RECORD
Space-time fractional diffusions in Gaussian noisy environment
Probability · Submitted 2 Aug 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^\beta + \frac{\nu}{2} (-\Delta)^{\alpha/2} \right) u(t,x)= \lambda u(t,x) \dot{W}(t,x)$, where $\dot W$ is a general Gaussian noise and $\beta\in (1/2, 2)$, $\alpha\in (0, 2]$. The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^\beta + \frac{\nu}{2} (-\Delta)^{\alpha/2} \right) u(t,x)= \lambda u(t,x) \dot{W}(t,x)$, where $\dot W$ is a general Gaussian noise and $\beta\in (1/2, 2)$, $\alpha\in (0, 2]$. | contribution 0.82 | 0.61 | 0.06 | asserted 0.97 |
| s1 | The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. | result 0.77 | 0.94 | 0.50 | asserted 1.00 |
| s2 | Along the way, we obtain some new properties of the fundamental solutions. | contribution 0.70 | 0.94 | 0.04 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^\beta + \frac{\nu}{2} (-\Delta)^{\alpha/2} \right) u(t,x)= \lambda u(t,x) \dot{W}(t,x)$, where $\dot W$ is a general Gaussian noise and $\beta\in (1/2, 2)$, $\alpha\in (0, 2]$. The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.
Human review
How the literature uses it
2 arXiv papers cite this in 9 sentences
Papers citing it · 6 sentences without a confident label
Builds on it 1 paper
-
These results generalize those obtained by Mainardi et al [ref] , Pskhu [ref] , and Chen et al [ref] .
Nonlinear stochastic time-fractional slow and fast diffusion equations on $\mathbb{R}^d$ · Introduction · 0.98
Compares with it 1 paper
-
When $γ=0$ or $γ=β-β$, the expressions for $Z$, $Y$ and $Z^*$ in Theorem [ref] recover the results in [ref] .
Nonlinear stochastic time-fractional slow and fast diffusion equations on $\mathbb{R}^d$ · Some special cases · 0.90
Cites it as background 1 paper
-
For related time fractional stochastic equations with different noise terms see [ref] .
Intermittency fronts for space-time fractional stochastic partial differential equations in $(d+1)$ dimensions · Introduction · 0.92
What it cites · 6 sentences without a confident label
Compares with 1 paper
-
Note that this condition on $β$ is more restrictive than the condition $β∈(0,2)$ in [ref] .
Nonlinear stochastic time-fractional diffusion equations on $\mathbb{R}$: moments, H\"older regularity and intermittency · The smoothed equation · 0.99
Cites as background 2 papers
-
In a series of papers [ref] , Nane and his coauthors studied the case $α∈(0,2]$.
Intermittence and time fractional stochastic partial differential equations · Introduction · 0.92
-
In a series of papers [ref] , Nane and his coauthors studied the case $α∈(0,2]$.
Asymptotic properties of some space-time fractional stochastic equations · Introduction · 0.92
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.