A BPE model for the Burgers’ equation

Shigeyoshi OGAWA^{∗}and Arturo KOHATSU-HIGA^{†}

August 12, 2003

Abstract We study the BPE (Brownian particle equation) model of the Burgers equation presented in the preceeding article [6]. More precisely, we are interested in establishing the existence and uniqueness properties of solutions using probabilistic techniques.

1 BPE Model for the Burgers equation

In this article, we are concerned with the stochastic representation for the Burgers equation. For this purpose we presented in the preceeding article [6] the two BPE models, one of which is as follows;

\begin{matrix}∂_{t}u(t, x) + ϵẆ ∂_{x}u(t, x) = \frac{1}{ϵ}f(\overline{u}(t, x))Ẇ , f(x) = x^{2} \\ u(0, x) = u_{0}(x). \end{matrix}

Here t ∈ [0, T], x ∈ IR, \overline{u}(t, x) = E(u(t, x)) and Ẇ denotes the white noise process. We already know by [6] (see Theorem 2.1) that for f regular (smooth with bounded derivatives) this problem has one and only one solution. This requirement is needed so that one can use classical results in the theory of PDEs.

Here, we would like to establish the result for f(x) = x^{2} via a probabilistic method so as to make the discussion self- contained and this is the principal objective of the present note. More precisely, we are to show that the solution can be constructed through a system of stochastic integral equations that can be obtained by applying the method of stochastic characteristics to the Cauchy problem (1). We will in general present a heuristic solution for the following equation

\begin{matrix}∂_{t}u(t, x) + (a + σẆ)∂_{x}u(t, x) = νf(\overline{u}(t, x))Ẇ , f(x) = x^{2} \\ u(0, x) = u_{0}(x). \end{matrix}  (1)

For two smooth coefficients a and σ. In particular the above example will follow from using a = 0, σ = ε and ν = 1/ε.

A different stochastic representation has already been obtained by Bossy and Talay [1], [2].

∗Kanazawa University, Faculty of Technology, s ogawa@t.kanazawa-u.ac.jp

†Universitat Pompeu Fabra, Department of Economics, arturo.kohatsu@econ.upf.es.

2 Preliminaries – Noncausal Stochastic Calculus

Since the stochastic integral of noncausal type, introduced by S.Ogawa in 1979 [11], plays an essential role in the BPE theory, we will briefly introduce it following [7] so that the discussion is self-contained. For more details of the noncausal calculus we refer the reader to [7], [9].

2.1 Causal functions and B-differentiability

In Itô’s theory, the stochastic integral, say with respect to the Brownian motion

{W_{t}(ω); t ∈ [0, T]}, denoted by ∫ f(t, ω)d^{+}W_{t}, is defined only for integrands f(t, ω)

that are causal (or non anticipative) with respect to the history of the Brownian motion. Namely, f(t, ω) is supposed to be adapted (sometimes also called “causal” ) to the filtration {F_{t}, t ≥ 0} where the F_{t} = σ{W_{s}; 0 ≤ s ≤ t}. In many situations one meets problems of noncausal character, therefore there is a need for another theory of stochastic calculus which is free from the restriction of causality. The noncausal calculus is one of such theories.

In what follows, we will fix the probability space (Ω, F, P) on which is defined the IR- valued Brownian motion. We denote by H the totality of all random functions f(t, ω), measurable in (t, ω) with respect to the field B(IR_{+}) ⊗ F, such that

P {∫_{0}^{T} |f(t, ω)|^{2}dt < ∞} = 1, and by M_{p}^{+} is the subset of H containing all L^{p}-

causal random functions, that is, the set of functions f(t, ω) such that they are;

(M.1) measurable in (t, ω) with respect to the field B(IR_{+}) ⊗ F, and especially

(M.2) adapted to the family of σ-fields {F_{t}}, where F_{t} = σ{W_{s}; 0 ≤ s ≤ t},

(M.3) and has finite moments of order p, E(∫_{0}^{T} |f(t, ω)|^{p}dt) < ∞.

An H-class random function g(t, ω) is said to be differentiable in L^{p}(Ω) with respect to the Brownian motion W_{t} (or B_{p}^{+}- differentiable) provided that sup_{t} E|g(t)|^{p} < ∞ and there exists an M_{p}^{+}-class random function say ĝ(t, ω) such that, for any small enough h > 0,

sup_{t,s,|t−s|<h}E|g(t, ω) − g(s, ω) − ∫_{s}^{t}ĝ(r, ω)d^{+}W_{r}|^{p} = o(h^{p/2})

where the integral ∫ ·d^{+}W stands for Itô’s stochastic integral. The function ĝ(r, ω)

is called the B_{p}^{+}- derivative^{1} of g and denoted by \frac{∂^{+}g}{∂W (r)}. It is not difficult to see that if the function g(t, ω) is B_{p}^{+}-differentiable then its B_{p}^{+}-derivative is uniquely determined as soon as its derivative is in L^{2}([0, T] × Ω)(see [12]) and that it satisfies a chain rule when g is adapted. That is, let f ∈ C^{2} with bounded second derivative and g be B_{p}^{+} differentiable for p ≥ 4, then the B_{p/2}^{+}-derivative of f(g) exists and we have

1The notion of the B_{p}^{+}-derivative is different from the ”stochastic derivative” which was introduced by A.Skorokhod

\frac{∂^{+}f(g)}{∂W (r)} = f′(g(r))\frac{∂^{+}g}{∂W (r)}.

