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 dXt = −b(Xt)dt + dWt, X(0) = x ∈ Rd, for a locally unbounded general b : Rd → Rd, 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 dXt = −b(Xt)dt + dWt, X(0) = x ∈ Rd, for a locally unbounded general b : Rd → Rd, d ≥ 3 Admissible singularities of b? Principal results:

• N. I. Portenko (1982): |b| ∈ Lp + 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 dXt = −b(Xt)dt + dWt, X(0) = x ∈ Rd, for a locally unbounded general b : Rd → Rd, d ≥ 3 Admissible singularities of b? Principal results:

• N. I. Portenko (1982): |b| ∈ Lp + L∞, p > d

(analytic proof or via Girsanov transform)

• R. Bass-Z.-Q. Chen (2003) for a larger class: b ∈ Kd+1

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 2 / 27

Kato class Kd+1

• f vector fields b : Rd → Rd

A vector field b is in the Kato class Kd+1

δ

, 0 < δ < 1, if |b|(λ − ∆)− 1

2 1→1 ≤ δ

(a way to say that b · ∇ ≤ −∆ in L1)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 3 / 27

Kato class Kd+1

• f vector fields b : Rd → Rd

A vector field b is in the Kato class Kd+1

δ

, 0 < δ < 1, if |b|(λ − ∆)− 1

2 1→1 ≤ δ

(a way to say that b · ∇ ≤ −∆ in L1) Then Kd+1 ≡ Kd+1 :=

• δ>0

Kd+1

δ

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 3 / 27

Kato class Kd+1

• f vector fields b : Rd → Rd

A vector field b is in the Kato class Kd+1

δ

, 0 < δ < 1, if |b|(λ − ∆)− 1

2 1→1 ≤ δ

(a way to say that b · ∇ ≤ −∆ in L1) Then Kd+1 ≡ Kd+1 :=

• δ>0

Kd+1

δ

There are b ∈ Kd+1 such that |b| ∈ L1+ε

loc (this excludes Girsanov transform)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 3 / 27

Kato class Kd+1 The role of Kato class: Qi S. Zhang (1996): Gaussian bounds on the heat kernel of −∆ + b · ∇, Kd+1

• R. Bass-Z.-Q. Chen (2003): SDEs with b ∈ Kd+1

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 4 / 27

Kato class Kd+1 The role of Kato class: Qi S. Zhang (1996): Gaussian bounds on the heat kernel of −∆ + b · ∇, Kd+1

• R. Bass-Z.-Q. Chen (2003): SDEs with b ∈ Kd+1

– This talk: larger class of drifts b

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 4 / 27

The class of drifts in this talk A vector field b is in the class F

1 2

δ , 0 < δ < 1, if

|b|

1 2 (λ − ∆)− 1 4 2→2 ≤ δ

(a way to say |b| ≤ (−∆)

1 2 in L2 (cf. −∆ + b · ∇, where, roughly, ∇ ≃ (−∆) 1 2 ) Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 5 / 27

The class of drifts in this talk A vector field b is in the class F

1 2

δ , 0 < δ < 1, if

|b|

1 2 (λ − ∆)− 1 4 2→2 ≤ δ

(a way to say |b| ≤ (−∆)

1 2 in L2 (cf. −∆ + b · ∇, where, roughly, ∇ ≃ (−∆) 1 2 )

Larger than Kato class: Kd+1

δ

F

1 2

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 The following are proper sub-classes of F

1 2

δ , but not of Kd+1 δ

(increasing order):

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk The following are proper sub-classes of F

1 2

δ , but not of Kd+1 δ

(increasing order): Ld + L∞ F

1 2

δ for any δ > 0 (Sobolev)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk The following are proper sub-classes of F

1 2

δ , but not of Kd+1 δ

(increasing order): Ld + L∞ F

1 2

δ for any δ > 0 (Sobolev)

Ld,∞ + L∞ F

1 2

δ (Strichartz)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk The following are proper sub-classes of F

1 2

δ , but not of Kd+1 δ

(increasing order): Ld + L∞ F

1 2

δ for any δ > 0 (Sobolev)

Ld,∞ + L∞ F

1 2

δ (Strichartz) e.g. b(x) = δ d−2 2 |x|−2x is in F

1 2

δ (Hardy)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk The following are proper sub-classes of F

1 2

δ , but not of Kd+1 δ

(increasing order): Ld + L∞ F

1 2

δ for any δ > 0 (Sobolev)

Ld,∞ + L∞ F

1 2

δ (Strichartz) e.g. b(x) = δ d−2 2 |x|−2x is in F

1 2

δ (Hardy)

