The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain

Correspondence should be addressed to Gerardo Rubio, grubioh@yahoo.com Received 20 December 2010; Revised 18 March 2011; Accepted 19 April 2011 Academic Editor: Lukasz Stettner Copyright q 2011 Gerardo Rubio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the Cauchy-Dirichlet problem in 0,∞ × D for a class of linear parabolic partial differential equations. We assume that D ⊂ R is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.


Introduction
In this paper, we study the existence and uniqueness of a classical solution to the Cauchy-Dirichlet problem for a linear parabolic differential equation in a general unbounded domain.Let L be the differential operator L u t, x : In the case of bounded domains, the Cauchy-Dirichlet problem is well understood see 1, 2 for a detailed description of this problem .Moreover, when the domain is unbounded and the coefficients are bounded, the existence of a classical solution to 1.2 is well known.For a survey of this theory see 3, 4 where the problem is studied with analytical methods and 5 for a probabilistic approach.
In the last years, parabolic equations with unbounded coefficients in unbounded domains have been studied in great detail.For the particular case when D R d , there exist many papers in which the existence, uniqueness, and regularity of the solution is studied under different hypotheses on the coefficients; see for example, 6-17 .In the case of general unbounded domains, Fornaro et al. in 18 studied the homogeneous, autonomous Cauchy-Dirichlet problem.They proved, using analytical methods in semigroups, the existence and uniqueness of a solution to the Cauchy-Dirichlet problem when the coefficients are locally C 1,α , with a ij bounded, b and c functions with a Lyapunov type growth; that is, there exists a function ϕ ∈ C 1,2  Using the theory of semigroups, Da Prato and Lunardi studied, in 22, 23 , the realization of the elliptic operator A 1/2 Δ − DU, D• , in the functions spaces L 2 D , L ∞ D and C b D , when U is an unbounded convex function defined in a convex set D. They proved existence and uniqueness for the elliptic and parabolic equations associated with A and studied the regularity of the semigroup generated by A. Geissert et al. in 24 , made a similar approach for the Ornstein-Uhlenbeck operator.
In the paper of Hieber et al. 25 , the existence and uniqueness of a classical solution for the autonomous, nonhomogeneous Cauchy-Dirichlet and Cauchy-Neumann problems is proved.The domain is considered to be an exterior domain with C 3 boundary.The coefficients are assumed to be C 3,α continuous functions with Lyapunov type growth.The continuity properties of the semigroup generated by the solution of the parabolic problem are studied in the spaces C b D and L p D .In all the papers cited above, the uniformly elliptic condition is assumed; that is, there exists λ > 0 such that for all t, x ∈ 0, ∞ × D we have that a ij t, x η i η j ≥ λ η 2 for all η ∈ R d .
In this paper, we prove the existence and uniqueness of a classical solution to 1.2 , when the coefficients are locally Lipschitz continuous in x and locally H ölder continuous in t, a ij has a quadratic growth, b i has linear growth, and c is bounded from above.We allow f, g, and h to have a polynomial growth of any order.We also consider the elliptic condition to be local; that is, for any 0, T × A ⊂ 0, ∞ × D, there exists λ T, A such that a ij t, x η i η j ≥ λ T, A η 2 for all t ∈ 0, T , x ∈ A and η ∈ R d .We assume that D is an unbounded, connected, open set with regular boundary see 1 Chapter III, Section 4, for a definition of regular boundary .Furthermore, we prove that the solution is locally H ölder continuous up to the second space derivative and the first time derivative.
Our approach is using stochastic differential equations and parabolic differential equations in bounded domains.For proving existence, many analytical methods construct the solution by solving the problem in nested bounded domains that approximate the domain D. In these cases, the convergence of the approximating solutions is always a very difficult task.Unlike these methods, first we propose, as a solution to 1.2 , a functional of the solution to a SDE, where

1.6
Using the continuity of the paths of the SDE, we prove that this function is continuous in 0, ∞ × D.Then, using the theory of parabolic equations in bounded domains, we study locally the regularity of the function v and prove that it is a C 1,2 function.Finally, with some standard arguments, we prove that it solves the Cauchy-Dirichlet problem.This kind of idea has been used for several partial differential problems see 5, 26, 27 .
In Section 2, we introduce the notation and the hypotheses used throughout the paper.Section 3 presents the main result.In this section we prove that if the function v is smooth, then it has to be the solution to the Cauchy-Dirichlet problem.Section 4 is devoted to prove the required differentiability for the candidate function.Finally, in Section 5, the reader will find some of the results used in the proof of our main theorem.

