Damir Kinzebulatov, U Laval, Qu ebec - PowerPoint PPT Presentation

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 1 / 27

Brownian motion with general drift Damir Kinzebulatov joint with Yuliy A. Sem¨ enov, Toronto Probability and Analysis 2019 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 1 / 27

This talk The subject of this talk: existence and uniqueness (in law) of weak solution to X (0) = x ∈ R d , dX t = − b ( X t ) dt + dW t , for a locally unbounded general b : R d → R d , d ≥ 3 Admissible singularities of b ? Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 2 / 27

This talk The subject of this talk: existence and uniqueness (in law) of weak solution to X (0) = x ∈ R d , dX t = − b ( X t ) dt + dW t , for a locally unbounded general b : R d → R d , d ≥ 3 Admissible singularities of b ? Principal results: N. I. Portenko (1982): | b | ∈ L p + L ∞ , p > d (analytic proof or via Girsanov transform) Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 2 / 27

This talk The subject of this talk: existence and uniqueness (in law) of weak solution to X (0) = x ∈ R d , dX t = − b ( X t ) dt + dW t , for a locally unbounded general b : R d → R d , d ≥ 3 Admissible singularities of b ? Principal results: N. I. Portenko (1982): | b | ∈ L p + L ∞ , p > d (analytic proof or via Girsanov transform) R. Bass-Z.-Q. Chen (2003) for a larger class: b ∈ K d +1 0 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 2 / 27

of vector fields b : R d → R d Kato class K d +1 0 A vector field b is in the Kato class K d +1 , 0 < δ < 1 , if δ �| b | ( λ − ∆) − 1 2 � 1 → 1 ≤ δ (a way to say that b · ∇ ≤ − ∆ in L 1 ) Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 3 / 27

of vector fields b : R d → R d Kato class K d +1 0 A vector field b is in the Kato class K d +1 , 0 < δ < 1 , if δ �| b | ( λ − ∆) − 1 2 � 1 → 1 ≤ δ (a way to say that b · ∇ ≤ − ∆ in L 1 ) Then ≡ K d +1 := K d +1 � K d +1 0 δ δ> 0 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 3 / 27

of vector fields b : R d → R d Kato class K d +1 0 A vector field b is in the Kato class K d +1 , 0 < δ < 1 , if δ �| b | ( λ − ∆) − 1 2 � 1 → 1 ≤ δ (a way to say that b · ∇ ≤ − ∆ in L 1 ) Then ≡ K d +1 := K d +1 � K d +1 0 δ δ> 0 There are b ∈ K d +1 such that | b | �∈ L 1+ ε loc (this excludes Girsanov transform) 0 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 3 / 27

Kato class K d +1 0 The role of Kato class: Qi S. Zhang (1996): Gaussian bounds on the heat kernel of − ∆ + b · ∇ , K d +1 0 R. Bass-Z.-Q. Chen (2003): SDEs with b ∈ K d +1 0 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 4 / 27

Kato class K d +1 0 The role of Kato class: Qi S. Zhang (1996): Gaussian bounds on the heat kernel of − ∆ + b · ∇ , K d +1 0 R. Bass-Z.-Q. Chen (2003): SDEs with b ∈ K d +1 – This talk: larger class of 0 drifts b Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 4 / 27

The class of drifts in this talk 1 A vector field b is in the class F δ , 0 < δ < 1 , if 2 1 2 ( λ − ∆) − 1 4 � 2 → 2 ≤ δ �| b | 2 in L 2 (cf. − ∆ + b · ∇ , where, roughly, ∇ ≃ ( − ∆) 1 1 2 ) (a way to say | b | ≤ ( − ∆) Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 5 / 27

The class of drifts in this talk 1 A vector field b is in the class F δ , 0 < δ < 1 , if 2 1 2 ( λ − ∆) − 1 4 � 2 → 2 ≤ δ �| b | 2 in L 2 (cf. − ∆ + b · ∇ , where, roughly, ∇ ≃ ( − ∆) 1 1 2 ) (a way to say | b | ≤ ( − ∆) Larger than Kato class: 1 K d +1 2 � F δ δ e.g. by interpolation between �| b | ( λ − ∆) − 1 2 � 1 → 1 ≤ δ and (by duality) � ( λ − ∆) − 1 2 | b |� ∞ ≤ δ Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 5 / 27

