Sticky-reflected stochastic heat equation driven by colored noise

UDC 519.21 We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line, but now the solution obeys the usual stochastic heat equation except for points where it reaches zero. The solution has no noise at zero and a drift pushes it to stay positive. The proof is based on a new approach that can also be applied to other types of SPDEs with discontinuous coefficients.

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Lp-Theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025721500107 ◽

2021 ◽

Vol 24 (02) ◽

pp. 2150010

Author(s):

Jingqi Han ◽

Litan Yan

Keyword(s):

Brownian Motion ◽

Heat Equation ◽

Fractional Brownian Motion ◽

Stochastic Heat Equation ◽

Index Formula ◽

Hurst Index ◽

Skorohod Integral ◽

Fractional Brownian Motions ◽

Infinite Dimensional ◽

Lp Theory

In this paper, we study the [Formula: see text]-theory of the fractional time stochastic heat equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] denotes the Caputo derivative of order [Formula: see text], and [Formula: see text] is a sequence of i.i.d. fractional Brownian motions with a same Hurst index [Formula: see text]. The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood–Paley inequality, and prove the existence and uniqueness of [Formula: see text]-solution to such equation.

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Fractional stochastic heat equation with piecewise constant coefficients

Stochastics and Dynamics ◽

10.1142/s0219493721500027 ◽

2020 ◽

Vol 21 (01) ◽

pp. 2150002

Author(s):

Yuliya Mishura ◽

Kostiantyn Ralchenko ◽

Mounir Zili ◽

Eya Zougar

Keyword(s):

Diffusion Coefficient ◽

Heat Equation ◽

Divergence Form ◽

Stochastic Heat Equation ◽

Piecewise Constant ◽

Constant Diffusion Coefficient ◽

Infinite Dimensional ◽

Order Elliptic Operator ◽

Constant Diffusion ◽

Constant Coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

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Gaussian fluctuations for the stochastic heat equation with colored noise

Stochastic Partial Differential Equations Analysis and Computations ◽

10.1007/s40072-019-00149-3 ◽

2019 ◽

Vol 8 (2) ◽

pp. 402-421

Author(s):

Jingyu Huang ◽

David Nualart ◽

Lauri Viitasaari ◽

Guangqu Zheng

Keyword(s):

Heat Equation ◽

Colored Noise ◽

Stochastic Heat Equation ◽

Gaussian Fluctuations ◽

The Stochastic Heat Equation

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Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

Stochastic Analysis and Applications ◽

10.1080/07362994.2020.1721301 ◽

2020 ◽

Vol 38 (4) ◽

pp. 747-768

Author(s):

Randall Herrell ◽

Renming Song ◽

Dongsheng Wu ◽

Yimin Xiao

Keyword(s):

Heat Equation ◽

Colored Noise ◽

Space Time ◽

Stochastic Heat Equation

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Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise

The European Physical Journal Plus ◽

10.1140/epjp/s13360-021-01864-4 ◽

2021 ◽

Vol 136 (9) ◽

Author(s):

Tran Thanh Binh ◽

Nguyen Huy Tuan ◽

Tran Bao Ngoc

Keyword(s):

Heat Equation ◽

Colored Noise ◽

Hölder Continuity ◽

Mild Solutions ◽

Space Time ◽

Stochastic Heat Equation ◽

Holder Continuity

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Erratum: A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand

Electronic Communications in Probability ◽

10.1214/ecp.v17-1774 ◽

2012 ◽

Vol 17 (0) ◽

Cited By ~ 2

Author(s):

Carl Mueller ◽

Zhixin Wu

Keyword(s):

Brownian Motion ◽

Heat Equation ◽

Fractional Brownian Motion ◽

Simple Proof ◽

Stochastic Heat Equation ◽

The Stochastic Heat Equation

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On the law of the solution to a stochastic heat equation with fractional noise in time

Random Operators and Stochastic Equations ◽

10.1515/rose-2014-0038 ◽

2015 ◽

Vol 23 (3) ◽

Author(s):

Solesne Bourguin ◽

Ciprian A. Tudor

Keyword(s):

Brownian Motion ◽

Heat Equation ◽

Fractional Brownian Motion ◽

Gaussian Noise ◽

Stochastic Heat Equation ◽

Bifractional Brownian Motion ◽

Fractional Noise ◽

The Stochastic Heat Equation ◽

The Law

AbstractWe study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the bifractional Brownian motion. Our result is an extension of a result by Swanson.

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Recent developments on stochastic heat equation with additive fractional-colored noise

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-014-0164-6 ◽

2014 ◽

Vol 17 (1) ◽

Cited By ~ 3

Author(s):

Ciprian Tudor

Keyword(s):

Comparative Study ◽

Heat Equation ◽

Gaussian Noise ◽

Colored Noise ◽

Covariance Structure ◽

Stochastic Heat Equation ◽

Basic Properties ◽

Recent Developments

AbstractWe expose some recent and less recent results related to the existence and the basic properties of the solution to the linear stochastic heat equation with additive Gaussian noise. We will make a comparative study of the behavior of the solution in function of the covariance structure of the driving noise.

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