Elliptic Operators with Unbounded Coefficients Federica Gregorio - PowerPoint PPT Presentation

Elliptic Operators with Unbounded Coefficients Federica Gregorio Universitá degli Studi di Salerno 8th June 2018 joint work with S.E. Boutiah, A. Rhandi, C. Tacelli

Motivation Consider the Stochastic Differential Equation � dX ( t , x ) = F ( X ( t , x )) dt + Q ( X ( t , x )) dB ( t ) , t > 0 , (SDE) X ( 0 ) = x ∈ R N , B ( t ) is a standard Brownian motion and Q ( x ) ∈ L ( R N ) . ◮ Probabilistic model of the physical process of diffusion ◮ Model in mathematical finance ◮ Model in biology

Motivation Consider the Stochastic Differential Equation � dX ( t , x ) = F ( X ( t , x )) dt + Q ( X ( t , x )) dB ( t ) , t > 0 , X ( 0 ) = x ∈ R N , ϕ ∈ C b ( R N ) u ( t , x ) := E ( ϕ ( X ( t , x )) , Then ⇒ u solves Kolomogorov/Fokker-Planck equation  N ∂ u �   ∂ t ( t , x ) = a ij ( x ) D ij u ( t , x ) + F ( x ) · ∇ u ( t , x ) , t > 0 , (FP) i , j = 1   x ∈ R N , u ( 0 , x ) = ϕ ( x ) , 2 Q ( x ) Q ( x ) ∗ can be unbounded. ( a ij ( x )) := 1

Examples ◮ The Black-Scholes equation:  ∂ t − σ 2 2 x 2 ∂ 2 v ∂ v ∂ x 2 − rx ∂ v  ∂ x + rv = 0 , 0 < t ≤ T , (BS)  v ( 0 , x ) = ( x − K ) + , x > 0 . ◮ v ( t , x ) = u ( T − t , x ) is the value of european type option on the asset price x at time t ; ◮ σ is the stock volatility; ◮ r is the risk-free rate; ◮ 0 < x is the underlying asset; ◮ K is the prescribed price.

Set y = log x . Then, � σ 2 � 2 e − ( α + 1 ) 2 v ( t , x ) = x − α − 1 σ 2 t w 2 t , log x , 8 where α = 2 r σ 2 and w solves the heat equation  ∂ t = ∂ 2 w ∂ w t > 0 , y ∈ R ,  ∂ y 2 α − 1 y ( e y − K ) +  w ( 0 , y ) = e y ∈ R . 2

Examples ◮ Ornstein-Uhlenbeck operator: � dX ( t , x ) = MX ( t , x ) dt + dB ( t ) , t > 0 , (OU) X ( 0 , x ) = x ∈ R N , � t 0 e sM e sM ∗ ds . Then M = ( m ij ) ∈ L ( R N ) . Set M t := u ( t , x ) : = E ( f ( X ( t , x ))) � 2 | M − 1 / 2 = ( 2 π ) − N 2 ( det M t ) − 1 R N e − 1 ( e tM x − y ) | 2 f ( y ) dy 2 t solves the Ornstein-Ulhenbeck equation  ∂ u ∂ t ( t , x ) = 1  2 ∆ u ( t , x ) + Mx · ∇ u ( t , x ) , t > 0 , (OU) x ∈ R N .  u ( 0 , x ) = f ( x )

Semigroups in short Let X be a Banach space. A family ( T ( t )) t ≥ 0 of bounded operators on X is called a strongly continuous or C 0 -semigroup if ◮ T ( 0 ) = I and T ( t + s ) = T ( t ) T ( s ) for all t , s ≥ 0; ◮ t �→ T ( t ) x ∈ X is continuous for every x ∈ X .

Semigroups in short The generator of a strongly continuous semigroup A is defined as D ( A ) := { x ∈ X : t �→ T ( t ) x is differentiable on [ 0 , ∞ ) } Ax := d 1 dt T ( t ) x | t = 0 = lim t ( T ( t ) x − x ) . t ↓ 0 T ( t ) � e tA

Semigroup approach to initial-boundary value problems Given X Banach space, A : D ( A ) ⊂ X → X with boundary conditions in D ( A ) � ∂ u ∂ t ( t ) = Au ( t ) ( ACP 1 ) u ( 0 ) = u 0 ◮ ( A , D ( A )) generates a C 0 -semigroup ( T ( t )) t ≥ 0 on X ⇔ ( ACP 1 ) well posed with solution u ( t ) = T ( t ) u 0 . ◮ Study qualitative properties: positivity, stability, regularity, ...

