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Rough Paths and Rough Partial Differential Equations

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Rough Paths and Rough Partial Differential Equations Weierstrass Institute for Applied Analysis and Stochastics Rough paths and rough partial differential equations Christian Bayer Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de March 18, 2016 Outline 1 Motivation and introduction 2 Rough path spaces 3 Integration against rough paths 4 Integration of controlled rough paths 5 Rough differential equations 6 Applications of the universal limit theorem 7 Rough partial differential equations Rough paths and rough partial differential equations · March 18, 2016 · Page 2 (48) Controlled differential equations Standard ordinary differential equation d y˙t = V(yt); y0 = ξ 2 R ; t 2 [0; 1] V : Rd ! Rd smooth Controlled differential equation d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] I V : Rd ! Rd×e smooth e I xt path taking values in R I xt may contain component t, i.e., includes dyt = V0(yt)dt + V(yt)dxt Rough paths and rough partial differential equations · March 18, 2016 · Page 3 (48) Controlled differential equations Standard ordinary differential equation d y˙t = V(yt); y0 = ξ 2 R ; t 2 [0; 1] V : Rd ! Rd smooth Controlled differential equation d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] I V : Rd ! Rd×e smooth e I xt path taking values in R I xt may contain component t, i.e., includes dyt = V0(yt)dt + V(yt)dxt Rough paths and rough partial differential equations · March 18, 2016 · Page 3 (48) Controlled differential equations Standard ordinary differential equation d y˙t = V(yt); y0 = ξ 2 R ; t 2 [0; 1] V : Rd ! Rd smooth Controlled differential equation d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] I V : Rd ! Rd×e smooth e I xt path taking values in R I xt may contain component t, i.e., includes dyt = V0(yt)dt + V(yt)dxt Rough paths and rough partial differential equations · March 18, 2016 · Page 3 (48) Examples of controlled differential equations d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] I xt smooth: y˙t = V(yt)x ˙t I xt = Wt(!) is a path of a Brownian motion, i.e., yt = yt(!) is pathwise solution of the stochastic differential equation dyt(!) = V(yt(!))dWt(!) (Ito, Stratonovich or some other sense?) I xt = Zt(!) for some other stochastic process, such as fractional Brownian motion, yt = yt(!) is pathwise solution of the corresponding stochastic differential equation Rough paths and rough partial differential equations · March 18, 2016 · Page 4 (48) Examples of controlled differential equations d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] I xt smooth: y˙t = V(yt)x ˙t I xt = Wt(!) is a path of a Brownian motion, i.e., yt = yt(!) is pathwise solution of the stochastic differential equation dyt(!) = V(yt(!))dWt(!) (Ito, Stratonovich or some other sense?) I xt = Zt(!) for some other stochastic process, such as fractional Brownian motion, yt = yt(!) is pathwise solution of the corresponding stochastic differential equation Rough paths and rough partial differential equations · March 18, 2016 · Page 4 (48) Examples of controlled differential equations d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] I xt smooth: y˙t = V(yt)x ˙t I xt = Wt(!) is a path of a Brownian motion, i.e., yt = yt(!) is pathwise solution of the stochastic differential equation dyt(!) = V(yt(!))dWt(!) (Ito, Stratonovich or some other sense?) I xt = Zt(!) for some other stochastic process, such as fractional Brownian motion, yt = yt(!) is pathwise solution of the corresponding stochastic differential equation Rough paths and rough partial differential equations · March 18, 2016 · Page 4 (48) Integral form d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] Assume that xt is not smooth, say ( ) α e e jxs − xtj x 2 C ([0; 1]; R ) B x 2 C [0; 1]; R sup α C kxkα < 1 ; α < 1 s,t js − tj I While x˙ does not “easily” make sense, maybe the integral form does: Z t yt = ξ + V(ys)dxs; t 2 [0; 1] 0 I Notice: If x 2 Cα, then generically y 2 Cα (and