Preliminaries and Notation
In this section, we present the hypotheses and the notation used in this paper.

Domain
Let D ⊂ R d be an unbounded, open, connected set with boundary ∂D and closure D. We assume that D has a regular boundary; that is, for any x ∈ ∂D, x is a regular point see 1 Chapter III, Section 4 or 28 Chapter 2, Section 4, for a detailed discussion of regular points .We denote the hypotheses on D by H0.

Stochastic Differential Equation
Let Ω, F, P, {F s } s≥0 be a complete filtered probability space and let {W} {W i } d i 1 be a ddimensional brownian motion defined in it.For t ≥ 0 and x ∈ D, consider the stochastic differential equation Although this process is the natural one for solving equation 1.2 , it does not posses many good properties.The continuity of the flow process does not imply the continuity with respect to t.Furthermore, although this process is a strong Markov process, it is not homogeneous in time, a very useful property for proving the results in this paper.
To overcome these difficulties, we augment the dimension considering the following process dξ s −ds, ξ 0 t.

2.2
Then the process {ξ s , X s } is solution to dX s b ξ s , X s ds σ ξ s , X s dW s ,

2.3
with ξ 0 , X 0 t, x .Throughout this paper we will use both processes, X s and ξ s , X s , in order to simplify the exposition.For the expectation, we use the notation when considering the process X defined as in 2.1 and the notation when working with the joint process ξ, X defined in 2.3 .
We need to define the following stopping times

2.6
Remark 2.1.Observe that τ is the exit time of the process ξ s , X s from the set 0, ∞ × D, that is, We cannot guarantee that the process X s leaves the set D in a finite time however, the process ξ s reaches the boundary s 0 at time t.Thus, the joint process ξ s , X s leaves the set 0, ∞ × D in a bounded time.
We assume the following hypotheses on the coefficients b and σ.We denote them by H1.The matrix norm considered is σ 2 : tr σσ i,j σ 2 ij .H1.
2 Linear-Growth.For each T > 0, there exists a constant 12 are continuous functions satisfying the hypotheses in H1 restricted to the set 0, ∞ × R d , then we can extend them to be defined for negative values of r as follows: let b and σ be defined as

2.13
It is easy to see that these functions satisfy H1 with the same constants L 1 and K 1 .
Remark 2.4.It follows, from the nondegeneracy the local ellipticity of the process X s , the regular boundary of the set D and Lemma 4.2, Chapter 2 in 28 , that, for any t, x ∈ 0, ∞ ×D, The next proposition presents some of the properties of the process ξ, X required in this work.Proposition 2.5.As a consequence of H1, ξ, X has the following properties.
i For all t, x ∈ 0, ∞ × R d , there exists a unique strong solution to 2.3 .
ii The process {ξ s , X s } s≥0 is a strong homogeneous Markov process.
iii The process {ξ s , X s } s≥0 does not explode in finite time a.s.. iv For all x ∈ R d , T > 0, and r ≥ 1,

The Cauchy-Dirichlet Problem
Consider the following differential operator: where a σσ .For the rest of the paper, we assume that the coefficients of L satisfy H1.The Cauchy-Dirichlet problem for a linear parabolic equation is

2.17
We assume the following hypotheses for the functions c, f, h, and g.We denote them by H2.

International Journal of Stochastic Analysis
There exists k > 0, such that for all T > 0, a constant K 2 T exists such that be continuous functions such that i Growth.There exists k > 0, such that, for all T > 0, there exists a constant for all r, x ∈ 0, T × D. ii Consistency.There exists consistency in the intersection of the space and the time boundaries, that is, for x ∈ ∂D.

Additional Notation
If μ is a locally Lipschitz function defined in some set R, then, for any bounded open set A for which A ⊂ R, we denote, by K μ A and L μ A , the constants

2.26
The space C 1,2,λ loc 0, ∞ ×D is the space of all functions such that they and all their derivatives up to the second order in x and first order in t, are locally H ölder continuous of order λ.