The proof of this assertion is not difficult to obtain if one uses Ito’s formula appropiately. The B_{p}^{+}-differentiability of the random function with respect to the multi-dimensional Brownian motion is defined in a similar way. Also one can define similarly the processes adapted to the filtration of the backward Wiener process, which we will denote by M_{p}^{−}. The backward stochastic integral will be denoted by ∫ ·d^{−}W_{s}. Then one defines the B_{p}^{−}-derivative with respect to backward integration and in such a case one denotes it by \frac{∂^{−}g}{∂W (r)}

Note: The notations \frac{∂^{+}}{∂W_{t}} , \frac{∂^{−}}{∂W_{t}} were first introduced by S. Ogawa in ([6],[7]).

But the definitions given in those papers do not coincide with the ones given here.

2.2 Noncausal stochastic integral

Given a random function f(t, ω) ∈ H and an arbitrary complete orthonormal system {ϕ_{n}} in L^{2}([0, T]), we consider the formal random series

∑_{n}^{∞} ∫_{0}^{T} f(t, ω)ϕ_{n}(t)dt × ∫_{0}^{T} ϕ_{n}(t)dW_{t}.

The stochastic integral of noncausal type was introduced by S. Ogawa in 1979 ([11]), in the following form:

Definition 1 A random function f(t, ω) ∈ H is said to be integrable with respect to the basis {ϕ_{n}} (or ϕ-integrable) when the random series above converges in probability

and the sum, denoted by ∫_{0}^{T} f(t, ω)d_{ϕ}W_{t}, is called the stochastic integral of noncausal

type with respect to the basis {ϕ_{n}}.

Obviously when f ∈ M_{p}^{+} for some p > 0, then the above integral and the stochastic Ito integral coincide.

2.3 Relation between symmetric and noncausal integrals

We call a random function, f(t, ω), a semimartingale, when it admits the decompo-

sition f(t, ω) = a(t, ω) + ∫_{0}^{t} f̂(s)d^{+}W_{s} where f̂ ∈ L^{p}([0, T] × Ω) for some p ≥ 2 and

a(·) is such that almost every sample path is of bounded variation in t over [0, T].

Notice that if sup_{t∈[0,T]} E|a(t)|^{p} < ∞ and sup_{t,s|t−s|<h}E|a(t) − a(s)|^{p} = o(h^{p/2}) then f is

B_{p}^{+}-differentiable. Note that a does not need to be adapted to the filtration in order to have that f is B_{p}^{+} -differentiable and \frac{∂^{+}f}{∂W (t)} = f̂(t).

Next we give some basic results about the relation between the symmetric integrals and the noncausal integral. Let H denote the Haar basis.

Theorem 2 ([10]) Every causal B_{p}^{+}- differentiable function (for some p > 0) is integrable in noncausal sense with respect to the system of Haar functions and the integral coincides with that of the symmetric integral of the same function. That is,

∫_{0}^{T} f(s)d_{H}W_{s} = ∫_{0}^{T} f(s)d^{+}W_{s} + \frac{1}{2} ∫_{0}^{T} \frac{∂^{+}f}{∂W (s)}ds.

A similar relationship is satisfied if f is adapted with respect to the backward filtration except that the stochastic integral becomes the backward stochastic integral which we denote by ∫ d^{−}W and the Lebesgue integral term has a negative sign in front. The advantage of the above representation is that the integral on the left side of the above equation is always well defined and that we can use it in various situations without having to explain which integral we are using. One can generalize the above integrals to a broader class of basis. We say that a c.o.n.s. {ϕ_{n}} is regular provided that it satisfies the next condition:

sup_{n} ∫_{0}^{T} |u_{n}(s)|^{2}ds < ∞, u_{n}(t) = ∑_{k≤n} ϕ_{k}(t) ∫_{0}^{t} ϕ_{k}(s)ds.

Theorem 3 ([10]) Every semimartingale (causal or not) is ϕ-integrable, iff the basis {ϕ_{n}} is regular. In this case, the noncausal integral coincides with the symmetric integral.

