Bismut formula for a stochastic heat equation with fractional noise

In this note, we establish the Bismut formula for stochastic heat equation ∂∂tu(t,x)=Δu(t,x)+ẆH(t,x),t≥0,x∈[0,1],∂∂xu(t,x)|x=0=∂∂xu(t,x)|x=1=0,t≥0,u(0,x)=f(x),x∈[0,1],where f(x)∈H≔L2([0,1]) and WH is the fractional noise with Hurst index H∈(12,1). As an application, we also introduce the Harnack inequality.