Rough paths and 1d SDE with a time dependent distributional drift: application to polymers

By François Delarue, R. Diel. 2014

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Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar–Parisi–Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a $$\alpha $$α-Hölder continuous function, $$\alpha >1/3$$α>1/3. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes the structure of the solutions more explicit than within the earlier framework of Dirichlet processes.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1dc24746a880dd27b2df5764a44523d07d566304

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Citations in the corpus

Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, background): “There are some results on weak well-posedness for measure-valued drifts, see [9], and distribution-valued drifts, see [44, 28, 15], but it is unclear whether they apply to the limit case p = d: for example, the result in [9], when restricted to measures with density b with respect to the Lebesgue me”
A stochastic sewing lemma and applications (2018, methodology): “They appear, for instance, in Davie’s [7, pg. 18] and in Delarue and Diel’s [8] in an attempt to identify the distributional drift of a diffusion.”