From now on we denote the above integral by ∫ ·d_{ϕ}W and we will always assume that the basis {ϕ_{n}} is regular. Furthermore, we also have the following two equalities

∫_{0}^{T} f(s)d_{ϕ}W_{s} = ∫_{0}^{T} f(s)d^{+}W_{s} + \frac{1}{2} ∫_{0}^{T} \frac{∂^{+}f}{∂W (s)}ds, if f ∈ M_{p}^{+} for some p > 0,

∫_{0}^{T} f(s)d_{ϕ}W_{s} = ∫_{0}^{T} f(s)d^{−}W_{s} − \frac{1}{2} ∫_{0}^{T} \frac{∂^{−}f}{∂W (s)}ds, if f ∈ M_{p}^{−} for some p > 0.

3 Description of the stochastic representation

In this section we give the main theorem where the method of stochastic characteristics is applied to the case of linear BPEs. This idea could eventually be used to solve a Cauchy-type stochastic partial differential equation. From now on we denote the essential supremum norm by | · |_{∞}.

Another important ingredient in the description of the stochastic representation is the solution of the stochastic equation

X_{s}^{(t,x)} = x − ∫_{s}^{t} a(u, X_{u}^{(t,x)})du − ∫_{s}^{t} σ(u, X_{u}^{(t,x)})d_{ϕ}W_{u} (0 ≤ s ≤ t ≤ T, x ∈ IR).

The solution X exists for Lipschitz coefficients. If the coefficients are smooth with bounded derivatives then its density function p exists, is smooth and has an upper bound of Gaussian type due to the uniform ellipticity of σ. That is, for s < t ≤ T and any α, β ∈ IN, there exists a positive constant M = M(α, β) such that we have (see [3], page 261 for details)

|\frac{∂^{α+β}p_{t,x}}{∂x^{α}∂y^{β}} (s, y)| ≤ \frac{M exp (−\frac{|x−y|^{2}}{M(t−s)})}{\sqrt{2π}(t − s)^{(α+β+1)/2}} ,

|\frac{∂^{α+β}p_{t,y+x}(s, y)}{∂x^{α}∂y^{β}} | ≤ \frac{M exp (−\frac{|x|^{2}}{M(t−s)})}{\sqrt{2π}(t − s)^{(α+1)/2}}.  (2)

We assume without loss of generality that the constant M is increasing in α and β. Furthermore p satisfies the parabolic PDE

\frac{∂p_{t,x}}{∂t} (s, y) = \frac{σ^{2}(t, x)}{2} \frac{∂^{2}p_{t,x}}{∂x^{2}} (s, y) + b(t, x)\frac{∂p_{t,x}}{∂x} (s, y),  (3)

where b(t, x) = a(t, x) − \frac{1}{2}σσ′(t, x).

Theorem 4 Suppose that the initial data u_{0}(x) satisfies the condition,

u_{0} ∈ C^{2} and that |u_{0}|_{∞} + |u′_{0}|_{∞} + |u′′_{0}|_{∞} < C(0).

Furthermore suppose that a(t, ·), σ(t, ·) ∈ C_{b}^{∞}(IR) uniformly in t ∈ [0, T] and that there exists positive constants c_{0} and c such that for all (t, x) ∈ [0, T]×IR , |σ(t, x)| ≥ c_{0} > 0 and |σ(t, ·)|_{∞} + |σ′(t, ·)|_{∞} < c. Then if cνC(0)M^{3/2}(2, 2) is small enough then the integral equations

u(t, x) = u_{0}(X_{0}^{(t,x)}) + ν∫_{0}^{t}\overline{u}(s, X_{s}^{(t,x)})^{2}d_{ϕ}W_{s}  (4)

have only one global solution u. The average of u, \overline{u}(t, x) = Eu(t, x) solves the following Cauchy problem for the Burgers equation:

\frac{∂\overline{u}}{∂t} + νσ(t, x)\frac{∂\overline{u}^{2}}{∂x} (t, x) = \frac{σ(t, x)^{2}}{2} \frac{∂^{2}\overline{u}}{∂x^{2}}(t, x) + b(t, x)\frac{∂\overline{u}}{∂x}(t, x)  (5)
\overline{u}(0, x) = u_{0}(x).

M is a constant that will depend on the smoothness of the coefficients a and σ as well as in the constant c_{0}. The restriction on the size of cνC(0)M^{3/2}(2, 2) can be explicitly characterized but as it depends on other constants that appear later in the proofs we will state it explicitly in the proof of Lemma 6.

Note that in the above theorems the integral ∫ ·d_{ϕ}W_{s} is the stochastic integral of noncausal type which, applied to the causal (or non anticipating) integrand, coincides with the integrals of symmetric type (i.e. the so called Stratonovich’s integral or the I_{1/2} integral introduced by S.Ogawa ([13])). Suppose that f(\overline{u})(t, x) = f(E(u(t, x))) is twice differentiable in x with bounded second derivative, then we have the next relation, as X_{·}^{(t,x)} ∈ M_{p}^{−}, for any p > 0

∫_{0}^{t}f(\overline{u})(s, X_{s}^{(t,x)})d_{ϕ}W_{s} = ∫_{0}^{t}f(\overline{u})(s, X_{s}^{(t,x)})d^{−}W_{s} − \frac{1}{2} ∫_{0}^{t} \frac{∂^{−}f(\overline{u})(·, X_{·}^{(t,x)})}{∂W (s)}ds.