Main Result
In this section, we present the main result of this work and some parts of the proof.Theorem 3.1.Assume H0, H1, and H2.Then, there exists a unique solution where X is the solution to the stochastic differential equation

3.2
Furthermore, for all T > 0, where c 0 , K 1 , K 2 , K 3 , and k are the constants defined in H1 and H2.
The proof of this theorem is given by several lemmas.The method we will use has the following steps: first we define a functional of the process X as a candidate solution.Let v : 0, ∞ × D → R be defined as then there exist some standard arguments see 27, chapter 4 to prove that v is the unique solution to 2.17 .The rest of this section is devoted to proving Theorem 3.1 in the case when v is a "regular" function.The proof is divided into two lemmas: the first one proves that if v ∈ C 0, ∞ × D ∩ C 1,2 0, ∞ × D , then v is a solution to equation.The second one proves that in case of existence of a classical solution, u, to problem 2.17 , then it is unique and has the form given by v in 3.4 .The regularity of v is proved in Section 4 below.

International Journal of Stochastic Analysis
The next proposition gives an extension of the boundary data to all the space 0, ∞ × R d .This extension is given to simplify the notation and is required in the proofs of Lemmas 4.2 and 4.3.Proposition 3.2.Assume H2.Then, there exists a continuous function

3.5
Proof.Thanks to the consistency condition in H2 and the continuity of g and h, we can extend by Tietze's extension theorem see 32, Section 2.6 the functions g, h from the closed set As a consequence of Proposition 3.2, we can write v in 3.4 as follows:

3.6
We are ready to prove both lemmas explained above.

3.7
Furthermore, for all T > 0, there exists C such that where c 0 , K 1 , K 2 , K 3 , and k are the constants defined in H1 and H2.
Proof.Let 0 ≤ α ≤ t, then, following the same argument used to prove 4.80 in the proof of Theorem 4.4 in Section 4 we have that

3.9
International Journal of Stochastic Analysis 11 Because of H1 and H2, we have that the random variable inside the conditional expectation is integrable and so the left-hand side of 3.9 is a F α -martingale, for α ∈ 0, t .Since v ∈ C 1,2 , we can apply Ito's formula to e α 0 cdr v to get

3.10
It follows from the continuity of Dv, σ and X • that 12 is a local martingale for 0 ≤ α ≤ t.So, combining 3.9 and 3.10 , we get that M α : For the boundary condition, it follows from the regularity of the set D and the local ellipticity see Remark 2.4 that From this, it is clear that the first addend of the right-hand side of 3.6 is zero.For the second addend, we get that the exponential term is equal to one and that , and so we conclude that v satisfies the boundary condition.The second statement of the theorem is proved with the same argument used to prove 4.9 and 4.42 in the proofs of Lemmas 4.2 and 4.3 in Section 4.

International Journal of Stochastic Analysis
The next Lemma proves the uniqueness of the solution.

3.15
such that, for all T > 0, there exists C for which for some μ > 0.Then, u has the following representation: 3.17 and hence the solution is unique.

3.23
By 2.15 and the dominated convergence theorem, letting n → ∞, we get

3.24
Letting α ↑ t, a similar argument and the boundary condition proof that and the proof is complete.
International Journal of Stochastic Analysis

Regularity of v
In this section, we prove that v ∈ C 0, ∞ × D ∩ C 1,2,λ loc 0, ∞ × D .First, we prove, using the continuity of the flow process X, that v is a continuous function in 0, ∞ × D. Since we are only assuming the continuity of the coefficients, then the flow is not necessarily differentiable and so we can not prove the regularity of v in terms of the regularity of the flow.To prove that v ∈ C 1,2 , we show that v is the solution to a parabolic differential equation in a bounded domain, for which we have the existence of a classical solution and hence v ∈ C 1,2 .

Continuity of v
Let ξ, X denote the solution to 2.3 and let G be defined as in Proposition 3.2, then v has the following form:

4.1
For simplicity, we write v v 1 v 2 , where The proof of this theorem is divided into two lemmas.
Proof.First, we prove the continuity on 0, ∞ × D. For that, let in 0, ∞ × D and > 0. We need to prove that there exists N ∈ N such that, for all n ≥ N Denote by ξ, X and ξ n , X n the solutions to 2.3 with initial conditions t, x and t n , x n , respectively.To simplify the notation in the proof, let τ : t ∧ τ D and τ n : t n ∧ τ Dn denote their corresponding exit times from 0, ∞ × D.

