A unified L^q(L^p) theory establishes weak differentiability of SDE flows, a Bismut-Elworthy-Li derivative formula, and endpoint weak well-posedness for singular locally integrable coefficients.
Brownian motion with general drift
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abstract
We construct and study the weak solution to stochastic differential equation $dX(t)=-b(X(t))dt+\sqrt{2}dW(t)$, $X_0=x$, for every $x \in \mathbb R^d$, $d \geq 3$, with $b$ in the class of weakly form-bounded vector fields, containing, as proper subclasses, a sub-critical class $[L^d+L^\infty]^d$, as well as critical classes such as weak $L^d$ class, Kato class, Campanato-Morrey class, Chang-Wilson-T. Wolff class.
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$L^q(L^p)$-theory of stochastic differential equations
A unified L^q(L^p) theory establishes weak differentiability of SDE flows, a Bismut-Elworthy-Li derivative formula, and endpoint weak well-posedness for singular locally integrable coefficients.