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Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential. (English) Zbl 1407.60109

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 [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.

MSC:

60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Citations:

Zbl 1333.60149

References:

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