Operators of Kolmogorov type and parabolic operators associated with non-commuting vector fields: obstacle problems and boundary behaviour Kaj Nyström Umeå University, Sweden Operators of Kolmogorov type and parabolic operators associated
Non-divergence form parabolic operators Let 1 ≤ m ≤ N and consider ( x , t ) ∈ R N + 1 . Uniformly elliptic operators N � L = a ij ( x , t ) ∂ x i x j − ∂ t i , j = 1 Operators of Kolmogorov type m m N � � � L = a ij ( x , t ) ∂ x i x j + a i ( x , t ) ∂ x i + b ij x i ∂ x j − ∂ t i , j = 1 i = 1 i , j = 1 Operators associated with non-commuting vector fields m � L = a ij ( x , t ) X i X j − ∂ t i , j = 1 Operators of Kolmogorov type and parabolic operators associated
Outline of the talk Operators of Kolmogorov type 1 Obstacle problems for operators of Kolmogorov type 2 (Optimal) interior regularity and regularity at the initial state Smooth and non-smooth obstacles Operators associated with non-commuting vector fields 3 Boundary behaviour (for non-negative solutions vanishing 4 on the boundary) for operators associated with non-commuting vector fields Backward Harnack inequality Boundary Hölder continuity of quotients Doubling property of the L -parabolic measure Work in progress 5 Operators of Kolmogorov type and parabolic operators associated
Collaborators Operators of Kolmogorov type Marie Frentz Chiara Cinti Andrea Pascucci Sergio Polidoro Operators associated with non-commuting vector fields Marie Frentz Nicola Garofalo Elin Götmark Isidro Munive Operators of Kolmogorov type and parabolic operators associated
Operators of Kolmogorov type - an example 1 ≤ m , N = 2 m . 1 L = X 2 1 + .. + X 2 m + Y , X i = ∂ x i , Y = x 1 ∂ x m + 1 + .. + x m ∂ x 2 m − ∂ t Generators for the following system of SDEs 2 dX 1 = dW 1 , .., dX m = dW m , dX m + 1 = X 1 dt , .., dX 2 m = X m dt . Hypoellipticity: [ X i , Y ] = ∂ x m + i . 3 Dilation at (0,0,0): ( x ′ , x ′′ , t ) := ( x 1 , .., x m , x m + 1 , .., x 2 m , t ) , 4 x ′ → λ x ′ , x ′′ → λ 3 x ′′ , t → λ 2 t . Group law: L is invariant w.r.t left translations of the Lie 5 group ( R N + 1 , ◦ ) , ( x ′ , x ′′ , t ) ◦ ( ξ ′ , ξ ′′ , τ ) = ( ξ ′ + x ′ , ξ ′′ + x ′′ − τ x ′ , τ + t ) . Operators of Kolmogorov type and parabolic operators associated
Operators of Kolmogorov type z = ( x , t ) ∈ R N + 1 , 1 ≤ m ≤ N . General equations of the form m m N � � � L = a ij ( z ) ∂ x i x j + a i ( z ) ∂ x i + b ij x i ∂ x j − ∂ t . (1) i , j = 1 i = 1 i , j = 1 [H.1] A 0 ( z ) = { a ij ( z ) } i , j = 1 ,..., m is symmetric and uniformly positive definite in R m : m Λ − 1 | ξ | 2 ≤ � a ij ( z ) ξ i ξ j ≤ Λ | ξ | 2 , ∀ ξ ∈ R m , z ∈ R N + 1 . i , j = 1 [H.2] The constant coefficient (frozen) operator m N m � � � X 2 K = a ij ( z 0 ) ∂ x i x j + b ij x i ∂ x j − ∂ t = i + Y (2) i , j = 1 i , j = 1 i = 1 is hypoelliptic. Operators of Kolmogorov type and parabolic operators associated
