Averaging along irregular curves and regularization of ODEs Rmi - PowerPoint PPT Presentation

Averaging along irregular curves and regularization of ODEs Rémi Catellier Centre Henri Lebesgue - IRMAR October 24, 2014 joint work with Massimiliano Gubinelli (CEREMADE) Rémi Catellier (CHL - IRMAR) October 24, 2014 1 / 21

Regularization of ODEs 1 The operator T w : ( ρ, γ ) -irregular functions 2 Stochastic processes 3 Rémi Catellier (CHL - IRMAR) October 24, 2014 2 / 21

Perturbed differential equations ? Standard ODEs d x t = b ( x t ) d t ◮ For non Lipschitz b , the equation is ill posed ◮ Example b ( x ) = 2 sign ( x ) � | x | , x 0 = 0. Pertubed ODEs, Brownian case d x t = b ( x t ) d t + d B t ◮ When B is a Brownian motion, for measurable bounded b it is known (Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution. ◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ? Standard ODEs d x t = b ( x t ) d t ◮ For non Lipschitz b , the equation is ill posed ◮ Example b ( x ) = 2 sign ( x ) � | x | , x 0 = 0. Pertubed ODEs, Brownian case d x t = b ( x t ) d t + d B t ◮ When B is a Brownian motion, for measurable bounded b it is known (Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution. ◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ? Standard ODEs d x t = b ( x t ) d t ◮ For non Lipschitz b , the equation is ill posed ◮ Example b ( x ) = 2 sign ( x ) � | x | , x 0 = 0. Pertubed ODEs, Brownian case d x t = b ( x t ) d t + d B t ◮ When B is a Brownian motion, for measurable bounded b it is known (Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution. ◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ? Standard ODEs d x t = b ( x t ) d t ◮ For non Lipschitz b , the equation is ill posed ◮ Example b ( x ) = 2 sign ( x ) � | x | , x 0 = 0. Pertubed ODEs, Brownian case d x t = b ( x t ) d t + d B t ◮ When B is a Brownian motion, for measurable bounded b it is known (Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution. ◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ? Standard ODEs d x t = b ( x t ) d t ◮ For non Lipschitz b , the equation is ill posed ◮ Example b ( x ) = 2 sign ( x ) � | x | , x 0 = 0. Pertubed ODEs, Brownian case d x t = b ( x t ) d t + d B t ◮ When B is a Brownian motion, for measurable bounded b it is known (Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution. ◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

Perturbed differential equations ? Standard ODEs d x t = b ( x t ) d t ◮ For non Lipschitz b , the equation is ill posed ◮ Example b ( x ) = 2 sign ( x ) � | x | , x 0 = 0. Pertubed ODEs, Brownian case d x t = b ( x t ) d t + d B t ◮ When B is a Brownian motion, for measurable bounded b it is known (Krylov-Röckner, Veretennikov, Davie) that the equation as a unique solution. ◮ What about other type of perturbation ? Rémi Catellier (CHL - IRMAR) October 24, 2014 3 / 21

For b : R d → R d et w : [ 0 , T ] → R d we want to study the following equation � t x t = x 0 + b ( x u ) d u + w s 0 Definition For continuous vector fields b from R d to R d we define the averaging operator along w , T w by � t T w s , t b ( x ) = b ( x + w r ) d r s By a change of variable θ t = x t − w t , we have the following equivalent equation � t � t ( ∂ u T w T w θ u = θ 0 + u b )( θ u ) d u =: θ 0 + d u b ( θ u ) 0 0 Rémi Catellier (CHL - IRMAR) October 24, 2014 4 / 21

For b : R d → R d et w : [ 0 , T ] → R d we want to study the following equation � t x t = x 0 + b ( x u ) d u + w s 0 Definition For continuous vector fields b from R d to R d we define the averaging operator along w , T w by � t T w s , t b ( x ) = b ( x + w r ) d r s By a change of variable θ t = x t − w t , we have the following equivalent equation � t � t ( ∂ u T w T w θ u = θ 0 + u b )( θ u ) d u =: θ 0 + d u b ( θ u ) 0 0 Rémi Catellier (CHL - IRMAR) October 24, 2014 4 / 21

For b : R d → R d et w : [ 0 , T ] → R d we want to study the following equation � t x t = x 0 + b ( x u ) d u + w s 0 Definition For continuous vector fields b from R d to R d we define the averaging operator along w , T w by � t T w s , t b ( x ) = b ( x + w r ) d r s By a change of variable θ t = x t − w t , we have the following equivalent equation � t � t ( ∂ u T w T w θ u = θ 0 + u b )( θ u ) d u =: θ 0 + d u b ( θ u ) 0 0 Rémi Catellier (CHL - IRMAR) October 24, 2014 4 / 21

� t T w s , t b ( x ) = b ( x + w r ) d r s d θ t = T w d t b ( θ t ) Questions What is the needed regularity of T w b for existence and uniqueness in the previous equation ? What are the admissible w for such a goal ? Is it possible to enhanced the averaging operator T w for more irregular vector fields ? Rémi Catellier (CHL - IRMAR) October 24, 2014 5 / 21

