Continuity modulus of stochastic homeomorphism flows for SDEs with non-Lipschitz coefficients

Let Xt(x)Xt(x) solve the following Itô-type SDE (denoted by EQ.(σ,b,x)EQ.(σ,b,x)) in RdRddXt=σ(Xt)⋅dWt+b(Xt)dt,X0=x∈Rd. Assume that for any N>0N>0 and some CN>0CN>0|b(x)−b(y)|+‖∇σ(x)−∇σ(y)‖⩽CN|x−y|(log|x−y|−1∨1),|x|,|y|⩽N, where ∇ denotes the gradient, and the explosion times of EQ.(σ,b,x)EQ.(σ,b,x) and EQ.(σ,tr(∇σ⋅σ)−b,x)EQ.(σ,tr(∇σ⋅σ)−b,x) are infinite for each x∈Rdx∈Rd. Then we prove that for fixed t>0t>0, x↦Xt−1(x) is α(t)α(t)-order locally Hölder continuous a.s., where α(t)∈(0,1)α(t)∈(0,1) is exponentially decreasing to zero as the time goes to infinity. Moreover, for almost all ω  , the inverse flow (t,x)↦Xt−1(x,ω) is bicontinuous.