We easily see from the equations above that the B_{p}^{−}-derivative of the integrand is given by the following (using the chain rule)

\frac{∂^{−}f(\overline{u})(·, X_{·}^{(t,x)} )}{∂W_{s}} = (f′(\overline{u})\frac{∂\overline{u}}{∂x}) (s, X_{s}^{(t,x)})\frac{∂^{−}X_{·}^{(t,x)}}{∂W_{s}}
= (f′(\overline{u})\frac{∂\overline{u}}{∂x}σ) (s, X_{s}^{(t,x)}).

Therefore

∫_{0}^{t} f(\overline{u})(s, X_{s}^{(t,x)})d_{ϕ}W_{s} = ∫_{0}^{t}f(\overline{u})(s, X_{s}^{(t,x)})d^{−}W_{s} − \frac{1}{2} ∫_{0}^{t}f′(\overline{u})\frac{∂\overline{u}}{∂x}σ(s, X_{s}^{(t,x)})ds.

4 Proof of the Main Theorem

From now we use the following notation for the constants which will be fixed troughout the proof. In particular, we fix |σ|_{∞} + |σ′|_{∞} < c, |u_{0}|_{∞} + |u′_{0}|_{∞} + |u′′_{0}|_{∞} < C(0). We carry out this proof in several steps:

4.1 Successive approximation

For the construction of the solution we apply the Picard’s method to our integral equations. Let {u_{k}} be a sequence of random functions defined inductively by

\begin{matrix}u_{k+1}(t, x) = u_{0}(X^{(t,x)}(0)) + ν∫_{0}^{t}\overline{u}_{k}^{2}(s, X^{(t,x)}(s))d_{ϕ}W_{s} (k ≥ 0), \\ where u_{0}(t, x) = u_{0}(x). \end{matrix}  (6)

The following lemma states some simple differentiability properties of ū_{k}(t, x) = E(u(t, x)) in order to be able to apply the chain rule. Later in Lemma 7 and 8 these properties will be refined.

Lemma 5 If u_{0} ∈ C^{2}(IR) with |u_{0}|_{∞}+|u′_{0}|_{∞}+|u′′_{0}|_{∞} < C(0) , then \overline{u}_{k} ∈ C^{1,2}([0, T]× IR), |\overline{u}_{k}|_{∞} + |\overline{u}′_{k}|_{∞} + |\overline{u}′′_{k}|_{∞} < C(k) ∀k for some positive constants C(k) .

Proof. The proof is done by induction. Suppose that the result is true for k. Then one has by the chain rule that for fixed (t, x)

\frac{∂^{−}\overline{u}_{k}^{2}(·, X^{(t,x)}(·))}{∂W_{s}} = 2\overline{u}_{k}\overline{u}′_{k}(s, X^{(t,x)}(s))\frac{∂^{−}X^{(t,x)}(·)}{∂W_{s}} = 2σ\overline{u}_{k}\overline{u}′_{k}(s, X^{(t,x)}(s))

where \overline{u}′_{k}(s, y) = \frac{∂\overline{u}_{k}}{∂y} (s, y). Therefore we have the following representation for the stochastic integral term in (6),

∫_{0}^{t}\overline{u}_{k}^{2}(s, X^{(t,x)}(s))d_{ϕ}W_{s} = ∫_{0}^{t}\overline{u}_{k}^{2}(s, X^{(t,x)}(s))d^{−}W_{s} − ∫_{0}^{t} σ\overline{u}_{k}\overline{u}′_{k}(s, X^{(t,x)}(s))ds.

Taking this into account, we get from equation (6) the following equation for the average \overline{u}_{k}(t, x) = Eu_{k}(t, x);

\begin{aligned}\overline{u}_{k+1}(t, x) & = Eu_{0}(X^{(t,x)}(0)) − νE∫_{0}^{t} σ\overline{u}_{k}\overline{u}′_{k}(s, X^{(t,x)}(s))ds \\ & = Eu_{0}(X^{(t,x)}(0)) − ν∫_{0}^{t} ∫_{IR} σ\overline{u}_{k}\overline{u}′_{k}(s, y)p_{t,x}(s, y)dyds \\ & = Eu_{0}(X^{(t,x)}(0)) + \frac{ν}{2}∫_{0}^{t} ∫_{IR}\overline{u}_{k}(s, y)^{2} \frac{∂(σp_{t,x})}{∂y}(s, y)dyds. \end{aligned}  (7)

The last equality follows because lim_{y→∞}\overline{u}_{k}^{2}(s, y)p_{t,x}(s, y) = 0, for each fixed k, s < t

and x. Therefore one has that

|\overline{u}_{k+1}|_{∞} ≤ (C(0) + νM^{3/2}C(k)^{2}c(t + \sqrt{t})).

