Well-posedness of the transport equation by stochastic perturbation
TLDR
In this paper, it was shown that a multiplicative stochastic perturbation of Brownian type is enough to render the linear transport equation well-posed. But it was not shown that multiplicative perturbations alone are sufficient to render a deterministic PDE wellposed.Abstract:
We consider the linear transport equation with a globally Holder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Ito-Tanaka type.read more
Citations
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Stochastic flows and stochastic differential equations
TL;DR: In this article, the authors consider continuous semimartingales with spatial parameter and stochastic integrals, and the convergence of these processes and their convergence in stochastically flows.
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Noise Prevents Singularities in Linear Transport Equations
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Pathwise uniqueness and continuous dependence for SDEs with non-regular drift
Ennio Fedrizzi,Franco Flandoli +1 more
TL;DR: In this article, a new proof of the pathwise uniqueness result of Krylov and Rockner is given, which concerns SDEs with drift having only certain integrability properties, and the proof is formulated in such a way that the only major tool is a good regularity theory for the heat equation forced by a function with the same regularity of the drift.
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Pathwise uniqueness for singular SDEs driven by stable processes
TL;DR: In this article, the authors prove pathwise uniqueness for stochastic dierential equations driven by non-degenerate symmetric L evy processes with values in R d having a bounded and -Holder continuous drift term.
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