Campanato-Morrey class F

1 2

δ (a proof in Fefferman-Phong)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk The following are proper sub-classes of F

1 2

δ , but not of Kd+1 δ

(increasing order): Ld + L∞ F

1 2

δ for any δ > 0 (Sobolev)

Ld,∞ + L∞ F

1 2

δ (Strichartz) e.g. b(x) = δ d−2 2 |x|−2x is in F

1 2

δ (Hardy)

Campanato-Morrey class F

1 2

δ (a proof in Fefferman-Phong)

S.Y.A. Chang-M. Wilson-T. Wolff class F

1 2

δ

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

The class of drifts in this talk The following are proper sub-classes of F

1 2

δ , but not of Kd+1 δ

(increasing order): Ld + L∞ F

1 2

δ for any δ > 0 (Sobolev)

Ld,∞ + L∞ F

1 2

δ (Strichartz) e.g. b(x) = δ d−2 2 |x|−2x is in F

1 2

δ (Hardy)

Campanato-Morrey class F

1 2

δ (a proof in Fefferman-Phong)

S.Y.A. Chang-M. Wilson-T. Wolff class F

1 2

δ

Remark: b ∈ F

1 2

δ ensures that the form-sum (λ − ∆)

1 2 ∔ |b| is well defined Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 6 / 27

Main result Theorem1 Let b ∈ F1/2

δ

with 0 < δ < m−1

d 4 d−2 (d−1)2 , where md := π

1 2 (2e)− 1 2 d d 2 (d − 1) 1−d 2 .

• 1D. Kinzebulatov, Yu.A. Sem¨

enov, arXiv:1710.06729

• 2D. Kinzebulatov, Annali SNS Pisa, 2017

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27

Main result Theorem1 Let b ∈ F1/2

δ

with 0 < δ < m−1

d 4 d−2 (d−1)2 , where md := π

1 2 (2e)− 1 2 d d 2 (d − 1) 1−d 2 .

There is Feller generator Λ ⊃ −∆ + b · ∇ on C∞ = {f ∈ Cb : limx→∞ f(x) = 0} (sup-norm)2

• 1D. Kinzebulatov, Yu.A. Sem¨

enov, arXiv:1710.06729

• 2D. Kinzebulatov, Annali SNS Pisa, 2017

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27

Main result Theorem1 Let b ∈ F1/2

δ

with 0 < δ < m−1

d 4 d−2 (d−1)2 , where md := π

1 2 (2e)− 1 2 d d 2 (d − 1) 1−d 2 .

There is Feller generator Λ ⊃ −∆ + b · ∇ on C∞ = {f ∈ Cb : limx→∞ f(x) = 0} (sup-norm)2 Let Px be the probability measures on D([0, ∞), ¯ Rd) determined by e−tΛ∞(b).

• 1D. Kinzebulatov, Yu.A. Sem¨

enov, arXiv:1710.06729

• 2D. Kinzebulatov, Annali SNS Pisa, 2017

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27

Main result Theorem1 Let b ∈ F1/2

δ

with 0 < δ < m−1

d 4 d−2 (d−1)2 , where md := π

1 2 (2e)− 1 2 d d 2 (d − 1) 1−d 2 .

There is Feller generator Λ ⊃ −∆ + b · ∇ on C∞ = {f ∈ Cb : limx→∞ f(x) = 0} (sup-norm)2 Let Px be the probability measures on D([0, ∞), ¯ Rd) determined by e−tΛ∞(b). Then:

• 1. Px are concentrated on C([0, ∞), Rd)
• 2. EPx

t

0 |b(X(s))|ds < ∞ and there exists a d-dimensional Brownian motion

Wt such that Px a.s. Xt = x − t b(Xs)ds + √ 2Wt, t ≥ 0

• 1D. Kinzebulatov, Yu.A. Sem¨

enov, arXiv:1710.06729

• 2D. Kinzebulatov, Annali SNS Pisa, 2017

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27

Main result Theorem1 Let b ∈ F1/2

δ

with 0 < δ < m−1

d 4 d−2 (d−1)2 , where md := π

1 2 (2e)− 1 2 d d 2 (d − 1) 1−d 2 .

There is Feller generator Λ ⊃ −∆ + b · ∇ on C∞ = {f ∈ Cb : limx→∞ f(x) = 0} (sup-norm)2 Let Px be the probability measures on D([0, ∞), ¯ Rd) determined by e−tΛ∞(b). Then:

• 1. Px are concentrated on C([0, ∞), Rd)
• 2. EPx

t

0 |b(X(s))|ds < ∞ and there exists a d-dimensional Brownian motion

Wt such that Px a.s. Xt = x − t b(Xs)ds + √ 2Wt, t ≥ 0

• 3. (Uniqueness) If {Qx}x∈Rd is another weak solution such that

Qx = w- lim

n Px(˜

bn) for every x ∈ Rd, where {˜ bn} ⊂ F

1/2

δ

∩ C∞ then {Qx}x∈Rd = {Px}x∈Rd.

