Multiscale Systems, Homogenization, and Rough Paths:VAR75 2016: Probability and Analysis in Interacting Physical Systems

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479-520], taking into account recent progress in $p$-variation and c\`adl\`ag rough path theory.Comment: 27 pages. Minor corrections. To appear in Proceedings of the Conference in Honor of the 75th Birthday of S.R.S. Varadha

Rough path metrics on a Besov–Nikolskii-type scale

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in q-variation, resp., 1/q-H¨older-type metrics on the space of rough paths, for any regularity 1/q ∈ (0, 1]. We extend this to a new class of Besov–Nikolskii-type metrics, with arbitrary regularity 1/q ∈ (0, 1] and integrability p ∈ [q, ∞], where the case p ∈ {q,∞} corresponds to the known cases. Interestingly, the result is obtained as a consequence of known q-variation rough path estimates

Rough path metrics on a Besov–Nikolskii-type scale

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in q-variation, resp., 1/q-H¨older-type metrics on the space of rough paths, for any regularity 1/q ∈ (0, 1]. We extend this to a new class of Besov–Nikolskii-type metrics, with arbitrary regularity 1/q ∈ (0, 1] and integrability p ∈ [q, ∞], where the case p ∈ {q,∞} corresponds to the known cases. Interestingly, the result is obtained as a consequence of known q-variation rough path estimates

Examples of renormalized SDEs

We demonstrate two examples of stochastic processes whose lifts to geometric rough paths require a renormalisation procedure to obtain convergence in rough path topologies. Our first example involves a physical Brownian motion subject to a magnetic force which dominates over the friction forces in the small mass limit. Our second example involves a lead-lag process of discretised fractional Brownian motion with Hurst parameter H∈(1/4,1/2), in which the stochastic area captures the quadratic variation of the process. In both examples, a renormalisation of the second iterated integral is needed to ensure convergence of the processes, and we comment on how this procedure mimics negative renormalisation arising in the study of singular SPDEs and regularity structures

A (rough) pathwise approach to a class of non-linear stochastic partial differential equations

We consider non-linear parabolic evolution equations of the form and#948;tu=F(t,x,Du,D2u), subject to noise of the form H(x,Du) dB where H is linear in Du and dB denotes the Stratonovich differential of a multi-dimensional Brownian motion. Motivated by the essentially pathwise results of [P.-L. Lions, P.E. Souganidis, Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sand#233;r. I Math. 326 (9) (1998) 1085-1092] we propose the use of rough path analysis [T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215-310] in this context. Although the core arguments are entirely deterministic, a continuity theorem allows for various probabilistic applications (limit theorems, support, large deviations, ...)

A (rough) pathwise approach to a class of non-linear stochastic partial differential equations

We consider non-linear parabolic evolution equations of the form δtu=F(t,x,Du,D2u), subject to noise of the form H(x,Du) dB where H is linear in Du and dB denotes the Stratonovich differential of a multi-dimensional Brownian motion. Motivated by the essentially pathwise results of [P.-L. Lions, P.E. Souganidis, Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math. 326 (9) (1998) 1085-1092] we propose the use of rough path analysis [T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215-310] in this context. Although the core arguments are entirely deterministic, a continuity theorem allows for various probabilistic applications (limit theorems, support, large deviations, ...)