An Itô Formula for an Accretive Operator

Abstract

first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessCommunication An Itô Formula for an Accretive Operator by Rémi Léandre Laboratoire de Mathématiques, Université de Franche-Comté, route de Gray, Besançon 25030, France Axioms 2012, 1(1), 4-8; https://doi.org/10.3390/axioms1010004 Received: 21 November 2011 / Revised: 12 March 2012 / Accepted: 13 March 2012 / Published: 21 March 2012 (This article belongs to the Special Issue Axioms: Feature Papers) Download Download PDF Download PDF with Cover Download XML Download Epub Versions Notes Abstract: We give an Itô formula associated to a non-linear semi-group associated to a m-accretive operator. Keywords: non-linear semi-group; Itô formula 1. IntroductionLet us recall the Itô formula in the Stratonovich Calculus [1]. Let B t be a one dimensional Brownian motion and f be a smooth function on R. Then f ( B t ) = f ( B 0 ) + ∫ 0 t f ′ ( B s ) d B s (1) where we consider the Stratonovich differential.In [2,3], we have remarked that the couple ( B t , f ( B t ) ) is a diffusion on R × R whose generator can be easily computed. This leads to an interpretation inside the semi-group theory of the Itô formula. Various Itô formulas were stated by ourself for various partial differential equations where there is no stochastic process [4,5,6,7,8,9]. See [9] for a review. For an Itô formula associated to a bilaplacian viewed inside the Fock space, we refer to [10].There is roughly speaking following Hunt theory a stochastic process associated to a linear semi-group when the infinitesimal generator of the semi-group satisfied the maximum principle.For nonlinear semi-group, the role of maximum principle is played by the notion of accretive operator. The goal of this paper is to state an Itô formula for a nonlinear semi-group associated to a m-accretive operator on C b ( T d ) , the space of continuous functions on the d-dimensional torus T d endowed with the uniform metric ∥ . ∥ ∞ . 2. Statement of the TheoremsLet ( E , ∥ . ∥ ) be a Banach space. Let L be a non-linear operator densely defined on E. We suppose L 0 = 0 . We recall that L is said to be accretive if for λ ≥ 0 ∥ e 1 − e 2 + λ ( L ( e 1 ) − L ( e 2 ) ) ∥ ≥ ∥ e 1 − e 2 ∥ (2) It is said to be m-accretive if for λ > 0 I m ( I + λ L ) = E (3) Let us recall what is a mild solution of the non-linear parabolic equation ∂ ∂ t u t + L u t = 0 ; u 0 = e (4) We consider a subdivision 0 ≤ t 1 < ⋯ < t N = 1 . We say that u t i is an ϵ-discretization of Equation (4) if: t i + 1 − t i < ϵ (5) u t i − u t i − 1 t i + 1 − t i + L u i = 0 (6) Definition 1. v is said to be a mild solution of Equation (4) if for all ϵ there exist an ϵ-discretization u of Equation (6) such that ∥ u t − v t ∥ ≤ ϵ .Let us recall the main theorem of [11,12]:Theorem 1. If L is m-accretive, there exists for all e in E a unique mild-solution of Equation (4). This generates therefore a non-linear semi-group exp [ − t L ] .We consider the d-dimensional torus. We consider E = C b ( T d ) and let L be an m-accretive operator whose domain contains C b ∞ ( T d ) , the space of smooth functions on T d with bounded derivatives at each order which is continuous from C b ∞ ( T d ) into C b ( T d ) .Let f ∈ C b ∞ ( T d ) . We consider g ∈ C b ( T d × R ) .We consider the diffeomorphism ψ f of T d × R : ψ f ( x , y ) = ( x , y + f ( x ) ) (7) It defines a continuous linear isometry Ψ f of C b ( T d × R ) Ψ f [ g ] ( x , y ) = g ∘ ψ f ( x , y ) (8) Definition 2. The Itô transform L f of L is the operator densely defined on C b ( T d × R ) L f = ( Ψ f ) − 1 ∘ ( L ⊗ I 1 ) ∘ Ψ f (9) Let us give the domain of L ⊗ I 1 . C b ( T d × R ) is constituted of function g ( x , y ) . L ⊗ I 1 [ g ] ( x , y ) = L x g ( x , y ) (10) where we apply the operator L on the continuous function x → g ( x , y ) supposed in the domain of L for all y. We suppose moreover that ( x , y ) → L x g ( x , y ) is bounded continuous. The domain contains clearly C b ∞ ( T d × R ) .Theorem 2. If L is m-accretive on C b ( T d ) , its Itô-transform is m-accretive on C b ( T d × R ) .We deduce therefore two non-linear semi-groups if L is m-accretive:- exp [ − t L ] acting on C b ( T d ) .