Operators of Kolmogorov type [H.2] is equivalent to the following structural assumption on B : there exists a basis for R N such that the matrix B has the form ∗ 0 · · · 0 B 1 ∗ ∗ B 2 · · · 0 . . . . ... . . . . (3) . . . . ∗ ∗ ∗ · · · B κ ∗ ∗ ∗ · · · ∗ where B j is a m j − 1 × m j matrix of rank m j for j ∈ { 1 , . . . , κ } , 1 ≤ m κ ≤ . . . ≤ m 1 ≤ m 0 = m and m + m 1 + . . . + m κ = N , while ∗ represents arbitrary matrices with constant entries. [H.3] a ij and a i belong to the space C 0 ,α K ( R N + 1 ) of Hölder continuous functions for some α ∈ ] 0 , 1 ] . Operators of Kolmogorov type and parabolic operators associated
Operators of Kolmogorov type Dilations : Based on the structure of B one can introduce a family of dilations ( δ λ ) λ> 0 on R N + 1 defined by δ λ := diag ( λ I m , λ 3 I m 1 , . . . , λ 2 κ + 1 I m κ , λ 2 ) , (4) where I k , k ∈ N , is the k -dimensional unit matrix. Group law : relevant Lie group related to the operator K in (2) ( x , t ) ◦ ( ξ, τ ) = ( ξ + exp ( − τ B T ) x , t + τ ) , ( x , t ) , ( ξ, τ ) ∈ R N + 1 . (5) [H.4] The operator K in (2) is δ λ -homogeneous of degree two, i.e. K ◦ δ λ = λ 2 ( δ λ ◦ K ) , ∀ λ > 0 . Remark : [H.4] is satisfied if (and only if) all the blocks denoted by ∗ in (3) are null. Operators of Kolmogorov type and parabolic operators associated
Obstacle problem for operators of Kolmogorov type Ω ⊂ R N + 1 : an open subset. g , f , ψ : ¯ Ω → R , continuous and bounded on ¯ Ω and g ≥ ψ on ¯ Ω . � max { Lu ( x , t ) − f ( x , t ) , ψ ( x , t ) − u ( x , t ) } = 0 , in Ω , (6) u ( x , t ) = g ( x , t ) , on ∂ P Ω . u ∈ S 1 loc (Ω) ∩ C (Ω) is a strong solution to problem (6) if the 1 differential inequality is satisfied a.e. in Ω and the boundary datum is attained at any point of ∂ P Ω . Existence and uniqueness of a strong solution to (6) have 2 been proved by DiFrancesco, Pascucci, Polidoro (2008). By Sobolev embedding: u ∈ C 1 ,α K , loc (Ω) . 3 Applications to American options: geometric average 4 Asian options, option pricing in the stochastic volatility model of Hobson-Rogers (1998). Operators of Kolmogorov type and parabolic operators associated
Main results - optimal interior regularity Theorem 1 Assume H1 - H4 . Let α ∈ ] 0 , 1 ] and let Ω , Ω ′ be domains of R N + 1 , Ω ′ ⊂⊂ Ω . Let u be a solution to problem (6) . Then i) if ψ ∈ C 0 ,α K (Ω) then u ∈ C 0 ,α K (Ω ′ ) , � u � C 0 ,α (Ω ′ ) bounded by K � � α, Ω , Ω ′ , L , � f � C 0 ,α c (Ω) , � g � L ∞ (Ω) , � ψ � C 0 ,α ; (Ω) K K ii) if ψ ∈ C 1 ,α K (Ω) then u ∈ C 1 ,α K (Ω ′ ) , � u � C 0 ,α (Ω ′ ) bounded by K � � α, Ω , Ω ′ , L , � f � C 0 ,α c (Ω) , � g � L ∞ (Ω) , � ψ � C 1 ,α ; (Ω) K K iii) if ψ ∈ C 2 ,α K (Ω) then u ∈ S ∞ (Ω ′ ) , � u � S ∞ (Ω ′ ) bounded by � � α, Ω , Ω ′ , L , � f � C 0 ,α c (Ω) , � g � L ∞ (Ω) , � ψ � C 2 ,α . (Ω) K K Operators of Kolmogorov type and parabolic operators associated
