Towards an analysis of parabolic Anderson models in very rough - PowerPoint PPT Presentation

Towards an analysis of parabolic Anderson models in very rough environments Samy Tindel Purdue University University of Wyoming – 2019 1 st Meeting for the Northern States Section of Siam Ongoing joint work with A. Deya, X. Chen, C. Ouyang Samy T. (Purdue) Rough PAM Siam 2019 1 / 28

Outline Parabolic Anderson model 1 Main results 2 Feyman-Kac representations 3 Samy T. (Purdue) Rough PAM Siam 2019 2 / 28

Outline Parabolic Anderson model 1 Main results 2 Feyman-Kac representations 3 Samy T. (Purdue) Rough PAM Siam 2019 3 / 28

Some history Philip Anderson: Born 1923 Wide range of achievements ֒ → In condensed matter physics Nobel prize in 1977 Still Professor at Princeton One of Anderson’s discoveries: For particles moving in a disordered media ֒ → Localized behavior instead of diffusion. Samy T. (Purdue) Rough PAM Siam 2019 4 / 28

Equation under consideration Equation: Stochastic heat equation in R d , with very rough environment: ∂ t u t ( x ) = 1 2∆ u t ( x ) + u t ( x ) ˙ W t ( x ) , (1) with t ≥ 0, x ∈ R d (we take d = 1 or d = 2 to simplify presentation). ˙ W space-time Gaussian noise ˙ W rougher than white in some directions. u t ( x ) ˙ W t ( x ) differential: Stratonovich or Skorohod sense. Aim: Define and solve the equation 1 Information on moments of the solution 2 Samy T. (Purdue) Rough PAM Siam 2019 5 / 28

Basic questions A formal decomposition of PAM: In the equation ∂ t u t ( x ) = 1 2∆ u t ( x ) + u t ( x ) ˙ W ( x ) , we have (here ˙ W is a spatial noise) ∂ t u t = 1 2 ∆ u t implies strong smoothing effect ∂ t u t = u t ˙ W implies large fluctuations → Formally we would have u t ( x ) = e t ˙ W ( x ) ֒ Basic question 1: Who wins the above competition? Effect of randomness on u ? Related question 2: Various aspects of localization Samy T. (Purdue) Rough PAM Siam 2019 6 / 28

Localization 1: intermittency phenomenon 2 ∆ u t ( x ) + λ u t ( x ) ˙ Equation: ∂ t u t ( x ) = 1 W t ( x ) Phenomenon: The solution u concentrates its energy in high peaks. Characterization: through moments ֒ → Easy possible definition of intermittency: for all k 1 > k 2 ≥ 1 � � E 1 / k 1 | u t ( x ) | k 1 lim E 1 / k 2 [ | u t ( x ) | k 2 ] = ∞ . t →∞ Results: White noise in time: Khoshnevisan, Foondun, Conus, Joseph Fractional noise in time: Balan-Conus, Hu-Huang-Nualart-T Analysis through Feynman-Kac formula Samy T. (Purdue) Rough PAM Siam 2019 7 / 28

Intemittency: illustration (by Daniel Conus) Simulations: for λ = 0 . 1, 0 . 5, 1 and 2. u(t,x) u(t,x) x x t t u(t,x) u(t,x) x x t t Samy T. (Purdue) Rough PAM Siam 2019 8 / 28

Localization 2: Eigenfunctions Equation with spatial noise: 2 ∆ u t ( x ) + u t ( x ) ˙ ∂ t u t ( x ) = 1 W ( x ), for x ∈ [ − M , M ] d Fact (discrete case): 2 ∆ + ˙ The operator 1 W ( x ) admits a discrete spectrum ( λ k ) ֒ → Corresponding eigenfunction is v k Localization 2: The v k ’s decay exponentially fast around a center x k This is reflected on λ k ֒ → λ k ≃ principal eigenvalue on a ball centered at x k Samy T. (Purdue) Rough PAM Siam 2019 9 / 28

