On a class of stochastic fractional kinetic equation with fractional noise

Abstract

Abstract In this article we study a class of stochastic fractional kinetic equations with fractional noise which are spatially homogeneous and are fractional in time with H > 1 / 2 $H>1/2$ . The diffusion operator involved in the equation is the composition of the Bessel and Riesz potentials with any fractional parameters. We prove the existence of the solution under some mild conditions which generalized some results obtained by Dalang (Electron. J. Probab. 4(6):1–29, 1999) and Balan and Tudor (Stoch. Process. Appl. 120:2468–2494 , 2010). We study also its Hölder continuity with respect to space and time variables with b = 0 $b=0$ . Moreover, we prove the existence for the density of the solution and establish the Gaussian-type lower and upper bounds for the density by the techniques of Malliavin calculus.

Keywords

• Stochastic fractional kinetic equation
• Bessel and Riesz potentials
• Fractional noise
• Malliavin calculus
• Density estimates