About Fokker-Planck equation with measurable coefficients and applications to the fast diffusion equation

The object of this paper is the uniqueness for a $d$-dimensional Fokker-Planck type equation with non-homogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic representation of the so called Barenblatt solution of the fast diffusion equation which is the partial differential equation $\partial_t u = \partial^2_{xx} u^m$ with $m\in(0,1)$. Together with the mentioned Fokker-Planck equation, we make use of small time density estimates uniformly with respect to the initial condition


Introduction
The present paper is divided into three parts.i) A uniqueness result on a Fokker-Planck type equation with measurable non-negative (possibly degenerated) multidimensional unbounded coefficients.
ii) An application to the probabilistic representation of a fast diffusion equation.
iii) Some small time density estimates uniformly with respect to the initial condition.
In the whole paper T > 0 will stand for a fixed final time.In a one dimension space, the Fokker-Planck equation is of the type where a, b : [0, T ] × R → R are measurable locally bounded coefficients and µ is a finite real Borel measure.The Fokker-Planck equation for measures is a widely studied subject in the literature whether in finite or infinite dimension.Recent work in the case of time-dependent coefficients with some minimal regularity was done by [9,16,30] in the case d ≥ 1.In infinite dimension some interesting work was produced by [8].
In this paper we concentrate on the case of measurable (possibly) degenerate coefficients.Our interest is devoted to the irregularity of the diffusion coefficient, so we will set b = 0.A first result in that direction was produced in [7] where a was bounded, possibly degenerated, and the difference of two solutions was supposed to be in L 2 ([κ, T ] × R), for every κ > 0 (ASSUMPTION (A)).This result was applied to study the probabilistic representation of a porous media type equation with irregular coefficients.We will later come back to this point.We remark that it is not possible to obtain uniqueness without ASSUMPTION (A).In particular [7,Remark 3.11] provides two measure-valued solutions when a is time-homogeneous, continuous, with 1 a integrable in a neighborhood of zero.
One natural question is about what happens when a is not bounded and x ∈ R d .
A partial answer to this question is given in Theorem 3.1 which is probably the most important result of the paper; it is a generalization of [7,Theorem 3.8] where the inhomogeneous function a was bounded.Theorem 3.1 handles the multidimensional case and it allows a to be unbounded.
An application of Theorem 3.1 concerns the parabolic problem: where δ 0 is the Dirac measure at zero and u m denotes u|u| m−1 .It is well known that, for m > 1, there exists an exact solution to (1.2), the so-called Barenblatt's density, see [3].Its explicit formula is recalled for instance in [34,Chapter 4] and more precisely in [4,Section 6.1].Equation (1.2) is the classical porous medium equation.
In this paper, we focus on (1.2) when m ∈]0, 1[: the fast diffusion equation.In fact, an analogous Barenblatt type solution also exists in this case, see [34,Chapter 4] and references therein; it is given by the expression where 1−m dx. (1.4) 2) is a particular case of the so-called generalized porous media type equation xx β (u(t, x)) , t ∈]0, T ], u(0, x) = u 0 (dx), x ∈ R, (1.5) where β : R → R is a monotone non-decreasing function such that β(0) = 0 and u 0 is a finite measure.When β(u) = u m , m ∈]0, 1[ and u 0 = δ 0 , two difficulties arise: first, the coefficient β is of singular type since it is not locally Lipschitz, second, the initial condition is a measure.Another type of singular coefficient is β(u) = H(u − u c )u, where H is a Heaviside function and u c > 0 is some critical value, see e.g.[2].Problem (1.2) with m ∈]0, 1[ was studied by several authors.For a bounded integrable function as initial condition, the equation in (1.2) is well-stated in the sense of distributions, as a by product of the classical papers [10,6] on (1.5) with general monotonous coefficient β.
