Stochastic heat equation with fractional Laplacian and fractional noise: existence of the solution and analysis of its density

2017 ◽  
Vol 37 (6) ◽  
pp. 1545-1566 ◽  
Author(s):  
Junfeng LIU ◽  
Ciprian A. TUDOR
Keyword(s):  
Heat Equation ◽  
Fractional Laplacian ◽  
Stochastic Heat Equation ◽  
Fractional Noise

scholarly journals Sample paths of the solution to the fractional-colored stochastic heat equation

2016 ◽  
Vol 17 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Ciprian A. Tudor ◽  
Yimin Xiao
Keyword(s):  
Spatial Structure ◽  
Heat Equation ◽  
Space Variable ◽  
Stochastic Heat Equation ◽  
Moduli Of Continuity ◽  
Fractional Noise ◽  
Sample Paths ◽  
Iterated Logarithm ◽  
Path Properties ◽  
Laws Of Iterated Logarithm

Let [Formula: see text] be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the process [Formula: see text] with respect to the time and to the space variable, respectively. In particular, we derive exact uniform moduli of continuity and Chung-type laws of iterated logarithm.

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

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.

scholarly journals Mixed Fractional Heat Equation Driven by Fractional Brownian Sheet and Lévy Process

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Dengfeng Xia ◽  
Litan Yan ◽  
Weiyin Fei
Keyword(s):  
Fixed Point ◽  
Heat Equation ◽  
White Noise ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Brownian Sheet ◽  
Fractional Noise ◽  
Fractional Heat Equation ◽  
Fixed Point Principle ◽  
Uniqueness Of The Solution

We consider the stochastic heat equation of the form∂u/∂t=(Δ+Δα)u+(∂f/∂x)(t,x,u)+σ(t,x,u)L˙+W˙H,whereW˙His the fractional noise,L˙is a (pure jump) Lévy space-time white noise,Δis Laplacian, andΔα=-(-Δ)α/2is the fractional Laplacian generator onR, andf,σ:[0,T]×R×R→Rare measurable functions. We introduce the existence and uniqueness of the solution by the fixed point principle under some suitable assumptions.

scholarly journals On the distribution and q-variation of the solution to the heat equation with fractional Laplacian

2019 ◽  
Vol 39 (2) ◽  
pp. 315-335
Author(s):  
Ciprian A. Tudor ◽  
Zeina Mahdi Khalil
Keyword(s):  
Probability Distribution ◽  
Heat Equation ◽  
White Noise ◽  
Fractional Laplacian ◽  
Stochastic Heat Equation ◽  
Correlated Noise

We study the probability distribution of the solution to the linear stochastic heat equation with fractional Laplacian and white noise in time and white or correlated noise in space. As an application, we deduce the behavior of the q-variations of the solution in time and in space.

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