scholarly journals Couplings via Comparison Principle and Exponential Ergodicity of SPDEs in the Hypoelliptic Setting

2020 ◽  
Vol 379 (3) ◽  
pp. 1001-1034
Author(s):  
Oleg Butkovsky ◽  
Michael Scheutzow
Keyword(s):  
Existence And Uniqueness ◽  
General Framework ◽  
Exponential Convergence ◽  
Transition Probabilities ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Optimal Conditions ◽  
Markov Semigroups ◽  
Exponential Ergodicity ◽  
Stochastic Reaction

Abstract We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise of the stochastic reaction–diffusion equation in the hypoelliptic setting. This refines and complements corresponding results of Hairer and Mattingly (Electron J Probab 16:658–738, 2011).

Gaussian estimates on networks with dynamic stochastic boundary conditions

2017 ◽  
Vol 20 (01) ◽  
pp. 1750001 ◽  
Author(s):  
Francesco Cordoni ◽  
Luca Di Persio
Keyword(s):  
Boundary Conditions ◽  
Stochastic Optimal Control ◽  
Existence And Uniqueness ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Type Estimate ◽  
Stochastic Optimal Control Problem ◽  
Dynamic Stochastic ◽  
Stochastic Reaction ◽  
Stochastic Boundary

In this paper we prove the existence and uniqueness for the solution to a stochastic reaction–diffusion equation, defined on a network, and subjected to nonlocal dynamic stochastic boundary conditions. The result is obtained by deriving a Gaussian-type estimate for the related leading semigroup, under rather mild regularity assumptions on the coefficients. An application of the latter to a stochastic optimal control problem on graphs, is also provided.

scholarly journals Regularity of a Stochastic Fractional Delayed Reaction-Diffusion Equation Driven by Lévy Noise

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Tianlong Shen ◽  
Jianhua Huang ◽  
Jin Li
Keyword(s):  
Fixed Point ◽  
Diffusion Equation ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Banach Fixed Point Theorem ◽  
Fractional Operator ◽  
Initial Value ◽  
Delayed Reaction

The current paper is devoted to the regularity of the mild solution for a stochastic fractional delayed reaction-diffusion equation driven by Lévy space-time white noise. By the Banach fixed point theorem, the existence and uniqueness of the mild solution are proved in the proper working function space which is affected by the delays. Furthermore, the time regularity and space regularity of the mild solution are established respectively. The main results show that both time regularity and space regularity of the mild solution depend on the regularity of initial value and the order of fractional operator. In particular, the time regularity is affected by the regularity of initial value with delays.

scholarly journals Existence and Uniqueness of Weak Solution for chemotaxis model coupled with heat equation

2021 ◽  
Author(s):  
Ali slimani ◽  
Amar Guesmia
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Heat Equation ◽  
Existence And Uniqueness ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Chemical Concentration ◽  
Chemotaxis Model ◽  
Convection Diffusion Equation ◽  
Global Existence And Uniqueness

Keller-Segel chemotaxis model is described by a system of nonlinear PDE : a convection diffusion equation for the cell density coupled with a reaction-diffusion equation for chemoattractant concentration. In this work, we study the phenomenon of Keller Segel model coupled with a heat equation, because The heat has an effect the density of the cells as well as the signal of chemical concentration, since the heat is a factor affecting the spread and attraction of cells as well in relation to the signal of chemical concentration, The main objectives of this work is the study of the global existence and uniqueness and boundedness of the weak solution for the problem defined in (8) for this we use the technical of Galerkin method.

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