A new approach to the L p -theory of + b , and its applications - PowerPoint PPT Presentation

A new approach to the L p -theory of − ∆ + b · ∇ , and its applications to Feller processes with general drifts. Damir Kinzebulatov (McGill and the CRM) www.math.toronto.edu/dkinz October 2015 1 / 34

The key component of many models of Math. Physics: Brownian motion A trajectory of a three-dimensional Brownian motion Brownian motion is modeled by Wiener process W t (where t � 0 , W 0 = 0 ) 2 / 34

The key component of many models of Math. Physics: Brownian motion A trajectory of a three-dimensional Brownian motion Brownian motion is modeled by Wiener process W t (where t � 0 , W 0 = 0 ) The simplest diffusion processes 2 / 34

Brownian motion perturbed by a (singular) drift b : R d → R d ? 3 / 34

Brownian motion perturbed by a (singular) drift b : R d → R d ? Varadhan, Strook, Albeverio, Krylov, Carlen and many others . . . 3 / 34

Brownian motion perturbed by a (singular) drift b : R d → R d ? Varadhan, Strook, Albeverio, Krylov, Carlen and many others . . . The long search for the cricial singularities of the drift b . . . 3 / 34

A bridge between Probability and Analysis Probability Analysis ← → − ∆ W t 4 / 34

A bridge between Probability and Analysis Precisely, given a realization W s ∈ R d , s < t , we have � � = e ( t − s )∆ 1 A P [ W t ∈ A ] W s � �� � � �� � Probability Analysis i.e. to find the probability we need to solve the heat equation 5 / 34

A bridge between Probability and Analysis Precisely, given a realization W s ∈ R d , s < t , we have � � = e ( t − s )∆ 1 A P [ W t ∈ A ] W s � �� � � �� � Probability Analysis i.e. to find the probability we need to solve the heat equation ⇒ Analytic methods in Probability 5 / 34

A bridge between Probability and Analysis By analogy: let X t be a Brownian motion perturbed by drift b : R d → R d Then we must have � � e − ( t − s )( − ∆+ b ·∇ ) 1 A P [ X t ∈ A ] = ( X s ) , s < t 6 / 34

A bridge between Probability and Analysis By analogy: let X t be a Brownian motion perturbed by drift b : R d → R d Then we must have � � e − ( t − s )( − ∆+ b ·∇ ) 1 A P [ X t ∈ A ] = ( X s ) , s < t e.g. take b ≡ 0 ⇒ X t = W t 6 / 34

A bridge between Probability and Analysis By analogy: let X t be a Brownian motion perturbed by drift b : R d → R d Then we must have � � e − ( t − s )( − ∆+ b ·∇ ) 1 A P [ X t ∈ A ] = ( X s ) , s < t e.g. take b ≡ 0 ⇒ X t = W t We can solve the heat equation for fairly singular b ’s. But . . . 6 / 34

A bridge between Probability and Analysis . . . will the solutions of the heat equation determine a diffusion process? 7 / 34

A bridge between Probability and Analysis . . . will the solutions of the heat equation determine a diffusion process? The research program started in 1980s, closely tied to the progress in PDEs, and continuing within emerging areas of Probability (SPDEs). . . 7 / 34

What singularities of the drift are admissible? 8 / 34

What singularities of the drift are admissible? d � 3 8 / 34

� The best possible result in terms of L p -spaces Denote L p = L p ( R d ) Stampacchia . . . Krylov, R¨ ockner, Stannat and many others L d + L ∞ L p + L ∞ ( p > d ) 9 / 34 ✶

Critical drifts? Example: A vector field having critical singularity b ( x ) := x | x | − 2 , x ∈ R 3 (clearly, b �∈ L d + L ∞ ) 10 / 34

Critical drifts? Example: A vector field having critical singularity b ( x ) := x | x | − 2 , x ∈ R 3 (clearly, b �∈ L d + L ∞ ) There is a diffusion process X t with drift b . In fact, a weak solution of dX t = x | x | − 2 dt + dW t , X 0 = 0 10 / 34

Critical drifts? Example: A vector field having critical singularity b ( x ) := x | x | − 2 , x ∈ R 3 (clearly, b �∈ L d + L ∞ ) There is a diffusion process X t with drift b . In fact, a weak solution of dX t = x | x | − 2 dt + dW t , X 0 = 0 Replace x | x | − 2 with (1 + ε ) x | x | − 2 and the process will cease to exist 10 / 34

In other words, singularities of b are critical if they are sensitive to multiplication by constants 11 / 34

In other words, singularities of b are critical if they are sensitive to multiplication by constants The singularities of a b ∈ L d are sub-critical 11 / 34

The classes of critical vector fields previously studied in the literature 12 / 34

Critical point singularities of the drift The class of form-bounded vector fields 1 � � √ � 2 � � | b | ( λ − ∆) − 1 b ∈ L 2 F δ := loc : lim δ L 2 → L 2 � � λ ↑∞ √ δ d − 2 2 x | x | − 2 Example: b ( x ) = (Hardy inequality) 1 In relatively elementary terms: Kerman-Sawyer, Chang-Wilson-Wolff 13 / 34