no better), as well. I Need to make sense of expressions of the form Z t α ysdxs; x; y 2 C ([0; 1]) 0 Rough paths and rough partial differential equations · March 18, 2016 · Page 5 (48) Integral form d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] Assume that xt is not smooth, say ( ) α e e jxs − xtj x 2 C ([0; 1]; R ) B x 2 C [0; 1]; R sup α C kxkα < 1 ; α < 1 s,t js − tj I While x˙ does not “easily” make sense, maybe the integral form does: Z t yt = ξ + V(ys)dxs; t 2 [0; 1] 0 I Notice: If x 2 Cα, then generically y 2 Cα (and no better), as well. I Need to make sense of expressions of the form Z t α ysdxs; x; y 2 C ([0; 1]) 0 Rough paths and rough partial differential equations · March 18, 2016 · Page 5 (48) Integral form d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] Assume that xt is not smooth, say ( ) α e e jxs − xtj x 2 C ([0; 1]; R ) B x 2 C [0; 1]; R sup α C kxkα < 1 ; α < 1 s,t js − tj I While x˙ does not “easily” make sense, maybe the integral form does: Z t yt = ξ + V(ys)dxs; t 2 [0; 1] 0 I Notice: If x 2 Cα, then generically y 2 Cα (and no better), as well. I Need to make sense of expressions of the form Z t α ysdxs; x; y 2 C ([0; 1]) 0 Rough paths and rough partial differential equations · March 18, 2016 · Page 5 (48) Integral form d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] Assume that xt is not smooth, say ( ) α e e jxs − xtj x 2 C ([0; 1]; R ) B x 2 C [0; 1]; R sup α C kxkα < 1 ; α < 1 s,t js − tj I While x˙ does not “easily” make sense, maybe the integral form does: Z t yt = ξ + V(ys)dxs; t 2 [0; 1] 0 I Notice: If x 2 Cα, then generically y 2 Cα (and no better), as well. I Need to make sense of expressions of the form Z t α ysdxs; x; y 2 C ([0; 1]) 0 Rough paths and rough partial differential equations · March 18, 2016 · Page 5 (48) Young integral Z t α ysdxs; x; y 2 C ([0; 1]) 0 Recall the Riemann-Stieltjes integral: Z 1 X ysdxs B lim ys (xt − xs) (∗) jPj!0 0 [s;t]2P | {z } Cxs;t P a finite partition of [0; 1] Theorem (Young 1936) (a) Let y 2 Cβ([0; 1]; R), x 2 Cα([0; 1]; R) with 0 < α, β < 1 and α + β > 1. Then (∗) converges and the resulting bi-linear map R 1 R 1 x; y 7! y dx is continuous, i.e., y dx ≤ C jy j kyk kxk . ( ) 0 s s 0 s s α+β( 0 ) β α (b) Let α + β ≤ 1. Then there are y 2 Cβ([0; 1]; R), x 2 Cα([0; 1]; R) such that (∗) does not converge, i.e., such that different sequences of partitions yield different limits (or none at all). Rough paths and rough partial differential equations · March 18, 2016 · Page 6 (48) Young integral Z t α ysdxs; x; y 2 C ([0; 1]) 0 Recall the Riemann-Stieltjes integral: Z 1 X ysdxs B lim ys (xt − xs) (∗) jPj!0 0 [s;t]2P | {z } Cxs;t P a finite partition of [0; 1] Theorem (Young 1936) (a) Let y 2 Cβ([0; 1]; R), x 2 Cα([0; 1]; R) with 0 < α, β < 1 and α + β > 1. Then (∗) converges and the resulting bi-linear map R 1 R 1 x; y 7! y dx is continuous, i.e., y dx ≤ C jy j kyk kxk . ( ) 0 s s 0 s s α+β( 0 ) β α (b) Let α + β ≤ 1. Then there are y 2 Cβ([0; 1]; R), x 2 Cα([0; 1]; R) such that (∗) does not converge, i.e., such that different sequences of partitions yield different limits (or none at all). Rough paths and rough partial differential equations · March 18, 2016 · Page 6 (48) Young integral Z t α ysdxs; x; y 2 C ([0; 1]) 0 Recall the Riemann-Stieltjes integral: Z 1 X ysdxs B lim ys (xt − xs) (∗) jPj!0 0 [s;t]2P | {z } Cxs;t P a finite partition of [0; 1] Theorem (Young 1936) (a) Let y 2 Cβ([0; 1]; R), x 2 Cα([0; 1]; R) with 0 < α, β < 1 and α + β > 1. Then (∗) converges and the resulting bi-linear map R 1 R 1 x; y 7! y dx is continuous, i.e., y dx ≤ C jy j kyk kxk . ( ) 0 s s 0 s s α+β( 0 ) β α (b) Let α + β ≤ 1. Then there are y 2 Cβ([0; 1]; R), x 2 Cα([0; 1]; R) such that (∗) does not converge, i.e., such that different sequences of partitions yield different limits (or none at all). Rough paths and rough partial differential equations · March 18, 2016 · Page 6 (48) Young integral II d dyt = V(yt)dxt; y0 = ξ 2 R ; t 2 [0; 1] α e 1 2 d d×e Let x 2 C ([0; 1]; R ), α > 2 and V 2 Cb(R ; R ).
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