Electron.J. Probab.0 (2012), no.0, 1-28.ejp.ejpecp.org When the initial data is locally integrable, existence was proved by [19].[11] extended the validity of this result when u 0 is a finite Radon measure in a bounded domain, [29] established existence when u 0 is a locally finite measure in the whole space.The Barenblatt's solution is an extended continuous solution as defined in [13,14]; [14,Theorem 5.2] showed uniqueness in that class.[23,Theorem 3.6] showed existence in a bounded domain of solutions to the fast diffusion equation perturbed by a right-hand side source term, being a general finite and positive Borel measure.As far as we know, there is no uniqueness argument in the literature whenever the initial condition is a finite measure in the general sense of distributions.Among recent contributions, [15] investigated the large time behavior of solutions to (1.2).
The present paper provides the probabilistic representation of the (Barenblatt's) solution of (1.2) and exploits this fact in order to approach it via a Monte Carlo simulation with an L 2 error around 10 −3 .We make use of the probabilistic procedure developed in [4, Section 4] and we compare it to the exact form of the solution U of (1.2) which is given by the explicit formulae (1.3)- (1.4).The target of [4] was the case β(u) = H(u − u c )u; in that paper those techniques were compared with a deterministic numerical analysis recently developed in [12] which was very performing in that target case.At this stage, the implementation of the same deterministic method for the fast diffusion equation does not give satisfying results; this constitutes a further justification for the probabilistic representation.
We define where W is a Brownian motion on some suitable filtered probability space (Ω, F, (F t ) t≥0 , P ).
To the best of our knowledge, the first author who considered a probabilistic representation of a solution of (1.5) was H. P. Jr. McKean ([26]), particularly in relation with the so-called propagation of chaos.In his case β was smooth, but the equation also included a first order coefficient.From then on, literature steadily grew and nowadays there is a vast amount of contributions to the subject, especially when the nonlinearity is in the first order part, as e.g. in Burgers' equation.We refer the reader to the excellent survey papers [33] and [18].A probabilistic interpretation of (1.5) when β(u) = u.|u|m−1 , m > 1, was provided for instance in [5].Recent developments related to chaos propagation when β(u) = u 2 and β(u) = u m , m > 1 were proposed in [28] and [17].The probabilistic representation in the case of possibly discontinuous β was treated in [7] when β is non-degenerate and in [2] when β is degenerate; the latter case includes the case β(u) = H(u − u c )u.
As a preamble to the probabilistic representation we make a simple, yet crucial observation.Let W be a standard Brownian motion.Proposition 1.1.Let β : R → R such that β(u) = Φ 2 (u).u,Φ : R → R + and u 0 be a probability real measure.
Let Y be a solution to the problem (1.7) Electron.J. Probab.0 (2012), no.0, 1-28.ejp.ejpecp.org Proof of the above result is based on the following lemma.
Consider the function t → ρ(t, •) from [0, T ] to the space of finite real measures M(R), defined as ρ(t, •) being the law of Y t .Then ρ is a solution, in the sense of distributions Proof of Lemma 1.2.This is a classical result, see for instance [32,Chapter 4].The proof is based on an application of Itô's formula to ϕ(Y t ), ϕ ∈ S(R).