Now we differentiate with respect to x the second equality in (7) to obtain that

\overline{u}′_{k+1}(t, x) = \frac{∂}{∂x}Eu_{0}(X^{(t,x)}(0)) − ν∫_{0}^{t} ∫_{IR} σ\overline{u}_{k}\overline{u}′_{k}(s, y)\frac{∂p_{t,x}}{∂x} (s, y)dyds

Therefore one obtains bounds for |\overline{u}′_{k+1}|_{∞} similarly. In preparation for the estimates of \overline{u}′′_{k+1} note that

\frac{∂p_{t,x}(s, u + x)}{∂x} = \frac{∂p_{t,x}}{∂x} (s, u + x) + \frac{∂p_{t,x}}{∂y} (s, u + x).

For this ”diagonal” derivative we will apply the estimate (2). Using a change of variables (y = u + x), the above equality and an integration by parts we have

∫_{0}^{t} ∫_{IR}σ\overline{u}_{k}\overline{u}′_{k}(s, u + x)\frac{∂p_{t,x}}{∂x} (s, u + x)duds = ∫_{0}^{t} ∫_{IR} \frac{∂(σ\overline{u}_{k}\overline{u}′_{k})}{∂y}(s, u + x)p_{t,x}(s, u + x)duds

+∫_{0}^{t} ∫_{IR} σ\overline{u}_{k}\overline{u}′_{k}(s, u + x)\frac{∂p_{t,x}(s, u + x)}{∂x}duds.

Differentiating this again we find that

|\overline{u}′′_{k+1}(t, x)| ≤ |\frac{∂^{2}}{∂x^{2}} Eu_{0}(X^{(t,x)}(0))| + |ν ∫_{0}^{t} ∫_{ℝ} \frac{∂(σ\overline{u}_{k}\overline{u}′_{k})}{∂y}(s, y)\frac{∂}{∂x}p_{t,x}(s, y)dyds|

+ |ν ∫_{0}^{t} ∫_{ℝ} \frac{∂(σ\overline{u}_{k}\overline{u}′_{k})}{∂y}(s, u + x)\frac{∂p_{t,x}(s, u + x)}{∂x}duds|

+ |ν ∫_{0}^{t} ∫_{ℝ} σ\overline{u}_{k}\overline{u}′_{k}(s, u + x)\frac{∂^{2}p_{t,x}(s, u + x)}{∂x^{2}}duds| .

On the other hand, using properties of the derivatives of the flow X^{(t,x)}(0) and the hypotheses on u_{0}, one obtains that there is a positive constant C such that |\frac{∂^{2}}{∂x^{2}} Eu_{0}(X^{(t,x)}(0))| ≤ C. Finally using inequality (2) we have

|\overline{u}′′_{k+1}(t, x)| ≤ C + ν ∫_{0}^{t} ∫_{ℝ} 3cC(k)^{2}M \frac{exp (−\frac{|x−y|^{2}}{M(t−s)})}{\sqrt{2π}(t − s)}dyds

+ν ∫_{0}^{t} ∫_{ℝ} 4cC(k)^{2}M \frac{exp (−\frac{|u|^{2}}{M(t−s)})}{\sqrt{2π}(t − s)^{1/2}}duds.

Therefore

|\overline{u}′′_{k+1}|_{∞} ≤ (C + νM^{3/2}C(k)^{2}c(4t + 6\sqrt{t})).

□

4.2 Lemmas

We will show the convergence of the sequence \{\overline{u}_{k}\}. The key is to show the next two lemmas.

Lemma 6 For any two positive constants C(0) and c such that cνM^{3/2}C(0) is small enough and any 0 < t_{0} < 1 the following Volterra equation has a unique bounded solution y(t) up to t ≤ t_{0}

y(t) = C(0) + \frac{cν}{2} M^{3/2} ∫_{0}^{t} y^{2}(s) (1 + \frac{1}{\sqrt{t − s}}) ds.  (8)

Proof. Put y(t) = ∑_{k=0}^{∞}α_{k}t^{k/2} and substitute this into the equation (8), then we

will have the equality as follows,

∑_{k=0}^{∞}α_{k}t^{k/2} = C(0) + \frac{cν}{2} M^{3/2} ∫_{0}^{t} (∑_{k=0}^{∞} α_{k}s^{k/2})^{2} (1 + \frac{1}{\sqrt{t − s}})ds ; \begin{matrix}= C(0) + \frac{cν}{2} M^{3/2} ∑_{k_{1},k_{2}=0}^{∞}α_{k_{1}}α_{k_{2}} B(\frac{k_{1} + k_{2}}{2} + 1, \frac{1}{2})t^{\frac{k_{1}+k_{2}+1}{2}} \\ +cνM^{3/2} ∑_{k_{1},k_{2}=0}^{∞} \frac{α_{k_{1}}α_{k_{2}}}{k_{1} + k_{2} + 2}t^{\frac{k_{1}+k_{2}}{2} +1}. \end{matrix}

