Applications of controlled paths

Transcription

1 Applications of controlled paths Massimiliano Gubinelli CEREMADE Université Paris Dauphine OxPDE conference. Oxford. September 1th 212 ( 1 / 16 )

2 Outline I will exhibith various applications of the idea of a "controlled path". Rough path theory and controlled distributions Averaging by oscillations Non-linear PDEs with random dispersion Stochastic Burgers equation with derivative white noise perturbation ( 2 / 16 )

3 Rough differential equations A central theme of stochastic analysis is the study of stochastic differential equations d t Y t = ϕ(y t )dx t = i ϕ i (Y t )d t X i t where for example X is a Brownian motion (M-dimensional), (V i : R d R d ) i=1,...,m a collection of vector fields on R d (smooth). Standard framework: Itô theory of stochastic integration: Y t = Y + ϕ(y t )dx t The integral on the r.h.s. is defined as a limit in L 2 (P). Rough path theory (T. Lyons) is a way to give a meaning to the above integral path-wise: take a sample x of the Brownian motion X and try to solve the equation y t = y + ϕ(y t )dx t in the space of continuous functions: y C([, T]; R d ). ( 3 / 16 )

4 Problems What is the meaning of the integral ϕ(y t)dx t? Fact: x is only C 1/2 ([, T]; R M ). We expect the same regularity from y. Then ϕ(y) C 1/2 and t x C 1/2 (Here for convenience C γ = B γ, ) The product ϕ(y) t x is not well defined. In Itô theory the product turn out to be defined (in some sense) due to the stochastic cancellations due to the independece of the increments of x (and the fact that y does not "look into the future"). ( 4 / 16 )

5 What goes wrong? Take f C γ (R), g C ρ (R), γ, ρ (, 1) The problem is how to define fdg when f, g are Holder functions (Dg(t) = g (t)). (Inhomogeneous) Littlewood-Paley decomposition f = i f i 1 where i f contains the oscillations of f on the scale 2 i : D n i f L 2 (n γ)i Paraproduct fdg = i,j i f j Dg = π < (f, Dg) + π (f, Dg) + π > (f, Dg) with π < (f, g) = i<j 1 if j g, π (f, g) = i j 1 if j g, π > (f, g) = π < (g, f ). ( 5 / 16 )

6 Area Fact: π < (f, Dg) and π > (f, Dg) are always well defined: π < (f, Dg) C ρ 1, π > (f, Dg) C γ+ρ 1 The problem is here: π (f, Dg). Well defined only if γ + ρ > and in this case π (f, Dg) C γ+ρ 1 Seems not enough for Brownian motion (γ = ρ < 1/2). Area process Take x, y two independent samples of Brownian motion, then it is possible to show that π (x, Dy) exists and belongs to C almost surely. Again: stochastic cancellations. So at least xdy well defined. What else? ( 6 / 16 )

7 Controlled Besov distributions [Joint work with N. Perkowski and P. Imkeller] Fix 1/3 < γ < 1/2 and assume x, y C γ with π (x, Dy) C 2γ 1. Let f be controlled by x in the following sense: f = π < (f, x) + f with f C γ and f C 2γ. (f looks like x in the small scales). Commutator estimate Set R(f, x, Dy) = π (π < (f, x), Dy) f π (x, Dy) But now R(f, x, Dy) 3γ 1 f γ x γ Dy γ 1 fdy = π < (f, Dy) + f π (x, Dy + π > (f, Dy) + π (f, Dy) + R(f, x, Dy) } {{ }} {{ } C 2γ 1 C 3γ 1 and all the objects in the r.h.s. are well defined. ( 7 / 16 )

8 Solving RDEs Reconsider f t = f + ϕ(f s )Dx s dt with x a sample from a M-dimensional Brownian motion. Then x C γ for some 1/3 < γ < 1/2 and π (x i, Dx j ) C 2γ 1 for all i, j = 1,..., M. We can now solve this equation in the space of f controlled by x: Paralinearization theorem: ϕ(f ) = π < ( ϕ(f ), f ) + smoother remainder Controlled hypothesis f π < (f, x) implies ϕ(f ) = π < ( ϕ(f )f, x) + smoother remainder Product: ϕ(f )Dx = π < (ϕ(f ), Dx) + ϕ(f )f π (x, Dx) + smoother remainder Integration: ϕ(f )Dx = π< (ϕ(f ), x) + ϕ(f )f π (x, Dx) + smoother remainder So the map Γ (f ) = f + ϕ(f s )Dx s dt remain in the space of controlled paths and we can set up a fixed point. ( 8 / 16 )