Proof of Proposition 1.1.We set a(s, y) = Φ 2 (u(s, y)).We apply Lemma 1.2 setting ρ(t, dy) = u(t, y)dy, t ∈]0, T ], and ρ(0, •) = u 0 .When u 0 is the Dirac measure at zero and β(u) = u m , with m ∈] 3  5 , 1[, Theorem 5.7 states the converse of Proposition 1.1, providing a process Y which is the unique (weak) solution of (1.6).The first step consists in reducing the proof of that Theorem to the proof of Proposition 5.3 where the Dirac measure, as initial condition of (1.2), is replaced by the function U(κ, •), 0 < κ ≤ T .This corresponds to the shifted Barenblatt's solution along a time κ, which will be denoted by U. Also, in this case Proposition 5.3 provides an unique strong solution of the corresponding non-linear SDE.That reduction is possible through a weak convergence argument of the solutions given by Proposition 5.3 when κ → 0. The idea of the proof of Proposition 5.3 is the following.Let W be a standard Brownian motion and Y 0 be a r.v.distributed as U(κ, •) admits a unique strong solution.The marginal laws of (Y t ) and U can be shown to be both solutions to (1.8) for a(s, y) = (U(s, y)) m−1 ; that a will be denoted in the sequel by ā.The leading argument of the proof is carried by Theorem 3.1 which states uniqueness for measure valued solutions of the Fokker-Planck type PDE (1.8) under some Hypothesis(B).More precisely, to conclude that the marginal laws of (Y t ) and U coincide via Theorem 3.1, we show that they both verify the so-called Hypothesis(B2).In order to prove that for U, we will make use of Lemma 4.2.The verification of Hypothesis(B2) for the marginal laws of Y is more involved.It makes use of a small time (uniformly with respect to the initial condition) upper bound for the density of an inhomogeneous diffusion flow with linear growth (unbounded) smooth coefficients, even though the diffusion term is non-degenerate and all the derivatives are bounded.This is the object of Proposition 5.1, the proof of which is based on an application of Malliavin calculus.In our opinion this result alone is of interest as we were not able to find it in the literature.When the paper was practically finished we discovered an interesting recent result of M. Pierre, presented in [20,Chapter 6], obtained independently.This result holds in dimension 1 when the coefficients are locally bounded, non-degenerate and the initial condition has a first moment.In this case, the hypothesis of type (B) is not needed.
In particular it allows one to establish Proposition 5.3, but not Theorem 5.7 where the coefficients are not locally bounded on [0, T ] × R.
Electron.J. Probab.0 (2012), no.0, 1-28. ejp.ejpecp.org The paper is organized as follows.Section 2 is devoted to basic notations.Section 3 is concentrated on Theorem 3.1 which concerns uniqueness for the deterministic, time inhomogeneous, Fokker-Planck type equation.Section 4 presents some properties of the Barenblatt's solution U to (1.2).The probabilistic representation of U is treated in Section 5. Proposition 5.1 performs small time density estimates for timeinhomogeneous diffusions, the proof of which is located in the Appendix.Finally, Section 6 is devoted to numerical experiments.

Preliminaries
We start with some basic analytical framework.In the whole paper d will be a strictly positive integer.If f : R d → R is a bounded function we will denote f ∞ = sup By S(R d ) we denote the space of rapidly decreasing infinitely differentiable functions ϕ : R d → R, by S (R d ) its dual (the space of tempered distributions).We denote by M(R d ) the set of finite Borel measures on R d .If x ∈ R d , |x| will denote the usual Euclidean norm.
For ε > 0, let K ε be the Green's function of ε − ∆, that is the kernel of the operator In particular, for all ϕ ∈ L 2 (R d ), we have For more information about the corresponding analysis, the reader can consult [31].If ϕ ∈ C 2 (R d ) S (R d ), then (ε−∆)ϕ coincides with the classical associated PDE operator evaluated at ϕ. Definition 2.1.We will say that a function ψ : Definition 2.2.We will say that a function ψ : [0, T ] × R → R has linear growth (with respect to the second variable) if there is a constant with initial condition z(0, •) = z 0 if, for every t ∈ [0, T ] and φ ∈ S(R), we have (2.2)

Uniqueness for the Fokker-Planck equation
We now state the main result of the paper which concerns uniqueness for the Fokker-Planck type equation with measurable, time-dependent, (possibly degenerated and unbounded) coefficients.It generalizes [7,Theorem 3.8] where the coefficients were bounded and one-dimensional.
