Diffusion Approximation for Transport Equations with Dissipative Drifts

We study stochastic differential equations(SDEs) with a small perturbation parameter. Under the dissipative condition on the drift coefficient and the local Lipschitz condition on the drift and diffusion coefficients we prove the existence and uniqueness result for the perturbed SDE, also the convergence result for the solution of the perturbed system to the solution of the unperturbed system when the perturbation parameter approaches zero.We consider the application of the above-mentioned results to the Cauchy problem and the transport equations.


Introduction
We consider Markov processes X ǫ t which arise from small random perturbations of dynamical systems, imposing specific conditions on the coefficients of the diffusion process, i.e., the dissipativity and dissipativity for differences for the drift and the local Lipschitz condition for all coefficients.These kind of processes arise in different areas of natural sciences.The concept of dissipativity comes, in particular, from physics.Dissipative systems are systems which absorb more energy from the external world than they supply and such systems are contrasted with energy conserving systems like Hamiltonian systems.The dissipativity of dynamical systems as it is known in modern system and control community was introduced by Willems in [7].
Freidlin and Wentzell in their book [1] have developed the theory for random perturbations assuming that the coefficients satisfy Lipschitz condition and have a linear growth bound.They study the random perturbations by direct probabilistic methods and then deduce consequences concerning the corresponding problems for partial differential equations.They consider mainly schemes of random perturbations of the form Ẋǫ where ξt(ω), t≥0 is a random process on a probability space with values in R l , its trajectories are right continuous, bounded and have at most a finite number of points of discontinuity on every interval [0, T ], T < ∞.At the points of discontinuity of ξt, where as a rule, (1) can not be satisfied, is imposed the requirement of continuity of X ǫ t .Additionally ǫ is a small numerical number and b(x, y) = (b 1 (x, y), ..., b r (x, y)), x ∈ R r , y ∈ R l is a vector field assumed to be jointly continuous in its variables.Let b(x, 0) = b(x), the random process X ǫ t is considered as a result of small perturbations of the system The following equation can be considered as a special case of (1) with b(x, y) = b(x) + σ(x)y.Here y is substituted by white noise process.
The precise meaning of (9) can be formulated in the language of stochastic integrals in the following way: Every solution of ( 4) is a Markov process (a diffusion process with drift vector b(x) and diffusion matrix ǫ 2 σ(x)σ * (x)).
Freidlin and Wentzell in their book [1] show that X ǫ t converges to the solution xt of the unperturbed system as ǫ → 0, moreover they discuss the application of this result to related partial differential equations.Particularly, they obtain results concerning the behaviour of solutions of boundary value problems as ǫ → 0 from the behaviour of X ǫ t (w) as ǫ → 0. In the theory of differential equations of parabolic type, much attention is devoted to the study of the behaviour, as ǫ → 0, for solutions of boundary value problems for equations of the form Here L ǫ is a differential operator with a small parameter at the derivatives of highest order: Every operator L ǫ (whose coefficients are assumed to be sufficiently regular) has an associated diffusion process X ǫ,x t .This diffusion process can be given by means of the stochastic equation where In particular , they consider the Cauchy problem: t > 0, x ∈ R r for ǫ > 0 and together with it the problem for the first-order operator which is obtained for ǫ = 0: A special case of Cauchy equation is the so called transport equation: which equals (6) in the case that g ≡0 and c ≡0.

The model
Let (Ω, F, (Ft) t∈[0,T ] , P) be the reference filtered probability space and w is a given ldimensional standard Brownian motion adapted to the defined filtration (Ft) t∈[0,T ] , 0 < T < +∞ being a finite horizon time.
Here Ω is a nonempty set, which is interpreted as the space of elementary events.The second object, F, is a σ-algebra of subsets of Ω.Finally, P is a probability measure on the σ-algebra F.
We consider and the perturbed stochastic differential equation Where ǫ is a small numerical number, b(x) = (b 1 (x), ..., b r (x)) is a vector field in R r and σ(x) = (σ i j (x)) is a matrix having l columns and r rows.By a solution of this equation we understand a random process Xt = Xt(w) which satisfies the relation with probability 1 for every t ∈ [0, T ].We usually assume that the coefficients of our diffusion fulfil a Lipschitz condition and have a linear growth bound.Under those conditions is proved that the solution to the stochastic differential equation exists and is unique.We modify the conditions on coefficients and prove that the existence and uniqueness result for the solution still holds (the proof of this result is based on the book by Gihman and Skorohod [3]) in one-dimensional case.For multi-dimensional situation it was analysed in classical book by Stroock and Varadhan [6].We will assume that σ increases no faster than linearly and b satisfies dissipativity, the coefficients of (9) satisfy a local Lipschitz condition: for some K for each N there exists an LN for which After proving the existence and uniqueness result, we will show that the zeroth approximation for the process (9) with dissipative drift and locally Lipschitz coefficients holds, i.e the solution of (9) X ǫ t converges to the solution of (8) xt as ǫ → 0 .The last approximation will be used to show that the solution to the Cauchy problem for ǫ > 0 converges to the solution for ǫ = 0 with weaker conditions, this convergence holds also for Transport equation.
Before stating and proving the main results, we would like to state a Gronwall-Lemma which is often used in the proofs.
3 Main results

