An Itô Formula for an Accretive Operator

We give an Itô formula associated to a non-linear semi-group associated to a m-accretive operator.


Introduction
Let 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 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 .∞ .

Statement of the Theorems
Let (E, .) be a Banach space.Let L be a non-linear operator densely defined on E. We suppose L0 = 0. We recall that L is said to be accretive if for λ ≥ 0 It is said to be m-accretive if for λ > 0 Let us recall what is a mild solution of the non-linear parabolic equation We consider a subdivision 0 We say that u t i is an -discretization of Equation ( 4) if: Definition 1. v is said to be a mild solution of Equation ( 4) if for all there exist an -discretization u of Equation ( 6) such that u t − v t ≤ .
Let us recall the main theorem of [11,12]: 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[−tL].
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 It defines a continuous linear isometry Definition 2. The Itô transform L f of L is the operator densely defined on Let us give the domain of 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 We deduce therefore two non-linear semi-groups if L is m-accretive: - 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 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 where t → B t is a Brownian motion on R starting from 0.

Proof of the Theorems
Proof 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 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, (x, y) close from g(x, y) for the supremum norm and with bounded derivative of each order.x → L x g n, is continuous in x and the joint function (x, y) → L x g n, (x, y) is bounded continuous in (x, y) by the hypothesis on L.
Since Ψ f is a linear isometry of C b (T d ×R) which transform a smooth function into a smooth function, is clearly still m-accretive.
Proof of Theorem 3. Let us consider t i = i/N to simplify the exposition.Let us consider an -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) such that By doing y = 0 in the previous equality, we deduce that Therefore u f t i is an -discretization to the parabolic equation associated to L.