The class of drifts in this talk 1 δ , but not of K d +1 The following are proper sub-classes of F 2 (increasing order): δ Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk 1 δ , but not of K d +1 The following are proper sub-classes of F 2 (increasing order): δ 1 L d + L ∞ � F δ for any δ > 0 (Sobolev) 2 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk 1 δ , but not of K d +1 The following are proper sub-classes of F 2 (increasing order): δ 1 L d + L ∞ � F δ for any δ > 0 (Sobolev) 2 L d, ∞ + L ∞ � F 1 δ (Strichartz) 2 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk 1 δ , but not of K d +1 The following are proper sub-classes of F 2 (increasing order): δ 1 L d + L ∞ � F δ for any δ > 0 (Sobolev) 2 L d, ∞ + L ∞ � F 1 1 δ (Strichartz) e.g. b ( x ) = δ d − 2 2 | x | − 2 x is in F δ (Hardy) 2 2 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk 1 δ , but not of K d +1 The following are proper sub-classes of F 2 (increasing order): δ 1 L d + L ∞ � F δ for any δ > 0 (Sobolev) 2 L d, ∞ + L ∞ � F 1 1 δ (Strichartz) e.g. b ( x ) = δ d − 2 2 | x | − 2 x is in F δ (Hardy) 2 2 1 Campanato-Morrey class � F δ (a proof in Fefferman-Phong) 2 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk 1 δ , but not of K d +1 The following are proper sub-classes of F 2 (increasing order): δ 1 L d + L ∞ � F δ for any δ > 0 (Sobolev) 2 L d, ∞ + L ∞ � F 1 1 δ (Strichartz) e.g. b ( x ) = δ d − 2 2 | x | − 2 x is in F δ (Hardy) 2 2 1 Campanato-Morrey class � F δ (a proof in Fefferman-Phong) 2 1 S.Y.A. Chang-M. Wilson-T. Wolff class � F 2 δ Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk 1 δ , but not of K d +1 The following are proper sub-classes of F 2 (increasing order): δ 1 L d + L ∞ � F δ for any δ > 0 (Sobolev) 2 L d, ∞ + L ∞ � F 1 1 δ (Strichartz) e.g. b ( x ) = δ d − 2 2 | x | − 2 x is in F δ (Hardy) 2 2 1 Campanato-Morrey class � F δ (a proof in Fefferman-Phong) 2 1 S.Y.A. Chang-M. Wilson-T. Wolff class � F 2 δ 1 1 2 ∔ | b | is well defined Remark: b ∈ F δ ensures that the form-sum ( λ − ∆) 2 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

Main result Theorem 1 d 1 − d Let b ∈ F 1 / 2 2 (2 e ) − 1 1 with 0 < δ < m − 1 d − 2 2 ( d − 1) 2 . 2 d d 4 ( d − 1) 2 , where m d := π δ 1 D. Kinzebulatov, Yu.A. Sem¨ enov, arXiv:1710.06729 2 D. Kinzebulatov, Annali SNS Pisa, 2017 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27

Main result Theorem 1 d 1 − d Let b ∈ F 1 / 2 2 (2 e ) − 1 1 with 0 < δ < m − 1 d − 2 2 ( d − 1) 2 . 2 d d 4 ( d − 1) 2 , where m d := π δ There is Feller generator Λ ⊃ − ∆ + b · ∇ on C ∞ = { f ∈ C b : lim x →∞ f ( x ) = 0 } ( sup -norm) 2 1 D. Kinzebulatov, Yu.A. Sem¨ enov, arXiv:1710.06729 2 D. Kinzebulatov, Annali SNS Pisa, 2017 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27

Main result Theorem 1 d 1 − d Let b ∈ F 1 / 2 1 2 (2 e ) − 1 with 0 < δ < m − 1 d − 2 2 ( d − 1) 2 . 2 d d 4 ( d − 1) 2 , where m d := π δ There is Feller generator Λ ⊃ − ∆ + b · ∇ on C ∞ = { f ∈ C b : lim x →∞ f ( x ) = 0 } ( sup -norm) 2 Let P x be the probability measures on D ([0 , ∞ ) , ¯ R d ) determined by e − t Λ ∞ ( b ) . 1 D. Kinzebulatov, Yu.A. Sem¨ enov, arXiv:1710.06729 2 D. Kinzebulatov, Annali SNS Pisa, 2017 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27

Main result Theorem 1 d 1 − d Let b ∈ F 1 / 2 1 2 (2 e ) − 1 with 0 < δ < m − 1 d − 2 2 ( d − 1) 2 . 2 d d 4 ( d − 1) 2 , where m d := π δ There is Feller generator Λ ⊃ − ∆ + b · ∇ on C ∞ = { f ∈ C b : lim x →∞ f ( x ) = 0 } ( sup -norm) 2 Let P x be the probability measures on D ([0 , ∞ ) , ¯ R d ) determined by e − t Λ ∞ ( b ) . Then: 1. P x are concentrated on C ([0 , ∞ ) , R d ) � t 2. E P x 0 | b ( X ( s )) | ds < ∞ and there exists a d -dimensional Brownian motion W t such that P x a.s. � t √ X t = x − b ( X s ) ds + 2 W t , t ≥ 0 0 1 D. Kinzebulatov, Yu.A. Sem¨ enov, arXiv:1710.06729 2 D. Kinzebulatov, Annali SNS Pisa, 2017 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27