The theorem below holds with two classes of hypotheses: (B1), operating in the multidimensional case, and (B2), more specifically in the one-dimensional case.Theorem 3.1.Let a be a Borel nonnegative function on [0, T ] × R d .Let z i : [0, T ] → M(R d ), i = 1, 2, be continuous with respect to the weak topology on finite measures on M(R d ).Let z 0 be an element of M(R d ).Suppose that both z 1 and z 2 solve the problem ∂ t z = ∆(az) in the sense of distributions with initial condition z(0, •) = z 0 .
Then z := (z 1 −z 2 )(t, •) is identically zero for every t under the following requirement.
(B2) We suppose d = 1.For every t 0 > 0, we have 1.If a is bounded then the first item of Hypothesis(B1) implies the second one.
1.If a is non-degenerate, the third assumption of Hypothesis(B2) implies the first one.
Proof of Theorem 3.1.Let z 1 , z 2 be two solutions of (2.2); we set z := z 1 − z 2 .We evaluate, for every t ∈ [0, T ], the quantity Similarly to the first part of the proof of [7, Theorem 3.8], assuming we can show that we are able to prove that z(t) ≡ 0 for all t ∈]0, T ].We explain this fact.Electron.J. Probab.0 (2012), no.0, 1-28. ejp.ejpecp.org Since the two terms of the above sum are non-negative, if (3.1) holds, then •) → 0, in the sense of distributions, as ε goes to zero.Therefore z ≡ 0.
(iii) The Dirac measure δ 0 is the initial trace of U, in the sense that for every γ : R → R, continuous and bounded.
Proof of Proposition 4.1.(i) This is a well known fact which can be established by inspection.
(ii) For M ≥ 1, we consider a sequence of smooth functions (ϕ M ), such that Letting M → +∞, by Lebesgue's dominated convergence theorem, the left-hand side of (4.3) converges to R U(t, x)dx.The integral on the right-hand side of (4.3) is bounded by The last integral on the right is finite as m 1 − m > 0, for every m ∈]0, 1[.Therefore the integral in the right-hand side of (4.3) goes to zero as M → +∞.This concludes the proof of the second item of Proposition 4.1.
Note that the second item of Proposition 4.1 determines the explicit expression of the constant D. (i) Suppose that 1 3 < m < 1 .Then there is p ≥ 2 and a constant C p (depending on T ) such that for 0 (ii) In particular, taking p = 2 in (4.4), we get again when m belongs to ] 1 3 , 1[.
Proof of Lemma 4.2.
(ii) is a particular case of (i) and (iii) follows by similar arguments as for the proof of (i).
In particular we have   ejp.ejpecp.org

The probabilistic representation of the fast diffusion equation
We are now interested in a non-linear stochastic differential equation rendering the probabilistic representation related to (1.2) and given by (1.6).Suppose for a moment that Y 0 is a random variable distributed according to δ 0 , so Y 0 = 0 a.s.We recall that, if there exists a process Y being a solution in law of (1.6), then Proposition 1.1 implies that u solves (1.2) in the sense of distributions.
In this subsection we shall prove existence and uniqueness of solutions in law for (1.6).In this respect we first state a tool, given by Proposition 5.1 below, concerning the existence of an upper bound for the marginal law densities of the solution Y of an inhomogeneous SDE with unbounded coefficients.This result has an independent interest. (5.1) Then, for every s > 0, the law of Y s admits a density denoted p s (x 0 , •).
Moreover, we have where K is a constant which depends on σ ∞ , b ∞ and T but not on x 0 .3. If σ and b have polynomial growth and are time-homogeneous, various estimates are given in [25].However the behavior is of type O(t Let Y κ be a random variable distributed according to u 0,κ .We are interested in the following result.Proposition 5.3.Assume that m ∈] 3  5 , 1[.Let B be a classical Brownian motion independent of Y κ .Then there exists a unique (strong In particular pathwise uniqueness holds.
Proof of Proposition 5.3.Let W be a classical Brownian motion on some filtered probability space.Given the function U, defined in (4.10), we construct below a unique process Y strong solution of (5.6) In fact, Φ(U) is continuous, smooth with respect to the space parameter and all the space derivatives of order greater or equal than one are bounded; in particular Φ(U) is Lipschitz and it has linear growth.Therefore (5.5) admits a strong solution.
