Weak and strong well-posedness of critical and supercritical SDEs with singular coefficients

Consider the following time-dependent stable-like operator with drift: -L-t phi(x) = integral(Rd) [phi (x +z) - phi (x ) - z((alpha)) center dot del phi(x)]sigma (t, x, z)nu(alpha)(dz) +b(t, x) center dot del phi(x), where d >= 1, nu(alpha) is an alpha-stable type Levy measure with alpha is an element of (0, 1] and z((alpha)) = 1(alpha)=1(1)|z|<= 1(z), sigma is a real -valued Borel function on R+ x Rd x Rd and b is an Rd-valued Borel function on R+ Chi R-d. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with L-t under the sharp balance condition alpha + beta >= 1, where beta is the Holder index of b with respect to x. Moreover, we also study a class of stochastic differential equations driven by Markov processes with generators of the form L-t. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces.

Automated summary

This paper investigates a time-dependent stable-like operator with drift and establishes the well-posedness for the martingale problem associated with it using the Littlewood-Paley theory. It also studies a class of stochastic differential equations driven by generators of the form L-t and proves the pathwise uniqueness of strong solutions for coefficients in certain Besov spaces.