2023/01/02 by Beomseok Han, Han, Beom-Seok
Computer Science · Economics, Econometrics and Finance · Mathematics · #26A33 #35R11 #35R60 #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2301.00536
openalex publication_date 2023/01/02 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28
We present the Lp-solvability for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise: ∂tαu = aijuxixj + biuxi + cu + bi u uxi + ∂tβ∫0t σ(u)dWt, tgt;0; u(0,⋅) = u0, where α∈(0,1), β< 3α/4+1/2, and d< 4 - 2(2β-1)+/α. The operators ∂tα and ∂tβ are the Caputo fractional derivatives of order α and β, respectively. The process Wt is an L2(ℝd)-valued cylindrical Wiener process, and the coefficients aij, bi, c and σ(u) are random. In addition to the existence and uniqueness of a solution, we also suggest the Hölder regularity of the solution. For example, for any constant T<∞, small ε>0, and almost sure ω∈Ω, we have supx∈ℝd|u(ω,⋅,x)|_C^[ \fracα2( ( 2-(2β-1)+/α-d/2 )\wedge1 )+\frac(2β-1)-2 ]\wedge 1-ε([0,T])lt;∞ \quadand supt≤ T|u(ω,t,⋅)|C( 2-(2β-1)+/α-d/2 )\wedge1 - ε(ℝd) lt; ∞. The Hölder regularity of the solution in time changes behavior at β= 1/2. Furthermore, if β≥1/2, then the Hölder regularity of the solution in time is α/2 times the one in space.