Abstract
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 \(H \in (1/4,1/2)\), 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.
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Acknowledgements
P.K.F. is partially supported by the European Research Council through CoG-683164 and DFG research unit FOR2402. I.C., affiliated to TU Berlin when this project was commenced, was supported by DFG research unit FOR2402, and is currently supported by a Junior Research Fellowship of St John’s College, Oxford. Y.B. thanks Martin Hairer for financial support from his Leverhulme Trust award.
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Bruned, Y., Chevyrev, I., Friz, P.K. (2018). Examples of Renormalized SDEs. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_19
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