To conclude it remains to prove that U(t, y)dy is the law of Y t , ∀t ∈ [0, T −κ]; in particular the law of the r.v.Y t admits a density.For this we will apply Theorem 3.1 for which we need to check the validity of Hypothesis(B2) when a = ā and for z := z 1 − z 2 , where z 1 := ρ and z 2 := U.By additivity this will be of course fulfilled if we prove it separately for z := ρ and z := U, which are both solutions to (5.7).
Lemma 5.5.Let ψ : [0, T ] × R → R + , continuous (not necessarily bounded) such that ψ(t, •) is smooth with bounded derivatives of orders greater or equal than one.We also suppose ψ to be non-degenerate.
If X 0 = x 0 , where x 0 is a real number, then Proposition 5.1 implies that, for every t ∈]0, T ], the law of X t admits a density p t (x 0 , •).Consequently, if the law of X 0 is u 0,κ (x)dx, for every t ∈]0, T ], the law of X t has a density given by ν(t, x) = R u 0,κ (x 0 )p t (x 0 , x)dx 0 . (5.9) Using (5.9) we get (5.10) the latter inequality is valid because of (4.7) in Lemma 4.2.In the sequel of the proof, the constants K 2 , K 3 , K 4 will only depend on t 0 , T and ψ.Furthermore Since ψ has linear growth, this expression is bounded by (5.11) (5.11) follows because of (5.10).Besides, by Burkholder-Davis-Gundy and Jensen's inequalities, taking into account the linear growth of ψ, it follows that Then, by Gronwall's lemma, there is another constant K 4 such that (5.12) Finally (5.11), (5.12) and (5.10) allow us to conclude the proof.
We are now ready to provide the probabilistic representation related to function U which in fact is only a solution in law of (1.6).
Definition 5.6.We say that (1.6) admits a weak (in law) solution if there is a probability space (Ω, F, P), a Brownian motion (W t ) t≥0 and a process (Y t ) t≥0 such that the system (1.6) holds.(1.6) admits uniqueness in law if, given (W 1 , Y 1 ), (W 2 , Y 2 ) solving (1.6) on some related probability space, it follows that Y 1 and Y 2 have the same law.Proof of Theorem 5.7.First we start with the existence of a weak solution for (1.6).
Let U be again the (Barenblatt's) solution of (1.2).We consider the solution (Y κ t ) t∈[κ,T ] provided by Corollary 5.4 extended to [0, κ], setting Y κ t = Y κ , t ∈ [0, κ].We prove that the laws of processes Y κ are tight.For this we implement the classical Kolmogorov's criterion, see [22,Section 2.4,Problem 4.11 ].We will show the existence of p > where C p will stand for a constant (not always the same), depending on p and T but not on κ.Let s, t ∈]0, T ].Let p > 2. By Burkholder-Davis-Gundy inequality we obtain Then, using Jensen's inequality and the fact that U(r, •) is the law density of Y κ r , r ≥ κ,  Consequently there is a subsequence Y n := Y κn converging in law (as C([0, T ])−valued random elements) to some process Y .Let P n be the corresponding laws on the canonical space Ω = C([0, T ]) equipped with the Borel σ-field.Y will denote the canonical process Y t (ω) = ω(t).Let P be the weak limit of (P n ).
1) We first observe that the marginal laws of Y under P n converge to the marginal law of Y under P .Let t ∈]0, T ].If the sequence (κ n ) is lower than t, then the law of Y t under P n equals the constant law U(t, x)dx.Consequently, for every t ∈]0, T ], the law of Y t under P is U(t, x)dx.