International Journal of Stochastic Analysis 15
Let α > 0, then there exists N 1 ∈ N such that, for all n ≥ N 1 t n , x n − t, x < α.

4.6
Observe that, for all n ≥ N 1 , we get where C C t, α, c 0 and K K t, α, k .We use 4.6 , 4.7 , 2.15 and the polynomial growth of f.Let M > 0, 0 < η < 1, and β > 0 and define the set Then, For simplicity of notation, we write the set E M,n,η,β as E and define and let be an open set such that A ⊂ B A .
On the set E, for all n ≥ N 1 and 0 ≤ s ≤ t α, it is satisfied that ξ n s , X n s , ξ s , X s ∈ A. 4.17 We have that

4.19
We first analyse 4.19 : To summarize, we get with the above estimations that
ii Let N 1 ≥ N such that, for all n ≥ N 1 , iii Let δ > 0 fulfil the uniformly integrability condition 4.12 .
For the continuity at the boundary we make a similar argument.Let where t n , x n ∈ 0, ∞ × D and t, x ∈ ∂ 0, ∞ × D , that is, either t 0 or x ∈ ∂D.In both cases we get that τ 0 a.s. and so v 1 t, x 0.Then, we need to prove that Let 0 < α 1 and N 1 ∈ N be such that t n , x n − t, x < α.

4.34
We get International Journal of Stochastic Analysis for all n ≥ N 1 .For the continuity, we have

4.36
The convergence follows from the uniform integrability of Proof.We use an analogous argument to the one in the proof of Lemma 4.2.First, we prove the continuity in 0, ∞ × D. Let with t n , x n , t, x ∈ 0, ∞ × D. Denote by ξ n , X n and ξ, X the solutions to 2.3 with initial conditions t n , x n and t, x , respectively, and let τ n : t n ∧ τ Dn and τ : t ∧ τ D be their corresponding exit times from 0, ∞ × D. Let 0 < α 1 and N 1 be such that, for all n ≥ N 1 , t n , x n − t, x < α.

4.39
This implies that

4.40
First, we prove that the sequence of random variables is uniformly integrable for all n ≥ N 1 .As in 4.9 ,

4.42
where C C t, α, c 0 and K K t, α, k .We use 4.39 , 4.40 , 2.15 , and the polynomial growth of G in ∂ 0, ∞ × D .As in Lemma 4.2, let > 0, then there exists δ > 0 such that sup for all E ∈ F, with P E < δ .Let E M,n,η,β be defined as in 4.10 , and choose M > 0 and N 2 ∈ N such that

International Journal of Stochastic Analysis
For simplicity of notation, denote E M,n,η,β as E.Then,

4.45
Let A, D t , and B A be defined as in Lemma 4.2 see 4.15 and 4.16 .Then, on the set E, we get that, for all n ≥ N 1 and 0 ≤ s ≤ t α, ξ n s , X n s , ξ s , X s ∈ A.

4.46
So, We study each addend of the right-hand side separately:

4.50
First, we get a bound for 4.49 .Since G is continuous, then it is uniformly continuous on A.
Then, for > 0, there exists γ c 0 , t, α, , M such that τ.This and the continuity of X • and G imply that On the set E, we have that t n − τ n , X τ n , t − τ, X τ ∈ A and so By the dominated convergence theorem, there exists N 3 ∈ N such that 4.50 < 8 4.56 for all n ≥ N 3 .
To give a bound for 4.48 , we observe that, on the set E, To summarize, we get with the above estimations that

International Journal of Stochastic Analysis
Hence, to prove continuity, we proceed as follows.
ii Let N 1 ∈ N be such that, for all n ≥ N 1 , t n , x n − t, x < α.
x Let N 4 ∈ N be such that We repeat the same arguments made for the estimates to 4.49 and 4.50 with 4.74 and 4.75 , respectively.Then, we can prove that