The resolvent The Resolvent Operator is the operator R ( λ, A ) := ( λ − A ) − 1 , λ ∈ ρ ( A ) := { λ ∈ C : λ − A → X is bijective } The semigroup is related to the Resolvent Operator T ( t ) is a C 0 -semigroup, � T ( t ) � ≤ e ω t then λ ∈ ρ ( A ) for Re λ > ω � ∞ 1 e − λ s T ( s ) f ds , R ( λ, A ) f = � R ( λ, A ) � ≤ Re λ − ω . 0

Generation Theorems ◮ Hille, Yosida, (1948). Let ( A , D ( A )) closed, densely defined and λ ∈ ρ ( A ) for every λ > ω 1 ⇒ A generates � T ( t ) � ≤ e ω t � R ( λ, A ) � ≤ λ − ω = ◮ Lumer, Phillips, (1961). Rg ( λ − A ) = X , � ( λ − A ) f � ≥ λ � f � , for all λ > 0 = ⇒ A generates � T ( t ) � ≤ 1

Some trivial example Let ω ∈ R consider the operator A : R → R , Ax = ω x  d  dt x ( t ) = Ax ( t ) = ω x ( t ) (1)  x ( 0 ) = x 0 R ( λ, A ) = ( λ − A ) − 1 solves the equation λ x − ω x = y ⇐ y ⇒ x = λ − ω y R ( λ, A ) y = λ − ω and ρ ( A ) = C \ { ω }  A closed, dense  H. Y. Theorem A generates � T ( t ) � ≤ e ω t λ ∈ ρ ( A ) if λ > ω = ⇒ 1  � R ( λ, A ) � = λ − ω For t > 0, T ( t ) : R → R Fixing x 0 , T ( t ) x 0 = x ( t ) is a function in t and solves (1)

Some trivial example T ( t ) x = e ω t x Semigroup laws i ) T ( t + s ) x = e ω ( t + s ) x = e ω t e ω s x = T ( t ) T ( s ) x ii ) T ( 0 ) x = e µ 0 x = x iii ) lim t → 0 T ( t ) x = x Generator e ω t x − x T ( t ) x − x Ax = lim = lim = ω x t t t ↓ 0 t ↓ 0 Resolvent � ∞ � ∞ y e − λ s T ( s ) yds = e − λ s e ω s yds R ( λ, A ) y = λ − ω = 0 0

Kernel Representation If the coefficients of the differential operator have suitable regularity, the semigroup has a kernel representation. � T ( t ) f ( x ) = R N k ( t , x , y ) f ( y ) dy . Consider the heat equation ∂ t u ( t , x ) = ∆ u ( t , x ) , u ( 0 , x ) = f ( x ) � 1 e − | x − y | 2 u ( t , x ) = T ( t ) f ( x ) = f ( y ) dy 4 t ( 4 π t ) N / 2 ( 4 π t ) N / 2 e − | x − y | 2 1 In this case k ( t , x , y ) = 4 t ◮ The behavior of the semigroup depends on the behavior of the kernel ◮ The kernel is related to the eigenvalues and the ground state of the problem

Elliptic operators with unbounded coefficients We consider elliptic operators with unbounded coefficients of the form N N � � A u ( x ) = a ij ( x ) D ij u ( x ) + b i ( x ) D i u ( x ) + V ( x ) u ( x ) . i , j = 1 i = 1 The realisation A of A in C b ( R N ) with maximal domain � W 2 , p D max ( A ) = { u ∈ C b ( R N ) ∩ loc ( R N ) : A u ∈ C b ( R N ) } 1 ≤ p < ∞ Au = A u .