• 1D. Kinzebulatov, Yu.A. Sem¨

enov, arXiv:1710.06729

• 2D. Kinzebulatov, Annali SNS Pisa, 2017

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 7 / 27

Example: Critical-order singularity Let b(x) = c|x|−2x (is in F

1 2

δ )

Xt = −c t |Xs|−2Xsds + √ 2Wt

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 8 / 27

Example: Critical-order singularity Let b(x) = c|x|−2x (is in F

1 2

δ )

Xt = −c t |Xs|−2Xsds + √ 2Wt (c > 0, i.e. it pushes BM towards the origin)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 8 / 27

Example: Critical-order singularity Let b(x) = c|x|−2x (is in F

1 2

δ )

Xt = −c t |Xs|−2Xsds + √ 2Wt (c > 0, i.e. it pushes BM towards the origin) A simple argument shows: if c ≥ d, then there is no weak solution By Theorem, if c < m−1

d 2 (d−2)2 (d−1)2 , weak solution exists

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 8 / 27

Example: Critical-order singularity Let b(x) = c|x|−2x (is in F

1 2

δ )

Xt = −c t |Xs|−2Xsds + √ 2Wt (c > 0, i.e. it pushes BM towards the origin) A simple argument shows: if c ≥ d, then there is no weak solution By Theorem, if c < m−1

d 2 (d−2)2 (d−1)2 , weak solution exists

(b has a critical-order singularity at x = 0, i.e. SDE “senses” c)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 8 / 27

Comment: Gaussian bounds are no longer available “b ∈ F

1 2

δ ” destroys Gaussian bounds on the heat kernel of Λ ⊃ −∆ + b · ∇

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 9 / 27

Comment: Gaussian bounds are no longer available “b ∈ F

1 2

δ ” destroys Gaussian bounds on the heat kernel of Λ ⊃ −∆ + b · ∇

For example, for critical drift b(x) = δ d−2

2 |x|−2x, via desingularizing weights of

Milman-Sem¨ enov: Metafune, Sobajima, Spina, 2017 established sharp two-sided estimates e−tΛ(x, y) ≈ et∆(x, y)ϕt(y), where ϕt(y) =

• |y|−σ,

|y| √ t ≤ 1, 1 2, |y| √ t ≥ 2,

with σ = d−2

2 (1 −

√ 1 − δ)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 9 / 27

Proof: The central analytic object is a Feller generator Suppose b ∈ Cb Then Λ = −∆ + b · ∇, D(Λ) = (1 − ∆)−1C∞ is a Feller generator By a classical result, e−tΛ determines probability measures Px on D([0, ∞), ¯ Rd) such that t → f(Xt) − f(x) − t (Λf)(Xs)ds is a Px martingale for all f ∈ D(Λ)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 10 / 27

Proof: The central analytic object is a Feller generator Suppose b ∈ Cb Then Λ = −∆ + b · ∇, D(Λ) = (1 − ∆)−1C∞ is a Feller generator By a classical result, e−tΛ determines probability measures Px on D([0, ∞), ¯ Rd) such that t → f(Xt) − f(x) − t (Λf)(Xs)ds is a Px martingale for all f ∈ D(Λ) Here: know everything about D(Λ)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 10 / 27

Proof: Can’t construct Feller generator as the algebraic sum for singular b Let b ∈ F

1 2

δ

Feller generator Λ = Λ(b) can’t be defined on C∞ as the algebraic sum −∆ + b · ∇ (the latter, in general, will not densely defined)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 11 / 27

Proof: Can’t construct Feller generator as the algebraic sum for singular b Let b ∈ F

1 2

δ

Feller generator Λ = Λ(b) can’t be defined on C∞ as the algebraic sum −∆ + b · ∇ (the latter, in general, will not densely defined) Implicit connection: Set bn = eεn∆(1nb), 1n = 1{|x|≤n,|b(x)|≤n}, define Tt := s-C∞- lim

n e−tΛ(bn)

(existence?) where Λ(bn) = −∆ + bn · ∇, D(Λ) = (1 − ∆)−1C∞ The generator of Tt =: e−tΛ is a realization of −∆ + b · ∇ on C∞ D(Λ) =? (can’t hope to have an exhaustive description; C∞

c

⊂ D(Λ))

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 11 / 27

Detour in L2 A simpler problem: −∆ + b · ∇ as generator on some Lp The Kato class Kd+1