4.76
So, v 2 ∈ C 0, ∞ × D and the proof is complete.

Differentiability of v
Let 0 ≤ T 0 < T 1 and A ⊂ D be a bounded, open, connected set with C 2 boundary.Consider the following Cauchy-Dirichlet problem: where the boundary data is v.If we assume H0, H1, and H2, then by the continuity of v Theorem 4.1 and Theorem 5.5, we can guarantee the existence of a unique classical solution to problem 4.77 .To prove the regularity of v, we show that it coincides with the solution to 4.77 in the set T 0 , T 1 × A and so v ∈ C 1,2 T 0 , T 1 × A .Since T 0 , T 1 , and A are arbitrary, we get the desired regularity.We are ready to prove the next theorem.
Proof.Let w be the solution to 4.77 .Define the following stopping times

4.78
Following the same arguments of Section 5 in Chapter 6 of 31 , we can prove that w has the following representation: Next, we prove that v satisfies the following equality:

4.80
Let v 1 and v 2 be defined as in 4.2 and 4.3 , where τ : t ∧ τ D is introduced to simplify the notation.We will use the following representation of v 1 and v 2 :

4.85
Since θ < τ D and θ is bounded, we get that see Remark 2.1 where Θ • denotes the shift operator.Since the process ξ, X is a homogeneous strong Markov process, we get that In both sides of 4.88 , we consider the sequence F ∨ −n ∧ n and apply the conditional dominated convergence Theorem.So,

4.90
Next, we study v 2 .Again, for the integral, we use a couple of changes of variables to get

4.92
Then, the expression inside the conditional expectation can be written as

4.93
Repeating the same argument used for 4.88 , we have that the conditional expectation is   −→ a↓0 τ.
Theorem 5.5.Assume H1 and H2.Consider the following differential equation: −u t t, x L u t, x c t, x u t, x −f t, x t, x ∈ T 0 , T 1 × A, u T 0 , x g T 0 , x for x ∈ A, u t, x g t, x for t, x ∈ T 0 , T 1 × ∂A.

5.26
If g is continuous, then there exists a classical solution w ∈ C T 0 , T 1 × A ∩ C 1,2,β T 0 , T 1 × A of 5.26 .
Remark 5.6.Let w be the solution of 5.26 and define z as w t, x e c 0 t z t, x in T 0 , T 1 × A. Then, z fulfils 5.26 with c c − c 0 and f t, x e −c 0 t f t, x .And so the hypotheses of Theorem 9 of Chapter 3 in 3 are satisfied.
−r,X r dr f t − s, X s ds

0 19 A0
r,X r dr u t − α ∧ τ D , X α ∧ τ D .3.18Applying Ito's rule, we gete α∧τ D 0 c t−r,X r dr u t − α ∧ τ D , X α ∧ τ D −r,X r dr −u t L u cu t − s, X s ds α∧τ D Du t − s, X s • σ t − s, X s dW s .3.similar argument as the one used in the proof of Lemma 3.3 shows that α∧τ D Du t − s, X s • σ t − s, X s dW s 3.20 is a local martingale.Due to 3.15 , we conclude that M α : e α∧τ D 0 c t−r,X r dr u t − α ∧ τ D , X α ∧ τ D α∧τ D 0 e s 0 c t−r,X r dr f t − s, X s ds 3.21 International Journal of Stochastic Analysis 13 is a local martingale for α ∈ 0, t .Let {θ n } n≥1 be a sequence of localization times for M α ; that is, θ n ↑ ∞ a.s. as n → ∞ and M α ∧ θ n is a martingale for all n ≥ 1.Then, for all n ≥ 1, u t, x E x e α∧τ D ∧θn 0 c t−r,X r dr u t − α ∧ τ D ∧ θ n , X α ∧ τ D ∧ θ n −r,X r dr f t − s, X s ds .
n r ,X n r dr f ξ n s , X n s ds − τ 0 e s 0 c ξ r ,X r dr f ξ s , X s ds .4.8The sequence {Y n } n≥N 1 is uniformly integrable.Thanks to Theorem 4.2, in Chapter 5 of 33 , it is sufficient to prove that sup n E Y 2 n < ∞.So,

18 E τ n ∨τ τ n ∧τ e s 0
n r ,X n r dr f ξ n s , X n s − e s 0 c ξ r ,X r dr f ξ s , X s ds dP 4.c ξ n r ,X n r dr f ξ n s , X n s 1 τ n >τ e s 0 c ξ r ,X r dr f ξ s , X s 1 τ n ≤τ ds dP.

