STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

doi:10.1016/S0252-9602(17)30091-7

Abstract

Abstract:

In this paper we study a fractional stochastic heat equation on R^d (d ≥ 1) with additive noise (∂/∂t)u(t, x)=D(α/δ)u(t, x) +b(u(t, x)) +?^H(t, x) where D(α/δ) is a nonlocal fractional differential operator and ?^H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d=1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.

Key words: stochastic partial differential equation, fractional Brownian motion, Malliavin calculus, Gaussian density estimates

Junfeng LIU, Ciprian A. TUDOR. STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE: EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY[J].Acta mathematica scientia,Series B, 2017, 37(6): 1545-1566.

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References

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