Fractional time stochastic partial differential equations

Chen, Zhen Qing; Kim, Kyeong Hun; Kim, Panki

doi:10.1016/j.spa.2014.11.005

Cited by 99 publications

(102 citation statements)

References 15 publications

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“…Solutions of time fractional Cauchy as well as Poisson equations have been attracted a lot of attentions in the community of analysis, PDEs and stochastic analysis, see [1,6,12,15,17,18,28,31,38,39] and the references therein. We note that most of quoted papers are concentrated on the Caputo derivative of fractional order.…”

Section: Introductionmentioning

confidence: 99%

Time fractional Poisson equations: Representations and estimates

Chen

Kim

Kumagai

et al. 2020

Journal of Functional Analysis

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In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator L and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel q(t, x, y), which we call the fundamental solution for the time fractional Poisson equation, if the semigroup for L has an integral kernel. We further show that q(t, x, y) can be expressed as a time fractional derivative of the fundamental solution for the homogenous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator. Moreover, when the Laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self-decomposable, we establish two-sided estimates for the fundamental solution q(t, x, y) through explicit estimates of transition density functions of subordinators. ∞ 0 min{1, x}(−dw(x)) < ∞, a generalized time fractional derivative with weight w is defined by∂ w t f (t) := d dt t 0 whenever the right hand side is well defined. See [8, 13, 24, 27]. In particular, when w(s) = 1 Γ(1−β) s −β for β ∈ (0, 1) (where Γ(t) = ∞ 0 s t−1 e −s ds is the Gamma function), ∂ w t f is just the Caputo derivative of order β, i.e., ∂ β t f (t) := 1 Γ(1 − β) d dt t 0 Zhen-Qing Chen

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Section: Introductionmentioning

confidence: 99%

Time fractional Poisson equations: Representations and estimates

Chen

Kim

Kumagai

et al. 2020

Journal of Functional Analysis

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“…with real numbers b, ̺, σ, c as parameters. It is well known that On other hand, a perturbation-type argument [4] shows that the following fractional version of (6.1),…”

Section: Stochastic Fractional Parabolicity Conditionsmentioning

confidence: 99%

“…∂ β t f (t) = f (0+) = lim t→0,t>0 f (t), seems to achieve the right balance between mathematical utility and physical relevance [8] and has been recently used in the study of large classes of stochastic partial differential equations [4,11].…”

Section: Introductionmentioning

confidence: 99%

Classical and generalized solutions of fractional stochastic differential equations

Lototsky

Rozovsky

2019

Stoch PDE: Anal Comp

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For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise, there can be a generalized solution in suitable weighted chaos spaces. Presence of fractional derivatives in time leads to various modifications of the stochastic parabolicity condition. Interesting new effects appear when the order of the time derivative in the noise term is less than or equal to one-half.November 1, 20182010 Mathematics Subject Classification. Primary 60G22; Secondary 26A33, 60G20.

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“…Let α ∈ (0, 2), β ∈ (0, ∞) and p(t, x) be the fundamental solution to the space-time fractional differential equation Equation (1.1) has been an important topic in the mathematical physics related to non-Markovian diffusion processes with a memory [26,27,28,29], in the probability theory related to jump processes [5,6] and in the theory of differential equations [7,8,17,32,37]. If α ∈ (0, 1), then the fractional time derivative of order α can be used to model the anomalous diffusion exhibiting subdiffusive behavior, due to particle sticking and trapping phenomena and the fractional spatial derivative describes long range jumps of particles.…”

Section: Introductionmentioning

confidence: 99%

“…Our second application is the L p -theory of the stochastic partial differential equations of the type and W t is a Wiener process defined on a probability space (Ω, dP ). One can show (see [5]) that the solution to this problem is given by the formula u(t, x) =ˆt 0 ˆR d P σ (t − s, x − y)g(s, y)dy dW s .…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behaviors of Fundamental Solution and Its Derivatives to Fractional Diffusion-Wave Equations

Kim¹,

Lim²

2016

Journal of the Korean Mathematical Society

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Abstract. Let p(t, x) be the fundamental solution to the problemIf α, β ∈ (0, 1), then the kernel p(t, x) becomes the transition density of a Lévy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of p(t, x) and its space and time fractional derivativesx is a partial derivative of order n with respect to x, (−∆x) γ is a fractional Laplace operator and D σ t and I δ t are Riemann-Liouville fractional derivative and integral respectively.

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Fractional time stochastic partial differential equations

Cited by 99 publications

References 15 publications

Time fractional Poisson equations: Representations and estimates

Time fractional Poisson equations: Representations and estimates

Classical and generalized solutions of fractional stochastic differential equations

Asymptotic Behaviors of Fundamental Solution and Its Derivatives to Fractional Diffusion-Wave Equations

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