21 E 1 ≤ e c 0 s exp s 0 1 ≤ e c 0 s exp s 0 L 1 ≤ 22 ≤
n r ,X n r dr f ξ n s , X n s − f ξ s , X s ds dP4.A |t n − t| λ X n s − X s ds dP ≤ e c 0 t α t α L f B A |t n − t| λ η .r , X n r − c ξ r , X r dr − |c ξ n r , X n r − c ξ r , X r |dr − c B A |t n − t| λ X n r − X r dr − e c 0 s exp L c B A s |t n − t| λ η − 1 ,4.24 since |e x − 1| ≤ e |x| − 1.If we choose N 3 ∈ N such that |t n − t| λ ≤ 1/ 2L c B A t α for all n ≥ N 3 and η ≤ 1/ 2L c B A t α , then we get by the mean value theorem that e s 0 c ξ n r ,X n r dr − e s 0 c ξ r ,X r dr ≤ e c 0 s eL c B A s |t n − t| λ η .K f A ∩ D t e c 0 t α eL c B A t α 2 |t n − t| λ η .4.26
argument as the one made in 4.24 and 4.25 , we get e τn 0 c ξ n r ,X n r dr − e τ 0 c ξ r ,X r dr
r ,X r dr E ξ θ ,X θ e τ 0 c ξ r ,X r dr G ξ τ , X τ E t,x e θ 0 c ξ r ,X r dr v 2 ξ θ , X θ .4.95Combining equations 4.90 and 4.95 we prove that 4.80 holds.So due to equations 4.79 and 4.80 we have that v w.Since w ∈ C 1,2,λ T 0 , T 1 × A see Theorem 5.5 below and T 0 T 1 and A are arbitrary we get thatv ∈ C 1,2 0, ∞ × R d ∩ C 1,2,λ loc 0, ∞ × Rd and the proof is complete.We are ready to proof the Main Theorem Proof of Theorem 3.1.The proof follows from Theorems 4.1 and 4.4 and Lemmas 3.3 and 3.4.

Theorem 5 . 1 . 1 2
Let {Z t } t≥0 be a stochastic process with continuous paths a.s. and A ⊂ R d an open, connected set with regular boundary.Letτ : inf t > 0 | Z t / ∈ A .5.Assume that P τ < ∞ | Z 0 z 1 and P τ τ | Z 0 z 1 for all z ∈ A, whereτ : inf {t > 0 | Z t / ∈ A}. 5.For a > 0, defineA a : x ∈ R d | d x, ∂A < a , A a : A ∪ A a , A a− : A \ A a 5.3and the corresponding exit timesτ a : inf t > 0 | Z t / ∈ A a , τ a− : inf t > 0 | Z t / ∈ A a− .
0, T × R d such that lim − ∂ ∂t L ϕ t, x − λϕ t, x < ∞. 1.4 It is also assumed that D has a C 2 boundary.Schauder-type estimates were obtained for the gradient of the solution in terms of the data.Bertoldi and Fornaro in 19 obtained analogous results for the Cauchy-Neumann problem for an unbounded convex domain.Later, in 20 Bertoldi et al. generalized the method to nonconvex sets with C 2 boundary.They studied the existence, uniqueness, and gradient estimates for the Cauchy-Neumann problem.For a survey of this results, see 21 .
Observe that the local ellipticity is only assumed on 0, ∞ × D. This condition is used to prove the existence of a classical solution to 1.2 and so is only needed in that set.The local Lipschitz condition and the linear growth are assumed on R × R d to ensure the existence of a strong solution to 2.3 for s ∈ 0, ∞ .
, where t n , x n ∈ 0, ∞ × D and t, x ∈ ∂ 0, ∞ × D , that is, either t 0 or x ∈ ∂D.In both cases, we get that τ 0 a.s., We need to prove that|v 2 t n , x n − G t, x | −→ ,X n dr −1| ≤ e c 0 tα 1, we have that 4.71 is uniformly integrable and repeating the same argument made with 4.48 , we can prove that n The last equality follows from a general form of the strong Markov property see Theorem 4.18 in Chapter 5 of 29 Theorem 4.6 in Chapter 5 of 27 , or Theorem 5.1 in Chapter V of 26 that states that if