� ∂ t u ( t , x ) = Au ( t , x ) x ∈ R N t > 0 , (2) x ∈ R N , u ( 0 , x ) = f ( x ) with f ∈ C b ( R N ) . To have solution we assume that for some α ∈ ( 0 , 1 ) , (1) a ij , b i , V ∈ C α loc ( R N ) , ∀ i , j = 1 , ..., N ; (2) a ij = a ji and � a ij ( x ) ξ i ξ j ≥ k ( x ) | ξ | 2 � a ( x ) ξ, ξ � = i , j x , ξ ∈ R N , k ( x ) > 0; (3) ∃ c 0 ∈ R s.t. x ∈ R N . V ( x ) ≤ c 0 ,

Consider the problem on bounded domains  ∂ t u R ( t , x ) = Au R ( t , x ) t > 0 , x ∈ B R  u R ( t , x ) = 0 t > 0 , x ∈ ∂ B R (3)  u R ( 0 , x ) = f ( x ) x ∈ B R , with f ∈ C b ( R N ) . Then A is uniformly elliptic on compacts of R N and (3) admits unique classical solution u R ( t , x ) = T R ( t ) f ( x ) , t ≥ 0 , x ∈ B R with T R ( t ) analytic semigroup in C ( B R ) . The infinitesimal generator of ( T R ( t )) is ( A , D R ( A )) , � W 2 , p ( B R ) , : Au ∈ C ( B R ) } . D R ( A ) = { u ∈ C 0 ( B R ) ∩ 1 < p < ∞

Theorem 1 (i) ( T R ( t )) has the integral representation � T R ( t ) f ( x ) = p R ( t , x , y ) f ( y ) dy , f ∈ C ( B R ) , t > 0 , x ∈ B R B R with strictly positive kernel p R ∈ C (( 0 , + ∞ ) × B R × B R ) . (ii) T R ( t ) ∈ L ( L p ( B R )) per ogni t ≥ 0 e per ogni 1 < p < + ∞ ; (iii) T R ( t ) is contractive in C ( B R ) ; (iv) for all fixed y ∈ B R , p R ( · , · , y ) ∈ C 1 + α 2 , 2 + α ([ s , t 0 ] × B R ) for all 0 < s < t 0 and ∂ t p R ( t , x , y ) = Ap R ( t , x , y ) , ∀ ( t , x ) ∈ ( 0 , + ∞ ) × B R .

f ∈ C ( B R ) + ( iv ) , gives u R ∈ C 1 + α 2 , 2 + α ([ s , t 0 ] × B R ) . Proposition 1 Let f ∈ C b ( R N ) and t ≥ 0 ; then it exists ∀ x ∈ R N T ( t ) f ( x ) = R → + ∞ T R ( t ) f ( x ) , lim (4) and ( T ( t )) is a positive semigroup in C b ( R N ) .

Schauder interior estimates � u R � C 1 + α/ 2 , 2 + α ([ ε, T ] × B R − 1 ) ≤ C � u R � L ∞ (( 0 , T ) × B R ) ≤ Ce λ 0 T � f � ∞ Theorem 2 Let f ∈ C b ( R N ) , then the function u ( t , x ) = T ( t ) f ( x ) 1 + α 2 , 2 + α (( 0 , + ∞ ) × R N ) and it solves belongs to C loc � t > 0 , x ∈ R N , u t ( t , x ) = Au ( t , x ) x ∈ R N . u ( 0 , x ) = f ( x )

The operator For c > 0, b ∈ R , α > 2 and β > α − 2 we consider on L p ( R N ) A := ( 1 + | x | α )∆ + b | x | α − 2 x · ∇ − c | x | β . (5) Aim. ◮ Solvability of λ u − Au = f ◮ Properties of the maximal domains ◮ Generation of positive analytic semigroup

Related results ◮ b = c = 0 : Unbounded Diffusion : A = ( 1 + | x | α )∆ N [ G. Metafune, C. Spina,’10 ] α > 2 , p > N − 2 Schrödinger-Type Operator: A = ( 1 + | x | α )∆ − c | x | β ◮ b = 0 : [ L. Lorenzi, A. Rhandi ,’15] 0 ≤ α ≤ 2 , β ≥ 0 [ A. Canale, A. Rhandi, C.Tacelli ,’16] α > 2 , β > α − 2 ◮ c = 0 : Unbounded Diffusion & Drift : A = ( 1 + | x | α )∆ + b | x | α − 2 x · ∇ [ S. Fornaro, L. Lorenzi ’07 ]: 0 ≤ α ≤ 2. N [ Metafune, Spina, Tacelli,’14 ] α > 2 , b > 2 − N → p > N − 2 + b A = | x | α ∆ + b | x | α − 2 x · ∇ x − c | x | α − 2 Complete : ◮ [G. Metafune, N. Okazawa, M. Sobajima, C. Spina,’16] N / p ∈ ( s 1 + min { 0 , 2 − α } , s 2 + max { 0 , 2 − α } ) , c + s ( N − 2 + b − s ) = 0