Noisy differential equations with power type coefficients Samy - PowerPoint PPT Presentation

Noisy differential equations with power type coefficients Samy Tindel Université de Lorraine AMS Sectional Meeting - Las Vegas 2015 Ongoing joint work with Jorge León and David Nualart Samy T. (Nancy) Power type coefficients Las Vegas 2015 1 / 22

Outline Introduction 1 Equation under consideration Examples and heuristics Results 2 Samy T. (Nancy) Power type coefficients Las Vegas 2015 2 / 22

Outline Introduction 1 Equation under consideration Examples and heuristics Results 2 Samy T. (Nancy) Power type coefficients Las Vegas 2015 3 / 22

Outline Introduction 1 Equation under consideration Examples and heuristics Results 2 Samy T. (Nancy) Power type coefficients Las Vegas 2015 4 / 22

Typical noisy differential equation Equation: � t d � 0 σ j ( y u ) dx j y t = a + u . (1) j =1 Assumptions: x ∈ C γ ([0 , τ ]; R d ), with γ > 1 / 2 a initial data in R m σ 1 , . . . , σ d vector fields on R m 1 γ then Classical result: If σ ∈ C ֒ → Existence and uniqueness result for (1) Samy T. (Nancy) Power type coefficients Las Vegas 2015 5 / 22

Equation with power type coefficient Equation: for x ∈ C γ ([0 , τ ]; R d ), � t d � 0 σ j ( y u ) dx j y t = a + u . (2) j =1 Further assumptions on σ : For κ ∈ (0 , 1) σ (0) = 0, σ smooth away from 0 | σ ( ξ 2 ) − σ ( ξ 1 ) | � | ξ 2 − ξ 1 | κ in a neighborhood of 0. Case 1: γ + γκ > 1 ֒ → Easily handled by Young integration techniques. Case 2: γ + γκ < 1 ֒ → Case of interest for us. Samy T. (Nancy) Power type coefficients Las Vegas 2015 6 / 22

Outline Introduction 1 Equation under consideration Examples and heuristics Results 2 Samy T. (Nancy) Power type coefficients Las Vegas 2015 7 / 22

Brownian case Feller’s diffusion: Historical process defined as Solution to dX t = β X t dt + σ √ X t ˙ W t , where W ≡ Wiener. Obtained as limit of Galton-Watson processes. 12 8 6 4 2 0 0 1 2 3 4 5 Figure: Simulation for dX t = . 02 X t dt + 2 √ X t dW t on [0 , 5] Samy T. (Nancy) Power type coefficients Las Vegas 2015 8 / 22

Brownian case (2) Related question: Pathwise existence and uniqueness for dY t = | Y t | κ ˙ W t , κ ∈ (0 , 1) . Yamada-Watanabe’s results: If κ ≥ 1 / 2, existence and pathwise uniqueness. If κ < 1 / 2, pathwise uniqueness fails. Samy T. (Nancy) Power type coefficients Las Vegas 2015 9 / 22

Brownian case (3) Heuristics for critical exponent: Reduction to the definition of | Y t | κ ˙ W t as a distribution 1 ˙ W t ∈ C − 1 / 2 − ε 2 Problem to define | Y t | κ ˙ W t : when Y t close to 0 3 ֒ → otherwise power function well-behaved When Y t close to 0: equation becomes noiseless 4 (i) Morally Y ∈ C 1 (ii) Morally | Y | κ ∈ C κ instead of | Y | κ ∈ C κ 2 (iii) Product | Y t | κ ˙ W t well-defined if κ − 1 2 − ε > 0 Conclusion: Critical κ is 1 2 Remark: Heuristic not apparent in Yamada-Watanabe’s proof. Samy T. (Nancy) Power type coefficients Las Vegas 2015 10 / 22

Stochastic PDE case Dawson-Watanabe process: Defined as � X t ( x ) ˙ Solution to ∂ t X t ( x ) = 1 2 ∆ X t ( x ) + W t ( x ) where ˙ W ≡ space-time white noise on R + × R . Obtained as limit of branching Brownian particles. Figure: A simulation by Nicolas Champagnat Samy T. (Nancy) Power type coefficients Las Vegas 2015 11 / 22

Stochastic PDE case (2) Related question: Pathwise existence and uniqueness for ∂ t Y t ( x ) = 1 2∆ Y t ( x ) + | Y t ( x ) | κ ˙ W t , κ ∈ (0 , 1) . Mytnik-Mueller-Perkins’ results: If κ > 3 / 4, existence and pathwise uniqueness. If κ < 3 / 4, pathwise uniqueness fails. Criticality of 3 / 4 can be seen from heuristics Mytnik-Perkin’s proof is a (terrible) elaboration of heuristics. Samy T. (Nancy) Power type coefficients Las Vegas 2015 12 / 22

Stochastic PDE case (3) Heuristics for critical exponent: Reduction to the definition of | Y t ( x ) | κ ˙ W t ( x ) as a distribution 1 W ∈ C − 3 / 2 − ε in parabolic scaling ˙ 2 Problem to define | Y t ( x ) | κ ˙ W t ( x ): when Y t ( x ) close to 0 3 ֒ → otherwise power function well-behaved When Y t ( x ) close to 0: equation becomes noiseless 4 (i) Morally Y ∈ C 2 in parabolic scaling (ii) Morally | Y | κ ∈ C 2 κ instead of | Y | κ ∈ C κ 2 in parabolic scaling (iii) Product | Y t | κ ˙ W t well-defined if 2 κ − 3 2 − ε > 0 Conclusion: Critical κ is 3 4 Samy T. (Nancy) Power type coefficients Las Vegas 2015 13 / 22