9 Averaging along a Brownian motion Take a bounded function b : R d R d and a d-dimensional Brownian motion (Bm) W. A. Davie has showed that the average of b along the Brownian trajectory w: satisfy from which follows σ w s,t(b)(x) = b(w r + x)dr s E σ W s,t(b)(y) σ W s,t(b)(x) 2p p b L x y 2p t s p σ w s,t(b)(y) σ w s,t(b)(x) w,b x y t s 1/2 (1 + log 1/2 + 1 x y + log1/2 + 1 t s ) From this it is possible to deduce that the ODE (not SDE) x t = x + b(x s )ds + w t has a unique solution in C(R + ; R d ) for almost every sample path w of the Brownian motion. ( 9 / 16 )

10 Fractional Brownian motion To have the freedom to vary the regularity of the driving paths and retain many nice features of the Brownian motion (Gaussian, stationary increments, scaling) a convenient model for noise is the fractional Brownian motion (fbm) B H of Hurst index H (, 1). (B H t ) t [,T] is a Gaussian process with stationary increments, zero mean and covariance E[(B H t B H s ) 2 ] = t s 2H Setting H = 1/2 gives Brownian motion back. The fbm B H has trajectories almost surely in any C γ for any γ < H. ( 1 / 16 )

11 Averaging along an fbm Let FL α the set of distribution b : R d R d such that N α (b) = (1 + ξ ) α ˆb(ξ) dξ < +. R d Then it is possible to show that if (w t ) t is the sample path of a d-dim. fractional Brownian motion and x Q w γ C(R; R d ) is controlled by w in the sense that x t x s = w t w s + O( t s ρ ) for some ρ > 1/2, for all b FL α with α > 1 1/2H the integral lim b n (x s )ds =: b(x s )ds n is well defined for any sequence of smooth function (b n ) n 1 such that N α (b b n ) and independent of the sequence. Moreover the map t b(x s)ds is C γ for some γ > 1/2. [joint work with R. Catellier] ( 11 / 16 )

12 Regularization by oscillations If α > 2 1/2H the averaging map σ x s,t(b)(y) = b(x r + y)dr s is Lipshitz: σ x s,t (b)(y) σ x s,t(b)(z) x,w N α (b) y z t s γ. The previous results allows to study the the ODE in R d where b FL α. x t = x + b(x s )ds + w t Existence in Q w γ for α > 1 1/2H Uniqueness in Q w γ for α > 2 1/2H + Lipshitz flow. If b is not random we can have uniqueness for α > 1 1/2H. ( 12 / 16 )

13 Nonlinear PDEs with random dispersion [joint work with K. Chouk] Consider (Stratonovich-) stochastic nonlinear PDEs of the form t φ t = Aφ t t B t + N(φ t ) for φ : [, T] T C or R where B is a (1d) Brownian motion. Various cases: NSE: φ complex, A = i 2 ξ and N(φ) = ±i φ 2 φ NSE: φ complex, A = i 2 ξ and N(φ) = ±i ξ( φ 2 φ) KdV: φ real, A = 3 ξ and N(φ) = ξφ 2 Recent work of [Debussche De Bouard] on randomly modulated NSE in T (motivated by dispersion management in optical fibers) Spaces φ α = (1 + ξ 2 ) α/2 ˆφ(ξ) L 2 ξ where ˆφ is the space Fourier transform of φ. Almost sure results (with a universal exceptional set): NSE: Global unique solution in L 2 + Lipshitz flow map KdV: Local unique solution in H 1+ + Lipshitz flow map ( 13 / 16 )

14 Formulation of the equation Let U t = e AB t so that then φ should solve t U t = AU t t B t φ t = U t (φ + Us 1 N(φ s )ds). The path φ C([, T], H α ) is controlled if φ t = U t ψ t with ψ t C ρ ([, T], H α ) for some ρ > 1/2. Introduce the map X s,t : H α H α given by X s,t (ψ) = Ur 1 N(U r ψ)dr s Key estimate for some γ > 1/2. X s,t (ψ) X s,t (ψ ) α F( ψ α + ψ α ) t s γ ψ ψ α ( 14 / 16 )

15 Formulation as a controlled path problem The mild equation take the form ψ t = ψ + Us 1 N(U s ψ s )ds = ψ + [ ] d ds X,s (ψ s ) = ψ + X ds (ψ s ) = ψ + lim i X ti,t i+1 (ψ ti ) The key estimate implies t X ds (ψ s ) = Us 1 N(φ s )ds is in C γ ([, T]; H α ) for any controlled path φ and coincide with the limit lim n U 1 s N(P n φ s )ds = X ds (ψ s ) (P n is the projector on the Fourier modes k n) and is γ-hölder in time for some γ > 1/2 and locally Lipshitz in φ (in the controlled path norm). By standard fixed-point argument we get a (unique) local solution to the PDE. In the NSE case the L 2 conservation law allow to extend the solution to a global one. ( 15 / 16 )

16 Thanks ( 16 / 16 )