Harnack inequality and derivative formula for stochastic heat equation with fractional noise

In this note, we establish the Harnack inequality and derivative formula for stochastic heat equation driven by fractional noise with Hurst indexH ∈ ( 1 4 , 1 2 ). As an application, we introduce a strong Feller property.


Introduction and main results
Harnack inequality for stochastic partial differential equations is a recent research direction in probability theory. For results on Harnack inequality and related Derivative formula for stochastic differential equations we refer to, among others, X. Fan [3], F. Wang [10], F. Wang and J. Wang [11], F. Wang and Yuan [13], F. Wang and Zhang [14], L. Wang and X. Zhang [15], T. Zhang [16], X. Zhang [17,18]. However, in contrast to the extensive studies on Harnack inequality for stochastic differential equations, there has been little systematic investigation on Harnack inequality for stochastic partial differential equations. The main reasons for this, in our opinion, are the complexity of dependence structures of solutions to SPDEs. We mention the works Bao et al [1], Liu [5], Wang [10]. On the other hand, SPDEs driven by fractional noise also is a recent research direction in probability theory, and it is very limited to study Harnack inequality for such SPDEs.
Motivated by these results, in this note we consider the Harnack inequality and the derivative formula associated with the following stochastic heat equation with Dirichlet boundary condition: 1]) and W H is the fractional noise. Clearly, the solution of the above equation depends on the initial value f . So, we write u(t, x) = u(t, f, x) and u(t, f ) = u(t, f, ·) for all t ≥ 0. Let B b (H) be the space of all bounded measurable functions on H and define the operators P t , t ≥ 0 by We also introduce the derivative operator D by provided the limit in the right-hand side exists, where G ∈ B b (H) and f, g ∈ H.
Our main aims are to expound and prove the next theorems.
with Γ(·) being the classical Gamma function.
The rest of the paper is organized as follows. In Section 2, we recall some basic results about fractional noise W H . In Section 3 and Section 4, we prove the above theorems.

Preliminaries
Consider the kernel function . With the help of K H , one can show the Cameron-Martin space H is the set of f which can be written as Then, we have is a space-time white noise, and and in particular, when 1 ECP 23 (2018), paper 35.
It is proved in [2] that the operator K H : is an isomorphism and it has the following expression: for any f ∈ L 2 ([0, T ]) The inverse operator K −1 H is given by If f is absolutely continuous and H < 1 2 , it can be proved that Recall that W H is an F t -fractional noise if it is a fractional noise such that the spacetime white noise W (t, A) defined above is for all t ≥ 0 and x ∈ [0, 1]. Clearly, the solution depends on the initial value f . So, for notational simplicity we write u(t, x) = u(t, f, x) and u(t, f ) = u(t, f, ·) for all t ≥ 0. We say u(t, f ) is a solution to (1.1) if and only if for all t ≥ 0 and ϕ ∈ C 2 ([0, 1]) satisfying the next conditions: • ϕ (0) = ϕ (1) = 0 and the integral Proof of Theorem 1.1. We will prove the theorem in the three steps.
Step 1. For f 1 , f 2 ∈ H, consider equation x)} . It follows from the proof of Theorem A.2 in [9] that the operator A(t, v) satisfies (A1)-(A4) in [9]. Then by a similar argument with Theorem II.2.1 and II.2.2 in [4], we can derive that the above equation has a unique solution By the chain rule, we have Notice that the operator ∆ is negative. We get for all t ∈ [0, T ], which implies u(T, f 1 ) = v(T, f 2 ) a.s. on {τ > T }, and on {τ ≤ T } we also have u(T, f 1 ) = v(T, f 2 ) and Step 2. Define For this purpose, we need to show the conditions of Theorem 2.1 hold. Observe that condition (i) is equivalent to , a.e. (3.6) and condition (ii) follows from Novikov criterion. Now, we verify (3.6), we have thus,

dyds.
Combining this and Minkowski's integral inequality, we get Using the Novikov criterion, we obtain EL T = 1.
Step 3. We can rewrite v(t, f 2 ) as v(t, f 2 ) = f 2 (x) + t 0 ∆v(s, f 2 )ds + B H (t), (3.9) where B H (t) =    for every ε > 0. We now prove that W H (t, A) is an fractional noise under the probability measure R ε dP. By Novikov criterion, it suffices to show that