Stochastic PDE case (4) Parabolic scaling: for ϕ : R × R → R set S t , x ϕ ( s , y ) = 1 � s − t δ 2 , y − x � δ 3 ϕ δ Distributional exponent: F ∈ C − α in parabolic scaling if � R × R [ S t , x ϕ ]( s , y ) F ( s , y ) dsdy ≤ c ϕ δ − α W ∈ C − 3 / 2 − ε since White noise irregularity: We have ˙ 2 dsdy �� 2 � � � ˙ � � � W ( S t , x ϕ ) = � [ S t , x ϕ ]( s , y ) E � � � � � � R × R 1 � R × R | ϕ ( s , y ) | 2 dsdy = δ 3 Samy T. (Nancy) Power type coefficients Las Vegas 2015 14 / 22

Back to our equation y t = | y t | κ ˙ x t , where x ∈ C γ Equation: ˙ Heuristics for critical exponent: Reduction to the definition of | y t | κ ˙ x t as a distribution 1 x t ∈ C − (1 − γ ) ˙ 2 Problem to define | y t | κ ˙ x t : when y t close to 0 3 ֒ → otherwise power function well-behaved When y t close to 0: equation becomes noiseless 4 (i) Morally y ∈ C 1 (ii) Morally | y | κ ∈ C κ instead of | y | κ ∈ C γ κ (iii) Product | y t | κ ˙ x t well-defined if κ − (1 − γ ) > 0 Conclusion: Critical κ should be κ c = 1 − γ Remark: Our story is in fact quite different! Samy T. (Nancy) Power type coefficients Las Vegas 2015 15 / 22

Outline Introduction 1 Equation under consideration Examples and heuristics Results 2 Samy T. (Nancy) Power type coefficients Las Vegas 2015 16 / 22

Setting Equation: for x ∈ C γ ([0 , τ ]; R d ) with γ > 1 / 2, � t d � 0 σ j ( y u ) dx j y t = a + u . (3) j =1 Further assumptions on σ : σ (0) = 0 | σ ( ξ 2 ) − σ ( ξ 1 ) | � | ξ 2 − ξ 1 | κ for κ ∈ (0 , 1) Case of interest: γ + γκ < 1 Remark: relation γ + κ > 1 does not show up! Noisy integral: Extended Young sense ֒ → See later for details Samy T. (Nancy) Power type coefficients Las Vegas 2015 17 / 22

1-dimensional case Additional set of assumptions: d = 1 and y is real-valued � ξ ds φ ( ξ ) = σ ( s ) well-defined 0 � τ ds | φ η ( x s ) | < ∞ for a certain η < 1 − κ 0 ֒ → Satisfied for x ≡ fBm with H > 1 / 2 Theorem 1. Under assumptions above, y = φ − 1 ( x ) solution to: � t y t = 0 σ ( y s ) dx s Remark: No uniqueness here ( y ≡ 0 is also a solution) Samy T. (Nancy) Power type coefficients Las Vegas 2015 18 / 22

d -dimensional case: setting Additional set of assumptions: | σ j ( ξ ) | � | ξ | κ | σ j ( ξ 2 ) − σ j ( ξ 1 ) | � | ξ 2 − ξ 1 | κ in a neighborhood of 0. | D σ j ( ξ ) | � | ξ | − (1 − κ ) . A stopping time: Set t ∗ = inf { t ; y t = 0 } ∧ τ Samy T. (Nancy) Power type coefficients Las Vegas 2015 19 / 22

d -dimensional case: result Theorem 2. Under assumptions of previous slide ֒ → There exists a solution y to equation (3) in following sense: (i) For t ∈ [0 , t ∗ ] we have � t d � 0 σ j ( y u ) dx j y t = a + u j =1 (ii) On [ t ∗ , τ ], we have y ≡ 0 (iii) Integral understood in extended Young sense Samy T. (Nancy) Power type coefficients Las Vegas 2015 20 / 22

Extended Young integral Proposition 3. Consider: x and y in C γ with γ > 1 / 2 κ such that γ + γκ < 1 σ of the form σ ( ξ ) ≍ | ξ | κ Assume y satisfies: β ≥ 1 − ( γ + γκ ) | y | − 1 ∈ L β ([0 , τ ]) , with γ 2 Then following integral well-defined as limit of Riemann sums: � t 0 σ ( y s ) dx s Samy T. (Nancy) Power type coefficients Las Vegas 2015 21 / 22

Sketch of the proof Strategy for existence result for (3): Proceed by regularization σ → σ n and y → y n y n ∈ C γ � t ∗ s | − β ds 0 | y n Main step: prove uniform bounds on n To prove these bounds: use regularity increase as | y s | close to 0 ֒ → Back to Mytnik-Perkins’ heuristics Samy T. (Nancy) Power type coefficients Las Vegas 2015 22 / 22