Existence and Uniqueness of a Solution
We aim to show that under the weaker conditions on the coefficients, i.e. dissipativity for the drift and the local Lipschitz condition for all the coefficients, (9) has a solution and the solution is unique.This fact may be find in [6] but we will include the proof because certain steps in this proof we will use later.To prove the existence and uniqueness result we will need the following theorem: Theorem 3.1.Assume that the coefficients b1(x), b2(x), σ1(x), σ2(x) of the equations satisfy Lipschitz condition and linear growth condition, i.e., there exists a constant K such and that for some N > 0 with |x j | ≤ N for all j ∈ N0 : 0 ≤ j ≤ r, b1(x) = b2(x) and If X ǫ t,1 and X ǫ t,2 are solutions of (11) with the same initial condition and τi is the largest t for which sup 0≤s≤t,0≤j≤r |X ǫ,j t,i | ≤ N , then P {τ1 = τ2} = 1 and Where the last step is possible, because from γ1(t) = 1 follows b j 1 (X ǫ s,1 ) = b j 2 (X ǫ s,1 ) and Taking into account that γ1(t) = 1 implies γ1(s) = 1 for s ≤ t we can write the γ1(s)'s inside the brackets.Taking the expectation and then using the Lipschitz condition and Cauchy-Schwarz-inequality, we can show that for ǫ ≤ 1 there exists a constant L for which Now we can use Lemma 2.1, for which C=0.It follows that Considering the continuity of X ǫ t,1 and X ǫ t,2 we can establish On the interval [0, τ1] the processes X ǫ t,1 and X ǫ t,2 coincide with probability 1. Hence P {τ2 ≥ τ1} = 1.Interchanging the indices 1 and 2 in the proof of the theorem, we can show analogously that P {τ1 ≥ τ2} = 1.Theorem 3.2.Let the coefficients of (9) be defined and measurable for t ∈ [0, 1], and satisfy the conditions 2. for each N there exists an LN for which Then (9) has a unique solution in the sense that for two solutions X ǫ t,1 and X ǫ t,2 Proof.We will first start by showing the existence and afterwards we move to the uniqueness of the solution.Define x i N (the i-th component of the vector xN ) as For this equation all conditions for existence are given, because we have a growth bound depending on N and for the coefficients we also have a global Lipschitz condition.
Let τN be the largest value of t for which sup 0≤s≤t If we can show that the probability on the right hand side converges to zero for N → ∞, then it will clearly follow that X ǫ t,N converges uniformly with probability 1 to some limit X ǫ t as N → ∞.Going to the limit in we convince ourselves that X ǫ t is equal with probability 1 to the continuous solution of (9).So to finish the proof of the existence of a solution it remains to show that lim To do this we first define the function ψ(y) = 1 1+|y| 2 and then we use the Ito formula.We obtain By having that we can use Lemma 2.1 to get Which means we have where C1 is independent of N.
We can moreover write where the last inequality follows from the Chebychev inequality.

Consequently lim
Since δ is an arbitrary positive number and P {ψ(xN ) = 0} = 0, (15) results from the preceding relation.This completes the proof of the existence of a solution to (9).
Next we want to prove the uniqueness of the solution.Let X ǫ t,1 and X ǫ t,2 be two solutions of (9).Denoting by φ(t) the variable equal to 1 if sup 0≤s≤t |X ǫ,i s,1 | ≤ N and sup 0≤s≤t |X ǫ,i s,2 | ≤ N and equal to 0 otherwise.Using our second condition we can write Where we first used that a 2 + b 2 ≥ 2ab, then the Cauchy-Schwarz inequality and the properties of an Ito integral and afterwards the local Lipschitz continuity.Then we need to use Lemma 2.1 with C = 0 to get M |X ǫ t,1 − X ǫ t,2 | 2 φ(t) = 0, which means From the continuity of X ǫ t,1 and X ǫ t,2 follows their boundedness.Hence the probability on the right side of this inequality tend to zero as N → ∞, i.e., for all t ∈ [0, T ]: P {X ǫ t,1 = X ǫ t,2 } = 1 from which the uniqueness follows in the sense that P {sup 0≤t≤T |X ǫ t,1 −X ǫ t,2 | = 0} = 1.