2) We now prove that Y is a (weak) solution of (1.6), under P .By similar arguments as for the classical stochastic differential equations, see [32,Chapter 6], it is enough to prove that Y (under P ) fulfills the martingale problem i.e., for every f ∈ C 2 b (R), the process is an (F s )-martingale, where (F s ) is the canonical filtration associated with Y .C 2 b (R) stands for the set {f ∈ C 2 (R)|f, f , f bounded }.Let E (resp.E n ) be the expectation operator with respect to P (resp.P n ).Let s, t ∈ [0, T ] with s < t and R = R(Y r , r ≤ s) be an F s −measurable, bounded and continuous (on C([0, T ])) random variable.In order to show the martingale property (MP) of Y , we have to prove that (5.15) Electron.J. Probab.0 (2012), no.0, 1-28.

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We first consider the case when s > 0. There is n ≥ n 0 , such that κ n < s.Let f ∈ C 2 b (R); since (Y s ) s≥κn , under P n , are still martingales we have (5. 16) We are able to prove that (5.15) follows from (5.16).Let ε > 0 and N > 0 such that U m (r, y)dy ≤ ε, (5.17 where C is the linear growth constant of Φ 2 • U in the sense of Definition 2.2.In order to conclude, passing to the limit in (5.16), we will only have to show that where F ( ) but not bounded.The left-hand side of (5.18) equals where Since κ n < s, for N large enough, we get taking into account (5.17) and the second item of Lemma 4.2.For fixed N , chosen in (5.17), we have lim n→+∞ E 2 (n, N ) = 0, since F N is bounded and continuous.Again, since the law density under P of Y t , t ≥ s , is U(t, •), similarly as for (5.20), we obtain since ε > 0 is arbitrary, (5.18) is established.So (5.15) is verified for s > 0.
3) Now, we consider the case when s = 0. We first prove that using Lebesgue's dominated convergence theorem and (5.21).Consequently we obtain (5.22) It remains to show that Y 0 = 0 a.s.This follows because Y t → Y 0 a.s., and also in law (to δ 0 ), by the third item of Proposition 4.1.Finally we have shown that the limiting process Y verifies (MP), which proves the existence of solutions to (1.6).4) We now prove uniqueness.Since U is fixed, only uniqueness for the first line of equation (1.6) has to be established.Let (Y i t ) t∈[0,T ] , i = 1, 2, be two solutions.In order to show that the laws of Y 1 and Y 2 are identical, according to [21, Lemma 2.5], we will verify that their finite marginal distributions are the same.For this we consider 0 = t 0 < t 1 < . . .< t N = T .Let 0 < κ < t 1 .Obviously we have Y i t 0 = 0 a.s., in the corresponding probability space, ∀i ∈ {1, 2}.

Numerical experiments
In order to avoid singularity problems due to the initial condition being a Dirac delta function, we will consider a time translation of U, denoted v, and defined by v still solves equation (1.2), for m ∈]0, 1[, but with now a smooth initial data given by Indeed, we have the following formula where α, k and D are still given by (1.4).
We now wish to compare the exact solution of problem (1.2) to a numerical probabilistic solution.In fact, in order to perform such approximated solutions, we use the algorithm described in Sections 4 of [4] (implemented in Matlab).We focus on the case   We start with some notations for the Malliavin calculus.The set D ∞ represents the classical Sobolev-Malliavin space of smooth test random variables.D 1,2 is defined in the lines after [27, Lemma 1.2.2] and L 1,2 is introduced in [27, Definition 1.3.2].See also [24] for a complete monograph on Malliavin calculus.We state a preliminary result.Proposition 7.1.Let N be a non-negative random variable.Suppose, for every p ≥ 1, the existence of constants C(p) and 0 (p) such that  x −p dF (x).
(7.1) implies that I 1 and I 2 are well-defined.Indeed, on one hand, applying integration by parts on I 1 , we get moreover, there is a constant C(p) such that On the other hand, again (7.1) says that Consequently, using (7.4) and (7.5) and coming back to (7.3), (7.2) is established.
