Space-time fractional stochastic partial differential equations

作者：Mijena Jebessa B; Nane Erkan^*

来源：Stochastic Processes and Their Applications, 2015, 125(9): 3301-3326.

DOI：10.1016/j.spa.2015.04.008

• Time-fractional stochastic partial differential equations
• Fractional Duhamel's principle
• Caputo derivatives
• Walsh isometry

We consider non-linear time-fractional stochastic heat type equation
partial derivative(beta)(t)u(t)(x) = -v(-Delta)(alpha/2)u(t)(x) + I-t(1-beta)[sigma(u) (W) over dot (t, x)]
in (d + 1) dimensions, where v > 0, beta is an element of (0, 1), alpha is an element of (0,2] and d < min{2, beta(-1)}alpha, partial derivative(beta)(t) is the Caputo fractional derivative, -(-Delta)(alpha/2) is the generator of an isotropic stable process, I-t(1-beta) is the fractional integral operator, (W) over dot (t, x) is space time white noise, and sigma : R -> R is Lipschitz continuous. Time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove existence and uniqueness of mild solutions to this equation and establish conditions under which the solution is continuous. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in Foondun and Khoshnevisan (2009), Walsh (1986). In sharp contrast to the stochastic partial differential equations studied earlier in Foondun and Khoshnevisan (2009), Khoshnevisan (2014) and Walsh (1986), in some cases our results give existence of random field solutions in spatial dimensions d = 1, 2, 3. Under faster than linear growth of sigma, we show that time fractional stochastic partial differential equation has no finite energy solution. This extends the result of Foondun and Parshad (in press) in the case of parabolic stochastic partial differential equations. We also establish a connection of the time fractional stochastic partial differential equations to higher order parabolic stochastic differential equations. Published by Elsevier B.V.

• 出版日期2015-9