Sharp space-time regularity of the solution to stochastic heat equation driven by fractional-colored noise

In this article, we study the following stochastic heat equation ∂tu(t,x)=Lu(t,x)+Ḃ, u(0,x)=0, 0≤t≤T, x∈Rd, where L is the generator of a Lévy process X in Rd, B is a fractional-colored Gaussian noise with Hurst index H∈(12, 1) in the time variable and spatial covariance function f which is the Fourier transform of a tempered measure μ. After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution {u(t,x), t≥0, x∈Rd} in both time and space variables. Under mild conditions, we establish the exact uniform modulus of continuity and a Chung-type law of iterated logarithm for the sample function (t,x)↦u(t,x). These results, to our knowledge, are new even for the classical stochastic heat equation (where L=Δ) with space-time white noise and they strengthen the corresponding results of Balan and Tudor (2008) and Tudor and Xiao (2017) where partial regularity results were obtained.