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

Let X_t (x) solve the following Itô-type SDE (denoted by EQ. (σ, b, x)) in R^dd X_t = σ (X_t) ṡ d W_t + b (X_t) d t, X₀ = x ∈ R^d . Assume that for any N > 0 and some C_N > 0| b (x) - b (y) | + {norm of matrix} ∇ σ (x) - ∇ σ (y) {norm of matrix} ≤ C_N | x - y | (log | x - y |^-1 ∨ 1), | x |, | y | ≤ N, where ∇ denotes the gradient, and the explosion times of EQ. (σ, b, x) and EQ. (σ, tr (∇ σ ṡ σ) - b, x) are infinite for each x ∈ R^d. Then we prove that for fixed t > 0, x {mapping} X_t^-1 (x) is α (t)-order locally Hölder continuous a.s., where 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) {mapping} X_t^-1 (x, ω) is bicontinuous.