Abstract
Existence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established.
Highlights
We consider the transport equation, here posed (w.l.o.g.) as terminal value problem
Our driving vector fields will be very smooth, to compensate for the the irregularity of the noise, which we here assumed to be very rough (This trade-off is typical in rough paths and regularity structures.)
Throughout the paper we say geometric rough path, when we really mean weakly geometric rough path
Summary
We consider the transport equation, here posed (w.l.o.g.) as terminal value problem. This is,. Fd are nice enough (C1b will do) to ensure a C1 solution flow for the ODE ⎧ ⎪⎪⎨Xts,x = d fi (Xts,x )Wti ≡ f (Xt )Wt , Solving this ODE with random initial data induces a natural evolution of measures, given by the continuity - or forward equation ⎧ ⎪⎪⎨∂t ρ = d divx ( fi (x)ρt ) dWti in (0, T ) × Rn,. DiPerna–Lions [9] and Ambrsosio [1], showed that the transport problem (weak solutions) is well-posed under bounds on div b (rather than Dx b) which in turn leads to a generalized flow. Our driving vector fields will be very smooth, to compensate for the the irregularity of the noise, which we here assumed to be very rough (This trade-off is typical in rough paths and regularity structures.). Throughout the paper we say geometric rough path, when we really mean weakly geometric rough path (since we only work with this type of rough path, the difference [14] will not matter to us)
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