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STABLE PROCESS WITH SINGULAR DRIFT

Let d ≥ 2. In this talk, we consider weak solutions for the following type of stochastic differential equation

X0 = x,  dXt = dSt + b(s + t, Xt)dt, t 0,

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

bLloc(R+; L(Rd)) + Lqloc(R+; L p(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.