On Weak Solutions of SDEs with Singular Time-Dependent Drift and Driven by Stable Processes

الأصل · EN

Let d ≥ 2. In this paper, we study weak solutions for the following type of stochastic differential equation dXₜ=dSₜ+b(s+t, Xₜ)dt, X₀=x, where (s,x)∈ R+ × Rᵈ is the initial starting point, b: R+ × Rᵈ → Rᵈ is measurable, and S=(Sₜ)ₜ ≥ ₀ is a d-dimensional α-stable process with index α∈ (1,2). We show that if the α-stable process S is non-degenerate and b ∈ Lloc∞(R₊;L∞(Rᵈ))+ Llocq(R₊;Lᵖ(Rᵈ)) for some p,q>0 with d/ p+α/q <α-1, then the above SDE has a unique weak solution for every starting point (s,x)∈ R+ × Rᵈ.