A sharp L_p-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) du=(a^ij(ω,t)u_xⁱx^j+f)dt+(σ^ik(ω,t)u_xⁱ+g^k)dw_t^k,u(0,x)=u₀, where {w_t^k:k=1,2,⋯} is a sequence of independent Brownian motions. The coefficients are merely measurable in (ω,t) and can be unbounded and fully degenerate, that is, coefficients a^ij, σ^ik merely satisfy [Formula presente] In this article, we prove that there exists a unique solution u to (0.1), and ‖u_xx‖_Hp^γ(τ,δ)≤N(d,p)(‖u₀‖_{Bp^{γ+2(1−1/p)}}+‖f‖_{Hp^γ(τ,δ^1−p)}+‖g_x‖_{Hp^γ(τ,|σ|^pδ^1−p,l2)}^p+‖g_x‖_{Hp^γ(τ,δ^1−p/2,l2)}), where p≥2, γ∈R, τ is an arbitrary stopping time, δ(ω,t) is the smallest eigenvalue of α^ij(ω,t), H_p^γ(τ,δ) is a weighted stochastic Sobolev space, and B_p^{γ+2(1−1/p)} is a stochastic Besov space.