Comparing the coefficients on both sides of the equation above, we can find the coefficients α_{k} as α_{0} = C(0),

α_{k} = \frac{cν}{2} M^{3/2}B(\frac{k + 1}{2}, \frac{1}{2}) ∑_{k_{1}+k_{2}+1=k}α_{k_{1}}α_{k_{2}} + \frac{cνM^{3/2}}{k} ∑_{k_{1}+k_{2}+2=k}α_{k_{1}}α_{k_{2}}. (9)

On the other hand, we can see (using Stirling’s approximation for the Gamma function, see the appendix) that

lim_{k→∞}\sqrt{k + 1} · B(\frac{k + 1}{2}, \frac{1}{2}) = \sqrt{2π}

and that

lim_{k→∞} ∑_{k_{1}+k_{2}+1=k} \frac{1}{\sqrt{k_{1} + 1}} \frac{1}{\sqrt{k_{2} + 1}} = lim_{k→∞} \frac{1}{k} ∑_{k_{1}+k_{2}+1=k} \frac{1}{\sqrt{\frac{k_{1}+1}{k}} \sqrt{\frac{k_{2}+1}{k}}} = ∫_{0}^{1} \frac{dx}{\sqrt{x(1 − x)}} = π.

Hence we see that there exists positive constants γ_{1}, γ_{2} such that for any k, we have the estimates,

B(\frac{k + 1}{2}, \frac{1}{2}) ≤ γ_{1} \frac{1}{\sqrt{k + 1}}, ∑_{k_{1}+k_{2}+1=k} \frac{1}{\sqrt{k_{1} + 1}} \frac{1}{\sqrt{k_{2} + 1}} ≤ γ_{2}π.

Given these, we claim that α_{k} ≤ \frac{C(0)}{\sqrt{k + 1}} for any k ≥ 0. In fact, for k = 0 and k = 1

the result is trivial if \sqrt{2}cνM^{3/2}C(0) < 1 (α_{1} = cνM^{3/2}C(0)^{2}).

Now suppose the result is valid up to k −1 and that cνM^{3/2}C(0)γ_{2}π(\frac{γ_{1}}{2} +\sqrt{\frac{3}{4}}) < 1 then we get from (9) the following inequality that proves the assertion;

α_{k} ≤ \frac{cν}{2} M^{3/2}C(0)^{2} B(\frac{k + 1}{2}, \frac{1}{2}) ∑_{k_{1}+k_{2}+1=k} \frac{1}{\sqrt{(k_{1} + 1) (k_{2} + 1)}}+

+ \frac{cνM^{3/2}C(0)^{2}}{k} ∑_{k_{1}+k_{2}+2=k} \frac{1}{\sqrt{(k_{1} + 1) (k_{2} + 1)}}

≤ (\frac{cνM^{3/2}C(0)^{2}γ_{1}}{2\sqrt{k + 1}} + \frac{cνM^{3/2}C(0)^{2}}{k} ) γ_{2}π ≤ \frac{C(0)}{\sqrt{k + 1}}.

Note that the restriction on the constant cνM^{3/2}C(0) is explicit and fixed by the constants γ_{1} and γ_{2}.

The series

∑_{k}α_{k}t^{k/2} ≤ C(0)(1 + ∑_{k≥1} \frac{t^{k/2}}{\sqrt{k + 1}}),

converges absolutely over the interval [0, t_{0}] with t_{0} < 1 and this completes the proof of existence. For uniqueness one follows the usual proof of taking the difference between the two solutions and using the boundedness together with Gronwall’s inequality. □

Lemma 7 For small enough t (≤ t_{0}) it holds that sup_{t∈[0,t_{0}],x,k}|\overline{u}_{k}(t, x)| ≤ C_{1} where

C_{1} = C(0)(1 + A_{β}(t_{0})) where A_{β}(t_{0}) = ∑_{k≥1} \frac{t_{0}^{k/2}}{\sqrt{k+1}}.

Proof. From the equation (7) we get,

|\overline{u}_{k+1}(t, x)| ≤ C(0) + \frac{ν}{2} ∫_{0}^{t} ∫_{IR} \overline{u}_{k}^{2}(s, y)|\frac{∂}{∂y} (σp_{t,x})(s, y)|dyds.

Since

| (σ′p_{t,x} + σ \frac{∂}{∂y} p_{t,x}) (s, y)| ≤ \frac{c}{\sqrt{2π(t − s)}}(M + \frac{M}{\sqrt{t − s}}) exp (−\frac{|x − y|^{2}}{M(t − s)}) ,

then one has that

∫_{IR} |\frac{∂}{∂y} (σp_{t,x})(s, y)|dy ≤ cM^{3/2} (1 + \frac{1}{\sqrt{t − s}}) .

