Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential
The authors consider a stochastic differential equation of the form \(dX_t=V(t, X_t)+dB_t, \,\, X_0=x\), where \(B\) is a \(d\)-dimensional Brownian motion, \(x\) is a point in \(\mathbb R^d\) and \(V\) is a function of time taking values in the space of distributions. They study the existence and uniqueness of the solution by giving a meaning to the Strook-Varadhan martingale problem associated with the equation. The paracontrolled distributions approach introduced in [\textit{M. Gubinelli} et al., Forum Math. Pi 3, Article ID e6, 75 p. (2015; Zbl 1333.60149)] is applied. The examples include the case when a 2 and 3-dimensional polymer measure with white noise potential can be constructed.
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