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A Sobolev Space theory for stochastic partial differential equations with time-fractional derivatives
Probability · Submitted 6 May 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^{\alpha}_tu=L(\omega,t,x)u+f(u)+\partial^{\beta}_t \sum_{k=1}^{\infty}\int^t_0 ( \Lambda^k(\omega,t,x)u+g^k(u))dw^k_t, $$ where $\alpha\in (0,2)$, $\beta <\alpha+\frac{1}{2}$, and $\partial^{\alpha}_t$ and $\partial^{\beta}_t$ denote the Caputo derivative of order $\alpha$ and $\beta$ respectively.
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In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^{\alpha}_tu=L(\omega,t,x)u+f(u)+\partial^{\beta}_t \sum_{k=1}^{\infty}\int^t_0 ( \Lambda^k(\omega,t,x)u+g^k(u))dw^k_t, $$ where $\alpha\in (0,2)$, $\beta <\alpha+\frac{1}{2}$, and $\partial^{\alpha}_t$ and $\partial^{\beta}_t$ denote the Caputo derivative of order $\alpha$ and $\beta$ respectively. The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are independent one-dimensional Wiener processes defined on a probability space $\Omega$, $L$ is a second order operator of either divergence or non-divergence type, and $\Lambda^k$ are linear operators of order up to two. The coefficients of the equations depend on $\omega (\in \Omega), t,x$ and are allowed to be discontinuous. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^{\alpha}_tu=L(\omega,t,x)u+f(u)+\partial^{\beta}_t \sum_{k=1}^{\infty}\int^t_0 ( \Lambda^k(\omega,t,x)u+g^k(u))dw^k_t, $$ where $\alpha\in (0,2)$, $\beta <\alpha+\frac{1}{2}$, and $\partial^{\alpha}_t$ and $\partial^{\beta}_t$ denote the Caputo derivative of order $\alpha$ and $\beta$ respectively. | contribution 1.00 | 0.90 | 0.38 | asserted 1.00 |
| s1 | The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are independent one-dimensional Wiener processes defined on a probability space $\Omega$, $L$ is a second order operator of either divergence or non-divergence type, and $\Lambda^k$ are linear operators of order up to two. | background 0.61 | 0.22 | 0.03 | asserted 1.00 |
| s2 | The coefficients of the equations depend on $\omega (\in \Omega), t,x$ and are allowed to be discontinuous. | contribution 0.41 | 0.34 | 0.03 | asserted 0.90 |
| s3 | This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping. | interpretation 0.65 | 0.12 | 0.03 | hedged possibility 0.75 |
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Original abstract
In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^{\alpha}_tu=L(\omega,t,x)u+f(u)+\partial^{\beta}_t \sum_{k=1}^{\infty}\int^t_0 ( \Lambda^k(\omega,t,x)u+g^k(u))dw^k_t, $$ where $\alpha\in (0,2)$, $\beta <\alpha+\frac{1}{2}$, and $\partial^{\alpha}_t$ and $\partial^{\beta}_t$ denote the Caputo derivative of order $\alpha$ and $\beta$ respectively. The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are independent one-dimensional Wiener processes defined on a probability space $\Omega$, $L$ is a second order operator of either divergence or non-divergence type, and $\Lambda^k$ are linear operators of order up to two. The coefficients of the equations depend on $\omega (\in \Omega), t,x$ and are allowed to be discontinuous. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping.
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What it cites · 12 sentences without a confident label
Uses the method or tool of 1 paper
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For the general case, it is enough to repeat Steps 5 and 6 of the proof of [ref] .
An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients · Introduction · 0.84
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If $γ=0$ then this lemma is proved in [ref] under the condition that $a^ij_n$ are uniformly continuous in $(t,x)$, but without the condition $|a^ij|_B^|γ|≤K_3$.
An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients · Introduction · 0.83
Cites as background 2 papers
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Also see [ref] for $BUC_1-β([0,T];X)$-type estimates, [ref] for Schauder estimates, [ref] for DeGirogi-Nash type estimate, and [ref] for Harnack inequality.
A weak Harnack inequality for fractional evolution equations with discontinuous coefficients · Introduction · 0.96
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For the results on deterministic equations, we refer the reader e.g. to [ref] ($L_2$-theory), [ref] ($L_p$-theory), and [ref] ($L_q(L_p)$-theory).
An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients · Introduction · 0.92
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