δ

condition says that b · ∇ is a Miyadera perturbation of −∆ in L1, so Λ1 := −∆ + b · ∇, D(Λ1) = D(−∆) in L1 generates a C0 semigroup in L1

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 12 / 27

Detour in L2 A simpler problem: −∆ + b · ∇ as generator on some Lp The Kato class Kd+1

δ

condition says that b · ∇ is a Miyadera perturbation of −∆ in L1, so Λ1 := −∆ + b · ∇, D(Λ1) = D(−∆) in L1 generates a C0 semigroup in L1 Moreover, since e−tΛ1f∞ ≤ f∞, f ∈ L1 ∩ L∞, one can define e−tΛp :=

• e−tΛ1 ↾L1∩Lp

clos

Lp→Lp ,

1 ≤ p < ∞

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 12 / 27

Detour in L2 A simpler problem: −∆ + b · ∇ as generator on some Lp If b ∈ F

1 2

δ : The standard tools of Perturbation Theory (such as Miyadera’s and

Hille’s theorems, form-method) are not applicable

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 13 / 27

Detour in L2 A simpler problem: −∆ + b · ∇ as generator on some Lp If b ∈ F

1 2

δ : The standard tools of Perturbation Theory (such as Miyadera’s and

Hille’s theorems, form-method) are not applicable e.g. form-method ∇u, ∇u + b · ∇u, u ≥ (1 − ε)∇u, ∇u − 1 4ε|b|2u, u, ε < 1 (but b is only in L1

loc!)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 13 / 27

Detour in L2 Well, not quite: We can construct Λ2 ⊃ −∆ + b · ∇ with b ∈ F

1 2

δ using ideas of Lions and Hille3

(think of H+ ⊂ H ⊂ H− but with 6 spaces) Interesting: In comparison with the Kato-Lions-Lax-Milgram-Nelson (KLMN) Theorem, our approach yields a larger class of vector fields + greater regularity

• 3D. Kinzebulatov, Yu. A. Semenov, “On the theory of the Kolmogorov operator in the spaces

Lp and C∞. I”, arXiv:1709.08598

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 14 / 27

Historical remark

• 1. Well developed regularity theory of −∆ + b · ∇ with b ∈ Kd+1

δ

(Feller semigroup, Gaussian bounds)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 15 / 27

Historical remark

• 1. Well developed regularity theory of −∆ + b · ∇ with b ∈ Kd+1

δ

(Feller semigroup, Gaussian bounds)

• 2. On the other hand, Kovalenko-Sem¨

enov (1990): For b ∈ Fδ, i.e. |b|(λ − ∆)− 1

2 2→2 ≤ δ

(contains e.g. Hardy drift b(x) = δ d−2

2 |x|−2x and Ld,∞, Campanato-Morrey,

but not Kd+1 ) constructed Feller semigroup via Moser-type iterations

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 15 / 27

Historical remark

• 1. Well developed regularity theory of −∆ + b · ∇ with b ∈ Kd+1

δ

(Feller semigroup, Gaussian bounds)

• 2. On the other hand, Kovalenko-Sem¨

enov (1990): For b ∈ Fδ, i.e. |b|(λ − ∆)− 1

2 2→2 ≤ δ

(contains e.g. Hardy drift b(x) = δ d−2

2 |x|−2x and Ld,∞, Campanato-Morrey,

but not Kd+1 ) constructed Feller semigroup via Moser-type iterations Relies on accretivity of −∆ + b · ∇ on Lp for b ∈ Fδ (don’t have it for b ∈ F

1 2

δ )

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 15 / 27

Historical remark

• 1. Well developed regularity theory of −∆ + b · ∇ with b ∈ Kd+1

δ

(Feller semigroup, Gaussian bounds)

• 2. On the other hand, Kovalenko-Sem¨

enov (1990): For b ∈ Fδ, i.e. |b|(λ − ∆)− 1

2 2→2 ≤ δ

(contains e.g. Hardy drift b(x) = δ d−2

2 |x|−2x and Ld,∞, Campanato-Morrey,

but not Kd+1 ) constructed Feller semigroup via Moser-type iterations Relies on accretivity of −∆ + b · ∇ on Lp for b ∈ Fδ (don’t have it for b ∈ F

1 2

δ )

Fδ1 + Kd+1

δ2

⊂ F

1 2

δ ,

δ = δ1 + δ2

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 15 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Start with an operator-valued function on Reζ ≥ λ Θp(ζ, b) := (ζ − ∆)−1 − (ζ − ∆)− 1

