Quantitative Driver-Only Capacity Cores and Simultaneous Scalar It\^o Flows

Abstract

We construct quantitative compact raw-driver cores for the maximal nondominated class $M_\Lambda$ of laws under which the coordinate process is a scalar continuous local martingale satisfying $d[X]_t\leq\Lambda\,dt$. Fix $0<\eta<1/2$ and $0<\gamma<\min\{1/2,2\eta\}$. Explicit compact sets $K_{A,B}$ have sub-Gaussian tails in $A$ and subexponential tails in $B$, while continuous dyadic realized variation converges uniformly on each core at rate $O(B\Lambda T2^{-\gamma n})$. The intrinsic dyadic quadratic variation is Holder continuous in the raw uniform topology on the cores. The parabolic family $K_R=K_{R,R^2}$ has complement of capacity at most $Ce^{-cR^2}$ and is stable under stopping and controlled deterministic concatenation. For every compact family of sufficiently smooth autonomous coefficients with uniformly positive diffusion, a deterministic Lamperti--Follmer equation constructs Borel causal solution and integral fields jointly in the coefficient, starting time, and initial value. The fields satisfy the solution and integral cocycles. The solution field forms an orientation-preserving $C^1$ flow, and both fields have quantitative raw-driver Holder moduli on each $K_R$. Under every law in $M_\Lambda$, each fixed section agrees with the corresponding classical Ito solution and stochastic integral. We also place the Bichteler--Karandikar construction on a fixed Borel pair-domain, with projective causality and full-sequence semimartingale identification.