Zeroth Order Approximation for Dissipative Case
After proving the existence and uniqueness of a solution to (9) with our conditions on coefficients we want to prove convergence of the solution of (9) X ǫ t to the solution of (8) xt as ǫ → 0 under dissipativity and dissipativity for differences for the drift vector and the local Lipschitz condition for all coefficients.
Theorem 3.3.Assume that the coefficients of (9) satisfy a local Lipschitz condition, σ increases no faster than linearly and b satisfies dissipativity and dissipativity for the differences: 2. for each N there exists an LN for which Then for all t > 0 and δ > 0 we have: where a(t) is a monotone increasing function, which is expressed in terms of |x| and K.
Proof.We start by showing that To show that we first apply Ito's formula to get Applying the mathematical expectation and adding (1 + |x| 2 ) on both sides we obtain: Using the Cauchy-Schwarz inequality, that σ in (9) increases no faster than linearly and the dissipativity for b, the last relation implies the estimate Next we use Lemma 2.1 we choose m(t . By doing this we obtain By the inequality we proved that In the next step we want to use our result to prove that M |X ǫ t − xt| ≤ ǫ 2 a(t).To do this we work through it very similarly to before.
We apply the Ito formula to the function |X ǫ t − xt| 2 , which works the same way as it did with 1 + |X ǫ t |, just that the starting term vanishes, because X ǫ 0 = x = x0.Next we apply the mathematical expectation on both sides of the equality to get In the proof of the existence we proved (15).Since X ǫ t,N converges to X ǫ t as N → ∞ we also know From this follows that there exists an N, such that and In the following calculations we first split up our mathematical expectation in two different cases, then we use the Cauchy-Schwarz inequality and ( 16).Afterwards we apply the local Lipschitz condition (18) and the dissipativity for the differences for b (17).Then we estimate the probabilities we used in the inequality: We use Lemma 2.1 again and this time we choose Where we used the result (19) and a(t) is chosen such that it is a monotone increasing function.
Now we want to prove the second assertion of the theorem.We will now use the Cheby-shev inequality that says that P {ξ(ω) ≥ a} ≤ M f (ξ) f (a) .By setting ξ(ω) = max 0≤s≤t |X ǫ s − xs|, a = δ, f (x) = x 2 and applying the first assertion of the theorem we obtain Taking limits on both sides in (24) , we get We aim to obtain results concerning the behavior of solutions of Cauchy problem as ǫ → 0 from the behavior of X ǫ t (w) as ǫ → 0.In the preceding section we have obtained a result concerning the the behavior of solutions X ǫ t (w) as ǫ → 0, which will be used in the present section.We consider the Cauchy problem: t > 0, x ∈ R r for ǫ > 0 and together with it the problem for the first-order operator which is obtained for ǫ = 0: Where L ǫ is a differential operator with a small parameter at the derivatives of highest order: Every operator L ǫ (whose coefficients are assumed to be sufficiently regular) has an associated diffusion process X ǫ,x t .This diffusion process can be given by means of the stochastic equation where σ(x)σ * (x) = (a ij (x)), b(x) = (b 1 (x), ..., b r (x)).
We assume the following conditions are satisfied: 1. the function c(x) is uniformly continuous and bounded for x ∈ R r ; 2. the coefficients of L 1 satisfy a local Lipschitz condition, b satisfies dissipativity and dissipativity for the differences; 3. k −2 λ 2 t ≤ r i,j=1 a ij (x)λiλj ≤ k 2 λ 2 i for any real λ1, λ2, ..., λr and x ∈ R r , where k 2 is a positive constant.Under these conditions, the solutions of problems (25) and (26) exist and are unique.
Having these conditions we obtain the following result:  The function on the right side of the equality is a solution of (26),this finishes the proof.
The special case is when c(x) ≡ g(x) ≡ 0, which gives us the Transport equation ∂v ǫ (t, x) ∂t = L ǫ v ǫ (t, x), v ǫ (0, x) = f (x), the solution of the transport equation can be written in the following form As in the case of Cauchy problem passing to the limit when ǫ → 0 we get lim ǫ→0 v ǫ (t, x) = v 0 (t, x) where v 0 (t, x) is the solution of ∂v 0 (t, x) ∂t = L 0 v 0 (t, x), v 0 (0, x) = f (x).

M
Proof.If condition (3) is satisfied, then there exists a matrix σ(x) with entries satisfying a local Lipschitz condition for which σ(x)σ * (x) = (a ij (x)).The solution of (25) can be represented in the following way ([1] Chap.1, Sec.5):v ǫ (t, x) = M [f (X ǫ,x t ) exp{ t 0 c(X ǫ,x s )ds}] + M [ ǫ,x u )du}ds].(28)This stays true for the changed conditions, because of the uniqueness of the solution.From Theorem 3.3 follows the convergence of X ǫ,x s to X 0,x s in probability on the interval [0, t] as ǫ → 0. Taking into account that there is a bounded continuous functional of X ǫ,x s (ω) under the sign of mathematical expectation in (29), by the Lebesgue dominated convergence theorem, which we can use because the functional is bounded, we obtain lim ǫ↓0 v ǫ (t, x) = lim ǫ↓0