Proof of Proposition 5.1.In this proof σ (resp.b ) stands for , be the solution of (5.1).According to [27, Theorem 2. [27,Theorem 2.3.1], the law of Y s admits a density that we denote by p s (x 0 , •).
The second step consists in a re-scaling, transforming the time s into a noise multiplicative parameter λ; ), where In particular, Y s ∼ Y λ 1 .By previous arguments, for every t > 0, Y λ t ∈ D ∞ and its law admits a density denoted by p λ t (x 0 , •).Our aim consists in showing the existence of a constant K such that where K is a constant which does not depend on x 0 and λ.In fact, we will prove that, for every λ ∈]0, We set In fact, we will have attained (7.7), if we show We express the equation fulfilled by Z; it yields where, for every (r, z) ∈ [0, 1] × R, we set σ λ (r, z) = σ(rλ 2 , λz + x 0 ), and b λ (r, z) = λb(rλ 2 , λz + x 0 ).
At this stage we state the following lemma.
1.For simplicity, in the whole proof of Proposition 5.1, we will set T = 1.
2. Since there is no more ambiguity, we will use again the letter s in the considered integrals.
Then, using Jensen's inequality, we get Then, coming back to (7.18) and using (7.23), we obtain where . Finally, using Proposition 7.1, the result follows.
We go on with the proof of Proposition 5.1 taking into account the considerations before Lemma 7.4.In fact [27, Proposition 2.1.1]allows us to express, for fixed t ∈]0, 1], using Cauchy-Schwarz inequality, it implies that (7.26) According to (1.48) in [27], (7.26) implies Now we state a result that estimates the two terms in the right-hand side of (7.27).Indeed, we have the following.
In fact, we have On the other hand, we get Therefore, coming back to (7.30) and using (7.31), we obtain that Electron.J. Probab.0 (2012), no.0, 1-28.ejp.ejpecp.orgwhere with D s1 G den given in (7.32).In the sequel we will enumerate constants K 1 to K 20 ; all those will not depend on x 0 or t, but eventually on T , σ and b.We start estimating J 1 .Since σ is bounded, by Cauchy-Schwarz inequality, we have .
Since σ has linear growth, Lemma 7.2 and a further use of Cauchy-Schwarz inequality imply that, J 1 is bounded by .
Therefore, by Proposition 7.5 and Lemma 7.4, we obtain that (7.34) We go on with the analysis of J 2 .Since σ , σ are bounded, we have Returning to the proof of Proposition 5.1 and substituting in (7.27) the right-hand side of the first and the second item of Proposition 7.6, the inequality (5.2) is verified.Finally this concludes the proof of Proposition 5.1.Electron.J. Probab.0 (2012), no.0, 1-28. ejp.ejpecp.org

Definition 2 . 3 .
Let a : [0, T ] × R d → R + be a Borel function, z 0 ∈ M(R d ).A (weakly measurable) function z : [0, T ] → M(R d ) is said to be a solution in the sense of distributions of ∂ t z = ∆(az)
Hypothesis(B2) is always verified by Remark 3.2; the first item of Hypothesis(B2) implies the third one.So Theorem 3.1 is a strict generalization of [7, Theorem 3.8].4. Let (z(t, •), t ∈ [0, T ]) be the marginal law densities of a stochastic process Y solving

Remark 5. 2 . 1 . 2 .
The proof of Proposition 5.1 above is given in Appendix 7.1.If σ and b is bounded, the classical Aronson's estimates implies that (5.2) holds even without the |x 0 | 4 multiplicative term.If σ and b are unbounded,[1] provides an adaptation of Aronson's estimates; unfortunately they first considered timehomogeneous coefficients, and also their result does not imply (5.2).
where C 1 and C 2 depend on T , σ ∞ and b ∞ , but not on x 0 .whereG den =< DZ t , DZ t > .Since σ has linear growth, by Lemma 7.2 and using again Cauchy-Schwarz inequality, there is a constant C 1 such that