2 − 1 2q Qp(1 + Tp)−1Gp(ζ − ∆)− 1 2r′ Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 16 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Start with an operator-valued function on Reζ ≥ λ Θp(ζ, b) := (ζ − ∆)−1 − (ζ − ∆)− 1

2 − 1 2q Qp(1 + Tp)−1Gp(ζ − ∆)− 1 2r′

where Qp := (λ − ∆)

−

1 2q′ |b| 1 p′

Gp := b

1 p · ∇(λ − ∆)− 1 2 − 1 2r ,

Tp := b

1 p · ∇(λ − ∆)−1|b| 1 p′ ,

b

1 p := b|b| 1 p −1

for r < p < q < ∞ (a “candidate for the resolvent”; formally, Neumann series)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 16 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Start with an operator-valued function on Reζ ≥ λ Θp(ζ, b) := (ζ − ∆)−1 − (ζ − ∆)− 1

2 − 1 2q Qp(1 + Tp)−1Gp(ζ − ∆)− 1 2r′

where Qp := (λ − ∆)

−

1 2q′ |b| 1 p′

Gp := b

1 p · ∇(λ − ∆)− 1 2 − 1 2r ,

Tp := b

1 p · ∇(λ − ∆)−1|b| 1 p′ ,

b

1 p := b|b| 1 p −1

for r < p < q < ∞ (a “candidate for the resolvent”; formally, Neumann series) Need: Tp ∈ B(Lp) (same for Qp, Gp)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 16 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Need: Tp ∈ B(Lp) (same for Qp, Gp) b ∈ F

1 2

δ ⇔ |b|

1 2 (λ − ∆)− 1 2 |b| 1 2 2→2 ≤ mdδ yields (principal step)

|b|

1 p (λ − ∆)− 1 2 |b| 1 p′ p→p ≤ cpmdδ

(via Lp-inequalities of Liskevich-Sem¨ enov (1996) between (λ − ∆)

1 2 and

“potential” |b|)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 17 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Need: Tp ∈ B(Lp) (same for Qp, Gp) b ∈ F

1 2

δ ⇔ |b|

1 2 (λ − ∆)− 1 2 |b| 1 2 2→2 ≤ mdδ yields (principal step)

|b|

1 p (λ − ∆)− 1 2 |b| 1 p′ p→p ≤ cpmdδ

(via Lp-inequalities of Liskevich-Sem¨ enov (1996) between (λ − ∆)

1 2 and

“potential” |b|) ⇒ Tpp→p < 1 (via |∇(ζ − ∆)−1(x, y)| md(κdReζ − ∆)− 1

2 (x, y)) iff

p ∈

• 2

1 + √1 − mdδ , 2 1 − √1 − mdδ

• Damir Kinzebulatov, U Laval, Qu´

ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 17 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Need: Tp ∈ B(Lp) (same for Qp, Gp) b ∈ F

1 2

δ ⇔ |b|

1 2 (λ − ∆)− 1 2 |b| 1 2 2→2 ≤ mdδ yields (principal step)

|b|

1 p (λ − ∆)− 1 2 |b| 1 p′ p→p ≤ cpmdδ

(via Lp-inequalities of Liskevich-Sem¨ enov (1996) between (λ − ∆)

1 2 and

“potential” |b|) ⇒ Tpp→p < 1 (via |∇(ζ − ∆)−1(x, y)| md(κdReζ − ∆)− 1

2 (x, y)) iff

p ∈

• 2

1 + √1 − mdδ , 2 1 − √1 − mdδ

• ⇒ the “candidate for the resolvent” Θp(ζ, b) is in B(Lp)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 17 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Need: Tp ∈ B(Lp) (same for Qp, Gp) b ∈ F

1 2

δ ⇔ |b|

1 2 (λ − ∆)− 1 2 |b| 1 2 2→2 ≤ mdδ yields (principal step)

|b|

1 p (λ − ∆)− 1 2 |b| 1 p′ p→p ≤ cpmdδ

(via Lp-inequalities of Liskevich-Sem¨ enov (1996) between (λ − ∆)

1 2 and

“potential” |b|) ⇒ Tpp→p < 1 (via |∇(ζ − ∆)−1(x, y)| md(κdReζ − ∆)− 1

2 (x, y)) iff

p ∈

• 2

1 + √1 − mdδ , 2 1 − √1 − mdδ

• ⇒ the “candidate for the resolvent” Θp(ζ, b) is in B(Lp)

(Don’t need any of this for the Kato class. For Kato p ∈ [1, ∞[)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 17 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp:

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp: In five steps:

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp: In five steps: for Re ζ κdλ and n = 1, 2, . . .

