated. For such a problem has been got the estimation for the distribution of the supremum solution.

Evaluations of absorption probabilities for the Wiener process on large intervals

1980 ◽  
Vol 17 (02) ◽  
pp. 363-372 ◽  
Author(s):  
C. Park ◽  
F. J. Schuurmann
Keyword(s):  
Stochastic Processes ◽  
Wiener Process ◽  
Standard Wiener Process ◽  
Computationally Efficient ◽  
The Past ◽  
Gaussian Stochastic Processes ◽  
Absorption Probabilities

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦T W(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.

Tolerant control for non-Gaussian stochastic processes with unknown faults via two-step fuzzy modeling

2018 ◽  
Vol 95 (4) ◽  
pp. 2703-2716 ◽  
Author(s):  
Yang Yi ◽  
Liren Shao ◽  
Xiangxiang Fan ◽  
Tianping Zhang
Keyword(s):  
Stochastic Processes ◽  
Fuzzy Modeling ◽  
Non Gaussian ◽  
Gaussian Stochastic Processes
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