Nelson-type Limit for a Particular Class of Lvy Processes Haidar - PowerPoint PPT Presentation

Nelson-type Limit for a Particular Class of Lévy Processes Haidar Al-Talibi School of Computer Science, Physics and Mathematics Linnaeus University March 15, 2010 DFM LNU 1 / 21 Haidar Al-Talibi

Outline The OU-process 1 OU-process with drift 2 The Nelson Limit 3 Modified OU-process 4 Time change 5 Time Change - α -stable case Approximation Theorem 6 7 Bibliography DFM LNU 2 / 21 Haidar Al-Talibi

The OU-process In the physical model x ( t ) describes the position of a Brownian particle at time t > 0 . It is assumed that the velocity dx dt = v exists and satisfies the Langevin equation. Mathematically the two ordinary differential equations combine to the initial value problem: dx t = v t dt (1) dv t = − β v t dt + β K ( x t ) dt + dB t , with initial value ( x 0 , v 0 ) = ( x ( 0 ) , v ( 0 )) , where B t , t ≥ 0 , is mathematical Brownian motion on the real line and β > 0 is a constant which physically represents the inverse relaxation time between two successive collisions. K ( x , t ) is an external field of force. Moreover sufficient conditions for the existence of a unique solution of ( 1 ) can be found in e.g. [Applebaum] and in [Kolokoltsov, Schilling, Tyukov]. DFM LNU 3 / 21 Haidar Al-Talibi

The OU-process For the physical Ornstein Uhlenbeck theory of motion, given by a second R d , the solution of the corresponding system on the cotangent order SDE on I R 2 ) is given by: bundle ( I � t � t e − β ( t − u ) K ( x u ) du + e − β ( t − u ) dB u , v t = e − β t v 0 + β 0 0 which is called Ornstein-Uhlenbeck velocity process, and � t � t � s � t � s e − β s v 0 ds + β e − β ( s − u ) K ( x u ) duds + e − β s e β u dB u ds , (2) x t = x 0 + 0 0 0 0 0 which is called Ornstein-Uhlenbeck position process. The initial values are given by ( x 0 , v 0 ) = ( x ( 0 ) , v ( 0 )) and t ≥ 0 . Remark We introduce this physical notation for the Ornstein-Uhlenbeck process since it is more adequate for our studies than the mathematical one. In Nelson’s notation the noise B is Gaussian with variance 2 β 2 D with 2 β 2 D = 2 β kT m and physical constants k , T , m in order to match Smolouchwsky’s constants. DFM LNU 4 / 21 Haidar Al-Talibi

OU-process with drift Let us modify the Ornstein-Uhlenbeck process (2) as in [Al-Talibi, Hilbert, Kolokoltsov]. We introduce a stochastic Newton equation driven by β X t , where { X t } t ≥ 0 is an α -stable Lévy process, with 0 < α < 2 and β is a scaling parameter. Sufficient conditions for the existence of a unique solution may be found in [Applebaum] and [Kolokoltsov, Schilling, Tyukov]. In this case the solution of this stochastic differential equation can be represented as given in the proposition below. Proposition Assume A : I R → I R is linear. Furthermore, let X be a Lévy process on I R . Let f : [ 0 , ∞ ] → I R be a continuous function. Then the solution of the stochastic differential equation dx t = Ax t dt + f ( t ) dt + dX t , t ≥ 0 with initial value x ( 0 ) = x 0 , is � t � t x t = e At x 0 + e A ( t − s ) f ( s ) ds + e A ( t − s ) dX s . 0 0 DFM LNU 5 / 21 Haidar Al-Talibi

OU-process with drift Proof. The equation in question has a unique solution, [Applebaum]. We can verify the solution using integration by parts respectively Itô formula, i.e. � t � t e − At x t = x 0 + � − Ae − As � e − As dx s , ds + x s 0 0 and inserting for dx t = Ax t dt + f ( t ) dt + dX t we obtain � t � t e − At x t = x 0 + e − As f ( s ) ds + e − As dX s , 0 0 and we are done. DFM LNU 6 / 21 Haidar Al-Talibi

The Nelson Limit Let ( x , v ) be the solution of the system dx ( t ) = v ( t ) dt , x ( 0 ) = x 0 , dv ( t ) = − β v ( t ) dt + β K ( x ( t ) , t ) dt + dB t , v ( 0 ) = v 0 . where the noise B is Gaussian with variance β 2 on I R ℓ . Theorem 10.1 Let ( x , v ) satisfy the equation above and assume that K is a R ℓ satisfying a global Lipschitz condition. Moreover assume that function in I B is standard BM and y solves the equation dy ( t ) = K ( y ( t ) , t ) dt + dB ( t ) y ( 0 ) = v 0 . Then for all v 0 with probability one β →∞ x ( t ) = y ( t ) , lim uniformly for t in compact subintervals of [ 0 , ∞ ) . DFM LNU 7 / 21 Haidar Al-Talibi

Modified OU-process Here we introduce a modified Ornstein-Uhlenbeck position process driven by β X t , where { X t } t ≥ 0 is an α -stable Lévy process, 0 < α < 2 and β > 0 is a scaling parameter as above. Let us focus on the position process { x t } t ≥ 0 . Due to Proposition 1, the solution has the form � t � t � s � t � s e − β ( s − u ) K ( x u ) duds + e − β s v 0 ds + β β e − β s e β u dX u ds , (3) x t = x 0 + 0 0 0 0 0 where K satisfies sufficient conditions to guarantee existenc and uniqueness of solutions see e.g. [Applebaum] and [Kolokoltsov, Schilling, Tyukov]. DFM LNU 8 / 21 Haidar Al-Talibi