Localization 2: illustration Image (Filoche-Mayboroda): First eigenvectors for a PAM in [0 , 1] 2 Figure: Discrete random potential Figure: First five eigenvectors Samy T. (Purdue) Rough PAM Siam 2019 10 / 28

From spectral localization to u t ( x ) Heuristics: u t (0) related to the Laplace transform at t > 0 2 ∆ + ˙ → for the spectral measure of 1 ֒ W Asymptotics of u t (0) for large t ֒ → Information on spectral measure close to 0 Conclusion: Limiting behavior of E [ | u t (0) | p ] for large p , t Related to 2 ∆ + ˙ Spectral information on 1 W Samy T. (Purdue) Rough PAM Siam 2019 11 / 28

Outline Parabolic Anderson model 1 Main results 2 Feyman-Kac representations 3 Samy T. (Purdue) Rough PAM Siam 2019 12 / 28

Model description Equation: For x ∈ R or x ∈ R 2 we consider  2 ∆ u t ( x ) + u t ( x ) ˙ ∂ t u t ( x ) = 1 W t ( x ) ,  u 0 ( x ) = 1  Model for the noise: We take W fBs with parameters ( H 0 , H 1 , H 2 ) with some H i ∈ (0 , 1 / 2) ˙ W t ( x ) = ∂ tx 1 x 2 W t ( x ) Samy T. (Purdue) Rough PAM Siam 2019 13 / 28

Description of the noise Covariance function for W : We have d � E [ W t ( x ) W s ( y )] = R 0 ( s , t ) R j ( x j , y j ) , j =1 with R j ( u , v ) = 1 | u | 2 H j + | v | 2 H j − | u − v | 2 H j � � , u , v ∈ R . (2) 2 Remarks: We have a fBm in each direction We are rougher than white noise if H j < 1 2 Samy T. (Purdue) Rough PAM Siam 2019 14 / 28

Description of the noise (2) Covariance function for ˙ W : We have formally � ˙ d W t ( x ) ˙ � � W s ( y ) = γ 0 ( t − s ) γ j ( y j − x j ) E j =1 with the following distributional relation: γ j ( u , v ) = ∂ uv R ( u , v ) ’ = ’ | u − v | 2 H j − 2 . (3) Remark: The covariance γ j is given in Fourier mode as � R e ıξ x | ξ | 1 − 2 H j d ξ γ j ( x ) = Samy T. (Purdue) Rough PAM Siam 2019 15 / 28

Skorohod solution Skorohod equation: Of the form  2 ∆ u t ( x ) + u t ( x ) ⋄ ˙ ∂ t u t ( x ) = 1 W t ( x ) ,  u 0 ( x ) = 1 ,  where ⋄ is the Wick product. Mild form: Written as � t � R d p t − s ( x − y ) u s ( y ) d ⋄ W s ( y ) , u t ( x ) = 1 + 0 where the stochastic integral is a Skorohod integral ֒ → extension of Itô from Malliavin calculus. Samy T. (Purdue) Rough PAM Siam 2019 16 / 28

Stratonovich solution Stratonovich equation: Of the form  2 ∆ u t ( x ) + u t ( x ) ˙ ∂ t u t ( x ) = 1 W t ( x ) ,  u 0 ( x ) = 1 ,  where the product is the usual product. Mild form: We have u = (renormalized) − lim ε → 0 u ε , where � t � u ε R d p t − s ( x − y ) u ε s ( y ) dW ε t ( x ) = 1 + s ( y ) , (4) 0 where W ε is a mollification of W and (4) is an ordinary PDE ֒ → Regularity structures. Samy T. (Purdue) Rough PAM Siam 2019 17 / 28

A subcritical zone Theorem 1. Let us assume d = 1 1 H 0 > 1 / 2 and H 1 < 1 / 2 2 H 0 + H 1 > 3 3 4 3 2 < 2 H 0 + H 1 ≤ 2 4 Then we have Global exist. and uniqu. for both u and u ⋄ For all t ≥ 0, x ∈ R and p ≥ 1 we have E [ | u ⋄ t ( x ) | p ] < ∞ , E [ | u t ( x ) | p ] < ∞ and Samy T. (Purdue) Rough PAM Siam 2019 18 / 28