Critical point singularities of the drift The class of form-bounded vector fields 1 � � √ � 2 � � | b | ( λ − ∆) − 1 b ∈ L 2 F δ := loc : lim δ L 2 → L 2 � � λ ↑∞ √ δ d − 2 2 x | x | − 2 Example: b ( x ) = (Hardy inequality) F δ is ‘responsible’ for dissipativity of − ∆ + b · ∇ in L p ⇒ a diffusion via a Moser-type iterative procedure of Kovalenko-Semenov 1 In relatively elementary terms: Kerman-Sawyer, Chang-Wilson-Wolff 13 / 34

Critical hypersurface singularities of the drift The Kato class of vector fields � � � 2 � � | b | ( λ − ∆) − 1 K d +1 b ∈ L 1 := loc : lim L 1 → L 1 � δ � δ λ ↑∞ 2 Yu. Semenov, Q. S. Zhang, B. Davies . . . earlier, J. Nash, . . . 14 / 34

Critical hypersurface singularities of the drift The Kato class of vector fields � � � 2 � � | b | ( λ − ∆) − 1 K d +1 b ∈ L 1 := loc : lim L 1 → L 1 � δ � δ λ ↑∞ Example: � � � − γ , | b ( x ) | = � | x | − 1 γ < 1 is in K d +1 0 2 Yu. Semenov, Q. S. Zhang, B. Davies . . . earlier, J. Nash, . . . 14 / 34

Critical hypersurface singularities of the drift The Kato class of vector fields � � � 2 � � | b | ( λ − ∆) − 1 K d +1 b ∈ L 1 := loc : lim L 1 → L 1 � δ � δ λ ↑∞ Example: � � � − γ , | b ( x ) | = � | x | − 1 γ < 1 is in K d +1 0 is ‘responsible’ for the Gaussian bounds 2 for − ∆ + b · ∇ K d +1 δ ⇒ the Gaussian bounds yield a diffusion 2 Yu. Semenov, Q. S. Zhang, B. Davies . . . earlier, J. Nash, . . . 14 / 34

Classes K d +1 and F δ play prominent role in Analysis δ Crucial features: – Defined in terms of the operators that constitute the problem 3 3 This intuition worked, in particular, in unique continuation for Schr¨ odinger operators with form-bounded potentials (Kinzebulatov-Shartser, JFA, 2010) 15 / 34

Classes K d +1 and F δ play prominent role in Analysis δ Crucial features: – Defined in terms of the operators that constitute the problem 3 – Are integral conditions, i.e. the geometry of the singularities is not important (well, . . . ) . . . realizations of random fields (SHE) 3 This intuition worked, in particular, in unique continuation for Schr¨ odinger operators with form-bounded potentials (Kinzebulatov-Shartser, JFA, 2010) 15 / 34

Classes K d +1 and F δ play prominent role in Analysis δ Crucial features: – Defined in terms of the operators that constitute the problem 3 – Are integral conditions, i.e. the geometry of the singularities is not important (well, . . . ) . . . realizations of random fields (SHE) – What really matters is the relative bound δ > 0 (as in δx | x | − 2 ; has to be small, so that b · ∇ ≤ − ∆ ). For instance, L d ⊂ F 0 := � δ> 0 F δ . 3 This intuition worked, in particular, in unique continuation for Schr¨ odinger operators with form-bounded potentials (Kinzebulatov-Shartser, JFA, 2010) 15 / 34

Classes K d +1 and F δ play prominent role in Analysis δ Crucial features: – Defined in terms of the operators that constitute the problem 3 – Are integral conditions, i.e. the geometry of the singularities is not important (well, . . . ) . . . realizations of random fields (SHE) – What really matters is the relative bound δ > 0 (as in δx | x | − 2 ; has to be small, so that b · ∇ ≤ − ∆ ). For instance, L d ⊂ F 0 := � δ> 0 F δ . – Are L 1 , L 2 -conditions (e.g. Kato class of measure-valued drifts: Bass-Chen, Kim-Song), cf. “ b ∈ L d ” 3 This intuition worked, in particular, in unique continuation for Schr¨ odinger operators with form-bounded potentials (Kinzebulatov-Shartser, JFA, 2010) 15 / 34

� � � The state of affairs (not so long ago) K d +1 F δ δ � ✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱✱ ✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒ L d, ∞ + L ∞ L d, ∞ + L ∞ ✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒ L d + L ∞ L d + L ∞ ✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒ ✒ L p + L ∞ ( p > d ) L p + L ∞ ( p > d ) 16 / 34 ✶

The search for ‘the right’ class of critical drifts b The next step: b 1 ∈ K d +1 b := b 1 + b 2 , , b 2 ∈ F δ δ (i.e. b combines critical point and critical hypersurface singularities) 17 / 34