Hölder estimates of mild solutions for nonlocal SPDEs

Tian, Rongrong; Ding, Liang; Wei, Jinlong; Zheng, S.M.

doi:10.1186/s13662-019-2097-1

Cited by 4 publications

(4 citation statements)

References 17 publications

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“…Comparing with the earlier results of [28] (Tian et al obtained the Hölder estimate to equation (3.1) locally in R d ), we find the Hölder continuous index in this paper is larger than that in [28]. More precisely, we obtain the index of time variable is closed to 1/2.…”

Section: Preliminariessupporting

confidence: 79%

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On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations

Gao

Wei

et al. 2023

DCDS-B

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<p style='text-indent:20px;'>In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Campanato estimates and Sobolev embedding theorem, we first show the Hölder continuity (locally in the whole state space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula>) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions belong to the space <inline-formula><tex-math id="M2">\begin{document}$ C^{\gamma}(D_T;L^p(\Omega)) $\end{document}</tex-math></inline-formula> with the optimal Hölder continuity index <inline-formula><tex-math id="M3">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> (which is given explicitly), where <inline-formula><tex-math id="M4">\begin{document}$ D_T: = [0, T]\times D $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ T>0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M6">\begin{document}$ D\subset\mathbb{R}^d $\end{document}</tex-math></inline-formula> being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in <inline-formula><tex-math id="M7">\begin{document}$ L^p(\Omega;C^{\gamma^*}(D_T)) $\end{document}</tex-math></inline-formula>. What's more, we give an explicit formula between the two indexes <inline-formula><tex-math id="M8">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \gamma^* $\end{document}</tex-math></inline-formula>. Moreover, we prove Hölder continuity for mild solutions on bounded domains. Finally, we present a new criterion to justify Hölder continuity for the solutions on bounded domains. The novelty of this paper is that our method is suitable to the case of space-time white noise.</p>

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Section: Preliminariessupporting

confidence: 79%

“…Kuksin-Nadirashvili-Piatnitski [22] obtained Hölder estimates for solutions of parabolic SPDEs on bounded domains. Most recently, Tian-Ding-Wei [28] derived the local Hölder estimates of mild solutions of stochastic nonlocal diffusion equations by using tail estimates [22]. Wang [30] generalized the results of [24] to nonlocal diffusion equation.…”

mentioning

confidence: 99%

On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations

Gao

Wei

et al. 2023

DCDS-B

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“…Comparing with the earlier results of [22] (Tian et al obtained the Hölder estimate to equation (3.1) locally in R d ), we find the Hölder continuous index in this paper is larger than that in [22]. More precisely, we obtain the index of time variable is closed to 1/2.…”

Section: Introductionsupporting

confidence: 79%

Regularity of stochastic nonlocal diffusion equations

Lv,

Gao,

Wei

et al. 2018

Preprint

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In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Companato estimates and Sobolev embedding theorem, we first show the Hölder continuity (locally in the whole state space R d ) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions u belong to the space C γ (D T ; L p (Ω)) with the optimal Hölder continuity index γ (which is given explicitly), where D T := [0, T ] × D for T > 0, and D ⊂ R d being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in L p (Ω; C γ * (D T )). What's more, we give an explicit formula between the two index γ and γ * . Moreover, we prove Hölder continuity for mild solutions on bounded domains. Finally, we present a new criteria to justify Hölder continuity for the solutions on bounded domains. The novelty of this paper is that our method are suitable to the case of time-space white noise.

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“…For more details in this direction, we refer to [10][11][12][13]. For some extensions and applications, we refer to [14][15][16][17][18] and the references cited therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Stochastic differential equations with singular coefficients on the straight line

2020

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Consider the following stochastic differential equation (SDE): $$ X_{t}=x+ \int _{0}^{t}b(s,X_{s})\,ds+ \int _{0}^{t}\sigma (s,X_{s}) \,dB_{s}, \quad 0\leq t\leq T, x\in \mathbb{R}, $$ X t = x + ∫ 0 t b ( s , X s ) d s + ∫ 0 t σ ( s , X s ) d B s , 0 ≤ t ≤ T , x ∈ R , where $\{B_{s}\}_{0\leq s\leq T}$ { B s } 0 ≤ s ≤ T is a 1-dimensional standard Brownian motion on $[0,T]$ [ 0 , T ] . Suppose that $q\in (1,\infty ]$ q ∈ ( 1 , ∞ ] , $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) , $b=b_{1}+b_{2}$ b = b 1 + b 2 , $b_{1}\in L^{q}(0,T;L^{p}(\mathbb{R}))$ b 1 ∈ L q ( 0 , T ; L p ( R ) ) such that $1/p+2/q<1$ 1 / p + 2 / q < 1 and $b_{2}$ b 2 is bounded measurable, with $\sigma \in L^{\infty }(0,T;{\mathcal{C}}_{u}(\mathbb{R}))$ σ ∈ L ∞ ( 0 , T ; C u ( R ) ) there being a real number $\delta >0$ δ > 0 such that $\sigma ^{2}\geq \delta $ σ 2 ≥ δ . Then there exists a weak solution to the above equation. Moreover, (i) if $\sigma \in \mathcal{C}([0,T];\mathcal{C}_{u}(\mathbb{R}))$ σ ∈ C ( [ 0 , T ] ; C u ( R ) ) , all weak solutions have the same probability law on 1-dimensional classical Wiener space on $[0,T]$ [ 0 , T ] and there is a density associated with the above SDE; (ii) if $b_{2}=0$ b 2 = 0 , $p\in [2,\infty )$ p ∈ [ 2 , ∞ ) and $\sigma \in L^{2}(0,T;{\mathcal{C}}_{b}^{1/2}({\mathbb{R}}))$ σ ∈ L 2 ( 0 , T ; C b 1 / 2 ( R ) ) , the pathwise uniqueness holds.

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Hölder estimates of mild solutions for nonlocal SPDEs

Cited by 4 publications

References 17 publications

On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations

On the Campanato and Hölder regularity of local and nonlocal stochastic diffusion equations

Regularity of stochastic nonlocal diffusion equations

Stochastic differential equations with singular coefficients on the straight line

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