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp: In five steps: for Re ζ κdλ and n = 1, 2, . . . Θp(ζ, bn)p→p, Θp(ζ, b)p→p c|ζ|−1

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp: In five steps: for Re ζ κdλ and n = 1, 2, . . . Θp(ζ, bn)p→p, Θp(ζ, b)p→p c|ζ|−1 Θp(ζ, bn) is a pseudo-resolvent

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp: In five steps: for Re ζ κdλ and n = 1, 2, . . . Θp(ζ, bn)p→p, Θp(ζ, b)p→p c|ζ|−1 Θp(ζ, bn) is a pseudo-resolvent Θp(ζ, bn) coincides with the resolvent4(ζ + Λp(bn))−1

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp: In five steps: for Re ζ κdλ and n = 1, 2, . . . Θp(ζ, bn)p→p, Θp(ζ, b)p→p c|ζ|−1 Θp(ζ, bn) is a pseudo-resolvent Θp(ζ, bn) coincides with the resolvent4(ζ + Λp(bn))−1 Θp(ζ, bn)

s

→ Θp(ζ, b) in Lp as n ↑ ∞

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp: In five steps: for Re ζ κdλ and n = 1, 2, . . . Θp(ζ, bn)p→p, Θp(ζ, b)p→p c|ζ|−1 Θp(ζ, bn) is a pseudo-resolvent Θp(ζ, bn) coincides with the resolvent4(ζ + Λp(bn))−1 Θp(ζ, bn)

s

→ Θp(ζ, b) in Lp as n ↑ ∞ µ Θp(µ, bn)

s

→ 1 as µ ↑ ∞ in Lp uniformly in n.

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Θp(λ, b) is the resolvent of the generator of a holomorphic semigroup on Lp: In five steps: for Re ζ κdλ and n = 1, 2, . . . Θp(ζ, bn)p→p, Θp(ζ, b)p→p c|ζ|−1 Θp(ζ, bn) is a pseudo-resolvent Θp(ζ, bn) coincides with the resolvent4(ζ + Λp(bn))−1 Θp(ζ, bn)

s

→ Θp(ζ, b) in Lp as n ↑ ∞ µ Θp(µ, bn)

s

→ 1 as µ ↑ ∞ in Lp uniformly in n. By the Trotter Approximation Theorem, the limit Θp(ζ, b) is the resolvent of the generator Λp of a holomorphic semigroup on Lp Λp ⊃ −∆ + b · ∇

4apriori the domain of ζ → (ζ + Λp(bn))−1 may depend on n Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 18 / 27

Proof: Lp theory of −∆ + b · ∇, b ∈ F

1/2

δ

Thus, Θp(ζ, b) = (ζ + Λp)−1 ⇒ The very definition of Θp(ζ, b) yields D(Λp) ⊂ W1+ 1

q ,p,

any q > p for p ∈

• 2

1+√ 1−mdδ , 2 1−√ 1−mdδ

• Damir Kinzebulatov, U Laval, Qu´

ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 19 / 27

Proof: The Lp theory of −∆ + b · ∇ is a “trampoline” to C∞ For p > d − 1, by the Sobolev Embedding Theorem Θp(µ, b)Lp ⊂ C∞ ∩ C0,γ Define (µ + Λ(b))−1 := [Θp(µ, b) ↾C∞∩Lp]clos

C∞

(appeal again to Trotter but now in C∞) Λ(b) ⊃ −∆ + b · ∇ is Feller generator5 Remarks: We have transferred the proof of convergence in C∞ to Lp, p > d − 1, a space having much weaker topology (locally) Earlier proofs, for b ∈ Kd+1 , verified convergence in C∞ directly (Arzela-Ascoli)

• 5D. Kinzebulatov, Annali SNS, 2017

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 20 / 27

A comment about Schr¨

• dinger operators . . .

−∆ + V is a Feller generator if V is in the Kato class Kd

0 (E.-M. Ouhabaz,

• P. Stollmann, K.-Th. Sturm, J. Voigt, 1994)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 21 / 27

A comment about Schr¨

• dinger operators . . .

−∆ + V is a Feller generator if V is in the Kato class Kd

0 (E.-M. Ouhabaz,

• P. Stollmann, K.-Th. Sturm, J. Voigt, 1994)

For potentials, can’t go beyond the Kato class Kd

0 (J. Voigt, 1995)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 21 / 27

−∆ + b · ∇ – classes of vector fields b studied in the literature [Lp + L∞]d (p > d) [Ld + L∞]d Fδ2 [Ld,∞ + L∞]d Kd+1

δ

Kd+1 F

1 2

δ

• ∗

✺✺✺✺✺✺✺✺

• ✈

✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺✺

• ✠

✠ ✠ ✠ ✠ ✠ ✠ ✠ ✺✺✺✺✺✺✺✺

• ✠

✠ ✠ ✠ ✠ ✠ ✠ ✠

Remark: For Schr¨

• dinger operators there is no this dichotomy

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 22 / 27

To the SDEs6 . . .