Modified OU-process For arbitrary Lévy processes Y the characteristic function is of the form φ Y t ( u ) = e t η ( u ) for each u ∈ I R , t ≥ 0 , η being the Lévy-symbol of Y ( 1 ) . We concentrate on α -stable Lévy processes with Lévy-symbol for α � = 1 : � πα η ( u ) = − σ α | u | α � �� 1 − i γ sgn ( u ) tan (4a) 2 and for α = 1 is: � 1 + i γ 2 � η 1 ( u ) = − σ | u | π sgn ( u ) log ( | u | ) (4b) for constant γ . DFM LNU 9 / 21 Haidar Al-Talibi

Time change Proposition (Lukacs) Assume that Y is an α -stable Lévy process, 0 < α < 2 , and g is a continuous function on the interval [ s , t ] ⊂ T � I R . Let η be the Lévy symbol of Y 1 and � t ξ be the Lévy symbol of ψ ( t ) = s g ( r ) dY r . Then we have � t ξ ( u ) = η ( ug ( r )) dr . s For g ( ℓ ) = e β ( ℓ − t ) , ℓ ≥ 0 and the α -stable process X in (3) the symbol of � t s e β ( r − t ) dX r is: Z t = � � t s e αβ ( r − t ) dr · η ( u ) for 0 < α < 2 , α � = 1 ξ ( u ) = � t s e αβ ( r − t ) dr · η 1 ( u ) for α = 1 with η , η 1 as in (4a) and (4b), respectively, and 0 ≤ s ≤ t . DFM LNU 10 / 21 Haidar Al-Talibi

Time change Time Change - α -stable case For g ( ℓ ) = e β ( ℓ − t ) , ℓ ≥ 0 and the α -stable process X in (3) the symbol of � t s e β ( r − t ) dX r is: Z t = � � t s e αβ ( r − t ) dr · η ( u ) for 0 < α < 2 , α � = 1 ξ ( u ) = � t s e αβ ( r − t ) dr · η 1 ( u ) for α = 1 with η , η 1 as in (4a) and (4b), respectively, and 0 ≤ s ≤ t . We are thus lead to introduce the time change τ − 1 ( t ) where � t e − αβ t e αβ u du = 1 1 − e − αβ t � � τ ( t ) = (5) αβ 0 which is actually deterministic. This means that X t and Z τ − 1 ( t ) have the same distribution. DFM LNU 11 / 21 Haidar Al-Talibi

Approximation Theorem Theorem Let t 1 < t 2 , t 1 , t 2 ∈ T , and T a compact subset of [ 0 , ∞ ) . Then there are N 1 and N 2 satisfying: ( i ) t 2 − t 1 ≥ N 1 ( ii ) β α ≥ N 2 v α (6) and 0 , β with 0 < α < 2 . Furthermore let dy t = K ( y t ) dt + dX t , (7) with y ( 0 ) = x 0 and K : R d → R d satisfy a global Lipschitz condition, then β →∞ x t = y t , lim for any t ∈ T where { x t } t ≥ 0 is the Ornstein-Uhlenbeck position process ( 3 ) and { y t } t ≥ 0 is the solution of ( 7 ) with { X t } t ≥ 0 as its driving α -stable Lévy noise. DFM LNU 12 / 21 Haidar Al-Talibi

Approximation Theorem For the increment of the OU position process � t 2 � t 2 � s � t 2 � s e − β ( s − u ) K ( x u ) duds + e − β ( s − u ) β dX u ds . e − β s v 0 ds + β x t 2 − x t 1 = t 1 t 1 0 t 1 0 (8) � t 2 e − β t 1 − e − β t 2 � t 1 e − β s v 0 ds = v 0 � The first integral of ( 8 ) is . β Taking the latter expression to the power α , where 0 < α < 2 and taking into account that e − β t 1 − e − β t 2 ≤ 1 we obtain that v α � α ≤ 1 � α . � e − β t 1 − e − β t 2 � 0 � e − α N 1 � � − ( 1 − e − N 1 ) � β α N 2 Where we used (( 6 )( i ) , ( ii )) and the fact that e − αβ t 1 ≤ e − αβ ∆ t ≤ e − α N 1 . If we � α tends to zero. N 2 e − α N 1 � 1 � − ( 1 − e − N 1 ) � choose N 1 and N 2 large enough then DFM LNU 13 / 21 Haidar Al-Talibi

Approximation Theorem The third part of ( 8 ) is estimated by first splitting the double integral into two integrals. We have �� t 2 � s � t 2 � t 1 � e − β s e β u dX u ds + e − β s e β u dX u ds (9) β . t 1 t 1 t 1 0 The double integral of the second part of ( 9 ) tends to zero as β and N 1 tend to infinity, � t 2 � t 1 e − β s e β u dX u ds = − Z τ ( t 1 ) e − β t 2 − e − β t 1 � e β t 1 � β t 1 0 1 1 − e − β ∆ t � 1 − e − β ∆ t � � � = αβ ( 1 − e − αβ t 1 ) = √ β Z Z 1 α ( 1 − e − αβ t 1 ) , 1 α where Z τ is an α -stable Lévy process. Moreover, the scaling property of Lévy processes we used in the last step, i.e. Z γτ = γ α Z τ , where γ > 0 , is actually a special case of Proposition 2. DFM LNU 14 / 21 Haidar Al-Talibi