Subcritical zone: illustration In the ( H 0 , H 1 ) plane: H 1 1 1 2 1 4 1 1 3 1 3 H 0 4 8 2 4 Samy T. (Purdue) Rough PAM Siam 2019 19 / 28

Subcritical zone: illustration In the ( H 0 , H 1 ) plane: H 1 1 Young equation well posed 1 2 1 4 1 1 3 1 3 H 0 4 8 2 4 Samy T. (Purdue) Rough PAM Siam 2019 19 / 28

Subcritical zone: illustration In the ( H 0 , H 1 ) plane: H 1 Skorohod equation well posed 1 Young equation well posed 1 2 1 4 1 1 3 1 3 H 0 4 8 2 4 Samy T. (Purdue) Rough PAM Siam 2019 19 / 28

Subcritical zone: illustration In the ( H 0 , H 1 ) plane: H 1 First 2 renormalization regions Skorohod equation well posed 1 Young equation well posed 1 2 1 4 1 1 3 1 3 H 0 4 8 2 4 Samy T. (Purdue) Rough PAM Siam 2019 19 / 28

Subcritical zone: illustration In the ( H 0 , H 1 ) plane: H 1 First 2 renormalization regions Skorohod equation well posed 1 Limit of the Young equation renormalization procedure well posed 1 2 1 4 1 1 3 1 3 H 0 4 8 2 4 Samy T. (Purdue) Rough PAM Siam 2019 19 / 28

A critical zone Theorem 2. Let us assume d = 2 1 W does not depend on time: W = W ( x ) 2 H 1 < 1 / 2 3 H 1 + H 2 = 1 4 Then we have Local exist. and uniqu. for the Skorohod solution u ⋄ Global exist. and uniqu. for the Stratonovich solution u Samy T. (Purdue) Rough PAM Siam 2019 20 / 28

A critical zone (2) Theorem 3. Under the same conditions as in Theorem 2 consider p > 1 Then There exists τ ⋄ p such that for all t > τ ⋄ p , x ∈ R we have t < τ ⋄  < ∞ , p ,   E [ | u ⋄ t ( x ) | p ] t > τ ⋄  = ∞ , p .  For p ≥ 2, exact expression for τ ⋄ p Upper bound for τ ⋄ p when 1 < p < 2 Finite moments for the Strato solution u t ( x ) for small t ’s Samy T. (Purdue) Rough PAM Siam 2019 21 / 28

Comments on the results Previous results on asymptotic behavior of moments: H 0 = 1 2 , Itô framework: Khoshnevisan, Conus, Foondun Young type cases, 2 H 0 + H 1 > 2: Balan-Conus, Hu-Huang-Nualart-T, X. Chen Rough Skorohod case: X. Chen Previous results on renormalization: Hairer-Labbé, Deya Our contribution: Existence of moments for renormalized versions of PAM Link between renormalized Skorohod and Stratonovich ֒ → Through Feyman-Kac representations Samy T. (Purdue) Rough PAM Siam 2019 22 / 28

Outline Parabolic Anderson model 1 Main results 2 Feyman-Kac representations 3 Samy T. (Purdue) Rough PAM Siam 2019 23 / 28

Feynman-Kac for the Skorohod equation Regularized Feynman-Kac potential: For ε > 0 and a Brownian motion B , set � t � V ε, B R 2 p ε ( B x ( x ) = t − r − y ) dW s ( y ) (5) t 0 Regularized Feynman-Kac compensator: � � R d e − ε | ξ | 2 e ı � ξ, B t − s 1 − B t − s 2 � γ 0 ( s 1 − s 2 ) µ ( d ξ ) β ε, B = t [0 , t ] 2 where d | ξ j | 1 − 2 H j d ξ � µ ( d ξ ) = j =1 Samy T. (Purdue) Rough PAM Siam 2019 24 / 28