This, in turn, implies the following inequality,

sup_{x}|\overline{u}_{k+1}(t, x)| ≤ C(0) + \frac{cνM^{3/2}}{2} ∫_{0}^{t}\{sup_{y} \overline{u}_{k}^{2}(s, y)\} (1 + \frac{1}{\sqrt{t − s}}) ds.

The next step consists of proving that for any k ∈ N, we have that

sup_{x} |\overline{u}_{k}(t, x)| ≤ y(t) (∀t ≤ t_{0}).

By applying induction we have first that sup_{x}|\overline{u}_{0}(t, x)| ≤ C(0) ≤ y(t). Now suppose the inequality is satisfied for k. Then using the above inequality and the previous Lemma 6 we have that the inequality is satisfied for k + 1. □

Following a similar discussion to that given in the proof of the Lemma 6, we can establish the following result;

Lemma 8 For small enough t(≤ t_{0}), there exists a positive constant C_{2} such that

sup_{t∈[0,t_{0}],x,k}|\overline{u}′_{k}(t, x)| < C_{2} .

Proof. First, one differentiates (7) to obtain

\overline{u}′_{k+1}(t, x) = E(u′_{0}(X^{(t,x)}(0))\frac{∂X^{(t,x)}(0)}{∂x}) − ν∫_{0}^{t} ∫_{IR}σ\overline{u}_{k}\overline{u}′_{k}(s, y)\frac{∂}{∂x}(p_{t,x})(s, y)dyds.

As before the proof follows by induction. First we have that sup_{x, t∈[0,t_{0}]}|\overline{u}′_{0}(t, x)| ≤ C(0). Then we have using the previous Lemma that there exists a positive constant A(C(0), t_{0}) such that

sup_{x} |\overline{u}′_{k+1}(t, x)| ≤ A(C(0), t_{0}) + cC_{1}νM^{3/2}∫_{0}^{t} sup_{x}|\overline{u}′_{k}(s, x)| (1 + \frac{1}{\sqrt{t − s}}) ds.

Using the Gronwall lemma in the Appendix, one finally obtains that

sup_{x} |\overline{u}′_{k+1}(t, x)| ≤ A(C(0), t_{0}) exp((t + 2\sqrt{t})cC_{1}νM^{3/2}).

□

Similarly, one obtains the same conclusion for the second derivative. That is, sup_{t∈[0,t_{0}],x,k}|\overline{u}′′_{k}(t, x)| is bounded. Without loss of generality we assume that C_{2} > C_{1}.

4.3 Convergence of the sequence \{\overline{u}_{k}\}

Proposition 9 The sequence \{\overline{u}_{k}(t, x)\} converges to \overline{u}(t, x) uniformly in (t, x) ∈ [0, t_{0} ∧1]×IR as k ⟶ ∞. Here \overline{u}(t, x) is the unique solution to the integral equation

\overline{u}(t, x) = Eu_{0}(X^{(t,x)}(0)) + \frac{ν}{2} ∫_{0}^{t} ∫_{IR} \overline{u}^{2}(s, y)\frac{∂}{∂y} (σp_{t,x})(s, y)dyds.  (10)

Proof. From the equation (7) and Lemma 7, we get for t ≤ t_{0} ∧ 1 the following,

|\overline{u}_{k}(t, x) − \overline{u}_{k−1}(t, x)| = \frac{ν}{2}|∫_{0}^{t} ∫_{IR} [\{(\overline{u}_{k−1}^{2})′ − (\overline{u}_{k−2}^{2})′\}σp_{t,x}] (s, y)dyds|

= \frac{ν}{2}|∫_{0}^{t} ∫_{IR} [\{\overline{u}_{k−1}^{2} − \overline{u}_{k−2}^{2}\}\frac{∂}{∂y} (σp_{t,x})] (s, y)dyds|

≤ C_{1}ν∫_{0}^{t} ∫_{IR}|\overline{u}_{k−1} − \overline{u}_{k−2}|(s, y)|\frac{∂}{∂y} (σp_{t,x})(s, y)|dyds.