� t T w s , t b ( x ) = b ( x + w r ) d r s d θ t = T w d t b ( θ t ) Questions What is the needed regularity of T w b for existence and uniqueness in the previous equation ? What are the admissible w for such a goal ? Is it possible to enhanced the averaging operator T w for more irregular vector fields ? Rémi Catellier (CHL - IRMAR) October 24, 2014 5 / 21

� t T w s , t b ( x ) = b ( x + w r ) d r s d θ t = T w d t b ( θ t ) Questions What is the needed regularity of T w b for existence and uniqueness in the previous equation ? What are the admissible w for such a goal ? Is it possible to enhanced the averaging operator T w for more irregular vector fields ? Rémi Catellier (CHL - IRMAR) October 24, 2014 5 / 21

� t � t T w T w s , t b ( x ) = b ( x + w u ) d u , θ t − θ 0 = d r b ( θ r ) 0 0 Theorem (Catellier, Gubinelli) Let ψ ( z ) = 1 + log 1 / 2 ( 1 + z ) , z > 0. Let us suppose that there exists C > 0 and γ > 1 2 such that i T w b has the following regularity | T w s , t b ( x ) − T w s , t b ( y ) | ≤ C | t − s | γ | x − y | ψ ( | x | + | y | ) then the equation has a solution in C γ ([ 0 , T ] , R d ) . ii If b ∈ C b ( R d ; R d ) it is enough to ask that s , t b ( y ) | ≤ C | t − s | γ | x − y | 1 / 2 ˜ |∇ T w s , t b ( x ) − ∇ T w ψ ( | x | + | y | ) iii Furthermore, when the solution is unique, it is locally Lipschitz continuous in the initial value, uniformly in time. Rémi Catellier (CHL - IRMAR) October 24, 2014 6 / 21

� t � t T w T w s , t b ( x ) = b ( x + w u ) d u , θ t − θ 0 = d r b ( θ r ) 0 0 Theorem (Catellier, Gubinelli) Let ψ ( z ) = 1 + log 1 / 2 ( 1 + z ) , z > 0. Let us suppose that there exists C > 0 and γ > 1 2 such that i T w b has the following regularity |∇ T w s , t b ( x ) − ∇ T w s , t b ( y ) | ≤ C | t − s | γ | x − y | ˜ ψ ( | x | + | y | ) then the equation has a unique solution in C γ ([ 0 , T ] , R d ) . ii If b ∈ C b ( R d ; R d ) it is enough to ask that s , t b ( y ) | ≤ C | t − s | γ | x − y | 1 / 2 ˜ |∇ T w s , t b ( x ) − ∇ T w ψ ( | x | + | y | ) iii Furthermore, when the solution is unique, it is locally Lipschitz continuous in the initial value, uniformly in time. Rémi Catellier (CHL - IRMAR) October 24, 2014 6 / 21

� t � t T w T w s , t b ( x ) = b ( x + w u ) d u , θ t − θ 0 = d r b ( θ r ) 0 0 Theorem (Catellier, Gubinelli) Let ψ ( z ) = 1 + log 1 / 2 ( 1 + z ) , z > 0. Let us suppose that there exists C > 0 and γ > 1 2 such that i T w b has the following regularity |∇ T w s , t b ( x ) − ∇ T w s , t b ( y ) | ≤ C | t − s | γ | x − y | ˜ ψ ( | x | + | y | ) then the equation has a unique solution in C γ ([ 0 , T ] , R d ) . ii If b ∈ C b ( R d ; R d ) it is enough to ask that s , t b ( y ) | ≤ C | t − s | γ | x − y | 1 / 2 ˜ |∇ T w s , t b ( x ) − ∇ T w ψ ( | x | + | y | ) iii Furthermore, when the solution is unique, it is locally Lipschitz continuous in the initial value, uniformly in time. Rémi Catellier (CHL - IRMAR) October 24, 2014 6 / 21

� t � t T w T w s , t b ( x ) = b ( x + w u ) d u , θ t − θ 0 = d r b ( θ r ) 0 0 Theorem (Catellier, Gubinelli) Let ψ ( z ) = 1 + log 1 / 2 ( 1 + z ) , z > 0. Let us suppose that there exists C > 0 and γ > 1 2 such that i T w b has the following regularity |∇ T w s , t b ( x ) − ∇ T w s , t b ( y ) | ≤ C | t − s | γ | x − y | ˜ ψ ( | x | + | y | ) then the equation has a unique solution in C γ ([ 0 , T ] , R d ) . ii If b ∈ C b ( R d ; R d ) it is enough to ask that s , t b ( y ) | ≤ C | t − s | γ | x − y | 1 / 2 ˜ |∇ T w s , t b ( x ) − ∇ T w ψ ( | x | + | y | ) iii Furthermore, when the solution is unique, it is locally Lipschitz continuous in the initial value, uniformly in time. Rémi Catellier (CHL - IRMAR) October 24, 2014 6 / 21

Regularization of ODEs 1 The operator T w : ( ρ, γ ) -irregular functions 2 Stochastic processes 3 Rémi Catellier (CHL - IRMAR) October 24, 2014 7 / 21