Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise

Tran Thanh Binh; Nguyen Huy Tuan; Tran Bao Ngoc

doi:10.1140/epjp/s13360-021-01864-4

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EPJ PLUS - Condensed Matter and Complex Systems

Focus Point on Uncertainty Quantification of Modelling and Simulation in Physics and Related Areas: From Theoretical to Computational Techniques

Eur. Phys. J. Plus (2021) 136: 935
https://doi.org/10.1140/epjp/s13360-021-01864-4

Regular Article

Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise

Tran Thanh Binh¹, Nguyen Huy Tuan²^b and Tran Bao Ngoc³^c

¹ Faculty of Mathematics and Applications, Sai Gon University, Ho Chi Minh City, Viet Nam
² Department of Mathematics and Computer Science, University of Science-VNUHCM, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Viet Nam
³ Department of Mathematics, Faculty of Science, Nong Lam University, Ho Chi Minh City, Viet Nam

^b nhtuan@hcmus.edu.vn
^c tranbaongoc@hcmuaf.edu.vn

Received: 14 May 2021
Accepted: 10 August 2021
Published online: 13 September 2021

Abstract

An initial value problem for space-time fractional stochastic heat equations driven by colored noise $\partial_{t}^{s} u_{t} + \frac{ν}{2} {(- Δ)}^{α / 2} u_{t} = σ (u_{t}) \dot{W}$ $\partial _t^s u_t + \frac{\nu }{2} (- \Delta )^{\alpha /2} u_t = \sigma (u_t) \dot{W}$ has been discussed in this work. Here, $\partial_{t}^{s}$ $\partial _t^s$ and ${(- Δ)}^{α / 2}$ $(- \Delta )^{\alpha /2}$ stand for the Caputo’s fractional derivative of order $s \in (0, 1)$ $s\in (0,1)$ and the fractional Laplace operator of order $α \in (0, 2)$ $\alpha \in (0,2)$ , where the second one is also the generator of a strict stable process $X_{t}$ $$X_t$$ of order $α$ $\alpha$ with $E e^{i ξ \cdot X_{t}} = e^{- {t | ξ |}^{α}}$ $E e^{i\xi \cdot X_t}=e^{-t|\xi |^\alpha }$ . The nonlinearity $σ$ $\sigma$ is assumed to be Lipschitz continuous. A formulation of mild random field solutions is obtained due to the called space-time fractional Green functions G, H, where H contains a singular kernel. We focus on studying the spatially–temporally Hölder continuity of mild random field solutions, which can be obtained by constructing relevant moment bounds for increments of the convolutions $H \otimes b (u)$ $H \otimes b(u)$ and $H ⊛ σ (u)$ $H \circledast \sigma (u)$ . Our techniques are based on connecting the space-time fractional Green functions G, H to the fundamental solution of $\partial_{t} P_{t} + {(- Δ)}^{α / 2} P_{t} = 0$ $\partial _t P_t + (-\Delta )^{\alpha /2}P_t = 0$ , $P_{0} = δ_{0}$ $P_0=\delta _0$ via the Wright-type function $M_{s}$ ${\mathcal {M}}_s$ .