Gradient and stability estimates of heat kernels for fractional powers of elliptic operator

Abstract

Gradient and stability type estimates of heat kernel associated with fractional power of a uniformly elliptic operator are obtained. L p -operator norm of semigroups associated with fractional power of two uniformly elliptic operators are also obtained.

Intermittency properties for a large class of stochastic PDEs driven by fractional space-time noises

“…where g(α, s) is a probability density function of s ≥ 0 and defined in (1.2) in [9]. By Proposition 2.2 there, we have…”

“…Fractional spatial equations: Space nonhomogeneous case. The next model is the generalized d (≥ 1)-spatial dimensional fractional stochastic α-heat equation (α-SHE) that has been considered in [1,2,9]:…”

“…and the corresponding Green's function G h,α t (x) satisfies the following Nash's Hölder estimates (see e.g. [9] for more details):…”

“…Presumably, we may use (7.14) to obtain the desired bounds. However, we will use Pollard's formula in [9] to prove this proposition.…”

“…The equations include stochastic wave equation (SWE, L = ∂ 2 t − ∆), stochastic heat equation which is inhomogeneous and fractional in space ((α, A)-SHE, L = ∂ t − (−∇(A(x)∇)) α/2 ), where A is a positive definite symmetric matrix, stochastic partial differential equations which is both fractional in time and in space (SFDE, L = ∂ β t − 1 2 (−∆) α/2 ) (e.g. [1,2,4,7,9,10] and references therein).…”