Main results - optimal regularity at the initial state Theorem 2 Assume H1 - H4 , α ∈ ] 0 , 1 ] , Ω , Ω ′ as in Theorem 1. u solution to problem (6) in Ω t 0 . Ω t 0 (Ω ′ t 0 ) := Ω(Ω ′ ) ∩ { t > t 0 } , t 0 ∈ R . Then i) if g , ψ ∈ C 0 ,α K (Ω t 0 ) then u ∈ C 0 ,α K (Ω ′ t 0 ) , � u � C 0 ,α t 0 ) bounded (Ω ′ K � � α, Ω , Ω ′ , L , � f � C 0 ,α by c (Ω t 0 ) , � g � C 0 ,α (Ω t 0 ) , � ψ � C 0 ,α ; (Ω t 0 ) K K K ii) if g , ψ ∈ C 1 ,α K (Ω t 0 ) then u ∈ C 1 ,α K (Ω ′ t 0 ) , � u � C 1 ,α t 0 ) bounded (Ω ′ K � � α, Ω , Ω ′ , L , � f � C 0 ,α by c (Ω t 0 ) , � g � C 1 ,α (Ω t 0 ) , � ψ � C 1 ,α ; (Ω t 0 ) K K K iii) if g , ψ ∈ C 2 ,α K (Ω t 0 ) then u ∈ S ∞ (Ω ′ t 0 ) , � u � S ∞ (Ω ′ t 0 ) bounded � � α, Ω , Ω ′ , L , � f � C 0 ,α by c (Ω t 0 ) , � g � C 2 ,α (Ω t 0 ) , � ψ � C 2 ,α . (Ω t 0 ) K K K Operators of Kolmogorov type and parabolic operators associated
Regularity at the initial state - the proof S + k ( u − F ) = sup | u − F | , (7) Q + 2 − k where F = P ( 0 , 0 ) g , n ∈ { 0 , 1 , 2 } , γ ∈ { α, 1 + α, 2 } . n Key estimate : ∃ c > 0 such that S + k + 1 ( u − F ) is bounded, for all k ∈ N , by S + c 2 − ( k + 1 ) γ , S + , . . . , S + k − 1 ( u − F ) � k ( u − F ) 0 ( u − F ) � max , . (8) 2 γ 2 2 γ 2 ( k + 1 ) γ Implication : by dilation, translation, Taylor’s formula, (8) it follows that if ( u , f , g , ψ ) ∈ P n ( L , Q + R ( x 0 , t 0 ) , α, M 1 , M 2 , M 3 , M 4 ) , then ∃ c = c ( L , α, M 1 , M 2 , M 3 , M 4 ) , | u − g | ≤ cr γ , r ∈ ] 0 , R [ . sup Q + r ( x 0 , t 0 ) Operators of Kolmogorov type and parabolic operators associated
Regularity at the initial state - the proof in case n = 0 Assume that (8) is false and F = P ( 0 , 0 ) g = 0: ∀ j ∈ N , ∃ a 0 positive integer k j , ( u j , f j , g j , ψ j ) ∈ P 0 ( L , Q + , α, M 1 , M 2 , M 3 , M 4 ) , such that S + S + � k j ( u j ) k j − 1 ( u j ) , . . . , S + � 0 ( u j ) j ( C α + M 3 ) S + k j + 1 ( u j ) > max , , . 2 α 2 ( k j + 1 ) α 2 2 α 2 ( k j + 1 ) α ∃ ( x j , t j ) ∈ Q + 2 − kj − 1 , | u j ( x j , t j ) | = S + k j + 1 ( u j ) for every j ≥ 1. x j , ˜ u j : Q + Let (˜ t j ) = δ 2 kj (( x j , t j )) and define ˜ 2 kj − → R as u j ( δ 2 − kj ( x , t )) ˜ u j ( x , t ) = . (9) S + k j + 1 ( u j ) x j , ˜ t j ) belongs to the closure of Q + Note that (˜ 1 / 2 and x j , ˜ ˜ u j (˜ t j ) = 1 . (10) Operators of Kolmogorov type and parabolic operators associated
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