Rough linear transport equation with an irregular drift

2015/01/13 by Rémi Catellier, Catellier, Rémi · 2 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories

paper · doi:10.48550/arxiv.1501.03000

openalex publication_date 2015/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the linear transport equation (∂)/(∂ t) u ( t,x ) +b ( t,x ) ⋅ ∇ u ( t,x ) + ∇ u ( t,x ) ⋅ (∂)/(∂ t) X ( t ) =0, \hspace2em u ( 0,x ) =u₀ ( x ) where b is a vectorfield of limited regularity and X a vector-valued Hölder continuous driving term. Using the theory of controlled rough paths we give a meaning to the weak formulation of the PDE and solve that equation for smooth vectorfields b. In the case of the fractional Brownian motion a phenomenon of regularization by noise is displayed.

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• Rough path continuity equations with discontinuous coefficients - regularization by fractional Brownian motion
• Stochastic regularization effects of semi-martingales on random functions

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