- exp [ − t L f ] acting on C b ( T d × R ) .Let g be an element of C b ( T d × R ) . We consider g f ( x ) = g ( x , f ( x ) ) . We get:Theorem 3. (Itô formula) We have the relation exp [ − t L ] [ g f ] ( x ) = exp [ − t L f ] [ g ] ( x , f ( x ) ) (11) This formula is an extension in the non-linear case of the classical Itô formula for the Brownian motion. If we take L = − 1 / 2 ∂ 2 ∂ x 2 acting densely on C b ( R ) , we have exp [ − t L ] [ g ] ( x ) = E [ g ( B t + x ) ] (12) where t → B t is a Brownian motion on R starting from 0. ( B t + x , f ( B t + x ) + y ) is a diffusion on R × R whose generator is L f . 3. Proof of the TheoremsProof of Theorem 2. L ⊗ I 1 is clearly m-accretive on C b ( T d × R ) . Let us show this result.- L ⊗ I 1 is densely defined. Let g be a bounded continuous function on T d × R . By using a suitable partition of unity on R, we can write g ( x , y ) = ∑ g n ( x , y ) (13) where g n ( x , y ) = 0 if y does not belong to [ − n − 1 , n + 1 ] . By an approximation by convolution we can find a smooth function g n , ϵ ( x , y ) close from g ( x , y ) for the supremum norm and with bounded derivative of each order. x → L x g n , ϵ is continuous in x and the joint function ( x , y ) → L x g n , ϵ ( x , y ) is bounded continuous in ( x , y ) by the hypothesis on L.-Clearly Equation (2) is satisfied.-It remains to show Equation (3). If g belong to C b ( T d × R ) we can find x → h ( x , y ) such that h ( x , y ) + λ L x h ( x , y ) = g ( x , y ) (14) ∥ g ( , y ) − g ( . , y ′ ) ∥ ∞ ≥ ∥ h ( . , y ) − h ( . , y ′ ) ∥ ∞ (15) Therefore ( x , y ) → h ( x , y ) is jointly bounded continuous.Since Ψ f is a linear isometry of C b ( T d × R ) which transform a smooth function into a smooth function, L f = ( Ψ f ) − 1 ∘ ( L ⊗ I 1 ) ∘ Ψ f (16) is clearly still m-accretive. ☐Proof of Theorem 3. Let us consider t i = i / N to simplify the exposition. Let us consider an ϵ-discretization u . of the parabolic equation associated to L f . This means that u t i ∈ ( Ψ f ) − 1 ( I d + 1 + 1 / N ( L ⊗ I 1 ) ) − i Ψ f g (17) I d + 1 is the identity on C b ( T d × R ) . But ( I d + 1 + 1 / N ( L ⊗ I 1 ) ) = ( I d + 1 / N L ) ⊗ I 1 (18) such that ( ( I d + 1 / N L ) i ⊗ I 1 ) Ψ f u t i = Ψ f g (19) By doing y = 0 in the previous equality, we deduce that ( 1 + L / N ) i u t i f = g f (20) Therefore u t i f is an ϵ-discretization to the parabolic equation associated to L. ☐ AcknowledgementsWe thank M. Mokhtar-Karroubi and B. Andreianov for helpful discussion.ReferencesDellacherie, C.; Meyer, P.A. Probability and Potential B: Theory of Martingales; North-Holland: Amsterdam, The Netherlands, 1982. [Google Scholar]Léandre, R. Wentzel-Freidlin Estimates in Semi-Group Theory. In Proceedings of the 10th International Conference on Control, Automation, Robotics and Vision (ICARCV ’08), Hanoi, Vietnam, 17–20 December 2008; pp. 2233–2235.Léandre, R. Large Deviation Estimates in Semi-Group Theory. In Proceedings of the International Conference on Numerical Analysis and Applied Mathematics, Kos, Greece, 16–20 September 2008; Volume 1048, pp. 351–355.Léandre, R. 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