• 6D. Kinzebulatov, Yu.A. Sem¨

enov, ”Brownian motion with general drift”, arXiv:1710.06729

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 23 / 27

Proof: The analytic results behind the SDE characterization Let b ∈ F

1/2

δ

. We need:

• 1. Weight

ρ(x) := (1 + l|x|2)−ν with l, ν > 0 to be chosen Technical Lemma Fix p > d − 1. Then for l > 0 small,

• ρ(µ + Λ(bn))−1h
• ∞ ≤ K1ρhp,

(∗)

• ρ(µ + Λ(bn))−1|bm|h
• ∞ ≤ K2|bm|

1 p ρhp,

h ∈ C∞

c

(∗∗) Proof: Commute ρ in resolvent representation via |∇ρ| ≤ C1 √ lρ, |∆ρ| ≤ C2lρ (∗) allows to control behaviour at ∞ (replaces upper Gaussian bound) (∗∗) allows to control t

0 Ex[bn · ∇g](Xs)ds (e.g. in martingale problem)

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 24 / 27

We need: 2. e−tΛ(b) = s-C∞- lim

n e−tΛ(bn)

(in fact, that’s how we construct e−tΛ(b)), where Λ(bn) = −∆ + bn · ∇, D(Λ(bn)) = (1 − ∆)−1C∞

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 25 / 27

Proof: Outline The approximation result and the Technical Lemma allow to prove:

• 1. Trajectories do not escape to ∞
• 2. Px solve martingale problem on D([0, ∞), Rd)

t → f(Xt) − f(x) − t (−∆f + b · ∇f)(Xs)ds for all f ∈ C∞

c

(not for D(Λ(b)) – don’t even know if it separates closed sets)

• 3. A standard argument yields that the trajectories are continuous
• 4. SDE (use weights to get f(x) = xi, f(x) = xixj)
• 5. Uniqueness: The resolvent representation + a standard argument

Remark: Uniqueness in law? ∇(λ + Λ(b))−1f ∈ L∞ if b ∈ F

1 2

δ

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 26 / 27

In conclusion: Discontinuous diffusion matrix, singular drift Recently: Itˆ

• SDE (also Stratonovich)

X(t) = x − t b(X(s))ds + t σ(X(s))dW(s), x ∈ Rd, 1) b ∈ Fδ, i.e. |b|2 ∈ L2

loc ≡ L2 loc(Rd) and

|b|(λ − ∆)− 1

2 2→2

√ δ (previous examples, e.g. weak Ld, but not the Kato class)

7D.Kinzebulatov, Yu.A.Semenov, “Stochastic differential equations with singular

(form-bounded) drift” arXiv:1904.01268

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

In conclusion: Discontinuous diffusion matrix, singular drift Recently: Itˆ

• SDE (also Stratonovich)

X(t) = x − t b(X(s))ds + t σ(X(s))dW(s), x ∈ Rd, 1) b ∈ Fδ, i.e. |b|2 ∈ L2

loc ≡ L2 loc(Rd) and

|b|(λ − ∆)− 1

2 2→2

√ δ (previous examples, e.g. weak Ld, but not the Kato class) 2) a := σσ⊺ is uniformly elliptic and ∇ra·ℓ ∈ Fγrℓ (1 ≤ r, ℓ ≤ d).

7D.Kinzebulatov, Yu.A.Semenov, “Stochastic differential equations with singular

(form-bounded) drift” arXiv:1904.01268

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

In conclusion: Discontinuous diffusion matrix, singular drift Recently: Itˆ

• SDE (also Stratonovich)

X(t) = x − t b(X(s))ds + t σ(X(s))dW(s), x ∈ Rd, 1) b ∈ Fδ, i.e. |b|2 ∈ L2

loc ≡ L2 loc(Rd) and

|b|(λ − ∆)− 1

2 2→2

√ δ (previous examples, e.g. weak Ld, but not the Kato class) 2) a := σσ⊺ is uniformly elliptic and ∇ra·ℓ ∈ Fγrℓ (1 ≤ r, ℓ ≤ d). Examples: a(x) = I + cx ⊗ x |x|2 , c > −1 a(x) = I + c(sin log(|x|))2e ⊗ e, e ∈ Rd, |e| = 1,

• r their infinite sum (geometry of discontinuities is not important)

7D.Kinzebulatov, Yu.A.Semenov, “Stochastic differential equations with singular

(form-bounded) drift” arXiv:1904.01268

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

In conclusion: Discontinuous diffusion matrix, singular drift Recently: Itˆ

• SDE (also Stratonovich)

X(t) = x − t b(X(s))ds + t σ(X(s))dW(s), x ∈ Rd, 1) b ∈ Fδ, i.e. |b|2 ∈ L2

loc ≡ L2 loc(Rd) and

|b|(λ − ∆)− 1

2 2→2

√ δ (previous examples, e.g. weak Ld, but not the Kato class) 2) a := σσ⊺ is uniformly elliptic and ∇ra·ℓ ∈ Fγrℓ (1 ≤ r, ℓ ≤ d). Examples: a(x) = I + cx ⊗ x |x|2 , c > −1 a(x) = I + c(sin log(|x|))2e ⊗ e, e ∈ Rd, |e| = 1,

• r their infinite sum (geometry of discontinuities is not important)

Proof7: A different technique (essentially, “differentiate the equation without differentiating the drift”) – better control over relative bound δ

7D.Kinzebulatov, Yu.A.Semenov, “Stochastic differential equations with singular

(form-bounded) drift” arXiv:1904.01268

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

Thank you http://archimede.mat.ulaval.ca/pages/kinzebulatov

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 29 / 27

Liskevich-Sem¨ enov: A, a symmetric Markov generator. Then r ∈]1, ∞[ 0 u ∈ D(Ar) ⇒ v := u

r 2 ∈ D(A 1 2 ) and c−1

r A

1 2 v2

2 Aru, ur−1.

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 30 / 27

Liskevich-Sem¨ enov: A, a symmetric Markov generator. Then r ∈]1, ∞[ 0 u ∈ D(Ar) ⇒ v := u

r 2 ∈ D(A 1 2 ) and c−1

r A

1 2 v2

2 Aru, ur−1.

Let Aru = |f|, f ∈ Lr. Note that ur µ− 1

2 fr. Since b ∈ F 1 2

δ , we have

(crδ)−1|b|

1 2 v2

2 Aru, ur−1,

and so |b|

1 r ur

r crδfrur−1 r

, |b|

1 r A−1

r |f|r r crδµ− r−1

2 fr

r.

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 30 / 27

Let Y be a (complex) Banach space. A pseudo-resolvent Rζ is a function defined on a subset O of the complex ζ-plane with values in B(Y ) such that Rζ − Rη = (η − ζ)RζRη, ζ, η ∈ O. Clearly, Rζ have common null-set. Theorem (E. Hille) If the null-set of Rζ is {0}, then Rζ is the resolvent of a closed linear

• perator A, the range of Rζ coincides with D(A), and A = R−1

ζ

− ζ. Proof: Put A := R−1

ζ

− ζ. Since Rζ is closed, so is R−1

ζ

and A. A straightforward calculation shows that (ζ + A)Rζf = f, f ∈ Y , and Rζ(ζ + A)g = g, g ∈ D(A), as needed. Theorem (E. Hille) If there exists a sequence of numbers {µk} ⊂ O such that limk |µk| = ∞ and supk µkRµkY →Y < ∞, then the set {y ∈ Y : limk µkRµky = y} is contained in the closure of the range of Rζ. Proof: Indeed, let limk µkRµky = y. That is, for every ε > 0, there exists k such that y − µkRµky < ε, so y belongs to the closure of the range of Rζ.

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 31 / 27

Consider a sequence of C0 semigroups e−tAk on a (complex) Banach space Y . Theorem (H.F. Trotter) Let sup

k

(µ + Ak)−1Y →Y ≤ µ−1, µ > ω,

• r

sup

k

(z + Ak)−1Y →Y ≤ C|z|−1, Re z > ω, and let s- limµ→∞ µ(µ + Ak)−1 = 1 uniformly in k. Let s- limk(ζ + Ak)−1 exist for some ζ with Re ζ > ω. Then there is a C0 semigroup e−tA such that (z + Ak)−1

s

→ (z + A)−1 for every Re z > ω, and e−tAk

s

→ e−tA uniformly in any finite interval of t ≥ 0.

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 32 / 27

Theorem (Hille Pertrubation Theorem) Let e−tA be a symmetric Markov semigroup, K a linear operator in Lr for some r ∈]1, ∞[. If for some λ > 0 K(λ + Ar)−1r→r < 1

2, then

−Λr := −Ar − K of domain D(Ar) is the generator of a quasi bounded holomorphic semigroup on Lr.

Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 33 / 27

md := π

1 2 (2e)− 1 2 d d 2 (d − 1) 1−d 2 Damir Kinzebulatov, U Laval, Qu´ ebec http://archimede.mat.ulaval.ca/pages/kinzebulatov 34 / 27