Put B_{k}(t) = sup_{x}|\overline{u}_{k}(t, x) − \overline{u}_{k−1}(t, x)|, then from the inequality above we obtain,

\begin{aligned}B_{k}(t) & ≤ C_{1}ν∫_{0}^{t} B_{k−1}(s) ∫_{IR} |\frac{∂}{∂y} (σp_{t,x})(s, y)|dyds \\ & ≤ C_{1}cνM^{3/2}∫_{0}^{t} B_{k−1}(s) (1 + \frac{1}{\sqrt{t − s}}) ds. \end{aligned}  (11)

Hence by induction we get the inequality as follows for some positive constant C_{3},

B_{k}(t) ≤ C_{3}^{k}\{∏_{j=2}^{k} B(\frac{j + 1}{2}, \frac{1}{2})\}t^{\frac{k}{2}} .  (12)

In fact, for k = 1, one has that

B_{1}(t) ≤ |E(u_{0}(X_{0}^{(t,x)}) − u_{0}(x))| + ∫_{0}^{t} sup_{x}|\overline{u}_{0}(s, x)|^{2} (1 + \frac{1}{\sqrt{t − s}}) ds
≤ C_{4}\sqrt{t}

for some positive constant C_{4}. Now we can rewrite inequality (11) as

B_{k}(t) ≤ 2C_{1}cνM^{3/2}∫_{0}^{t} \frac{B_{k−1}(s)}{\sqrt{t − s}}ds.

Here we take C_{4} ∨ (2C_{1}cνM^{3/2}) ≤ C_{3}. Supposing that (12) is satisfied up to k − 1, we have

B_{k}(t) ≤ 2C_{1}cνM^{3/2}∫_{0}^{t} \frac{C_{3}^{k−1}\{∏_{j=2}^{k−1} B(\frac{j+1}{2}, \frac{1}{2})\}s^{\frac{k−1}{2}}}{\sqrt{t − s}} ds

≤ C_{3}^{k}\{∏_{j=2}^{k−1} B(\frac{j + 1}{2}, \frac{1}{2})\}t^{\frac{k}{2}} ∫_{0}^{1} \frac{s^{\frac{k−1}{2}}}{\sqrt{1 − s}}ds.

Since

lim_{k→∞} \frac{C_{3}^{k}\{∏_{j=2}^{k} B(\frac{j+1}{2}, \frac{1}{2})\}}{C_{3}^{k+1}\{∏_{j=2}^{k+1} B(\frac{j+1}{2}, \frac{1}{2})\}} = lim_{k→∞} \frac{1}{C_{3}B(\frac{k+2}{2}, \frac{1}{2})} = ∞,

we see that the series ∑_{k} B_{k}(t) converges uniformly for t < t_{0} ∧ 1. This finishes the proof. □

Following the similar discussion and taking the Lemma 8 into account, we also establish the next result.

Proposition 10 The sequence \{\overline{u}′_{k}(t, x)\} converges uniformly on any finite slab [0, t_{0}] × IR to \overline{u}′ as k tends to ∞.

Proof. As before,

|\overline{u}′_{k}(t, x) − \overline{u}′_{k−1}(t, x)| = ν|∫_{0}^{t} ∫_{IR} [σ\{\overline{u}_{k−1}\overline{u}′_{k−1} − \overline{u}_{k−2}\overline{u}′_{k−2}\}\frac{∂}{∂x}(p_{t,x})] (s, y)dyds|
≤ C_{2}cν ∫_{0}^{t} ∫_{IR} |\overline{u}_{k−1} − \overline{u}_{k−2}|(s, y)|\frac{∂}{∂x}(p_{t,x})(s, y)|dyds

+C_{1}cν∫_{0}^{t} ∫_{IR}|\overline{u}′_{k−1} − \overline{u}′_{k−2}|(s, y)|\frac{∂}{∂x}(p_{t,x})(s, y)|dyds.

Define B′_{k}(t) = sup_{x}|\overline{u}′_{k}(t, x) − \overline{u}′_{k−1}(t, x)|. Then we have

B′_{k}(t) ≤ 2C_{2}cνM^{3/2} (∫_{0}^{t} \frac{B_{k−1}(s)}{\sqrt{t − s}}ds+∫_{0}^{t} \frac{B′_{k−1}(s)}{\sqrt{t − s}}ds) .

As before one also proves in this case that ∑_{k} B′_{k}(t) converges uniformly for t < t_{0}∧1. Denote the limit of \overline{u}′_{k} by \overline{v}. Then it satisfies the equation

\overline{v}(t, x) = Eu_{0}(X^{(t,x)}(0)\frac{∂X^{(t,x)}(0)}{∂x}) + ν ∫_{0}^{t} ∫_{IR} (\overline{u} \overline{v}) (s, y)\frac{∂}{∂x}(σp_{t,x})(s, y)dyds.

The convergence being uniform also gives that \overline{u} is differentiable in x. The above equation has a unique solution which is also satisfied by \overline{u}′ if one differentiates (10). This finishes the proof. □

Based on these Propositions, we let k tend to infinity on both sides of the first equation in (7) to find that the lim_{k} \overline{u}_{k} = \overline{u} is the solution of the following,

\overline{u}(t, x) = Eu_{0}(X^{(t,x)}(0)) − ν∫_{0}^{t} ∫_{IR}\overline{u} \overline{u}′(s, y)σp_{t,x}(s, y)dyds.  (13)

Proposition 11 The solution, bounded in x, of the equation (13) is unique.

Proof Let \overline{u}, \overline{v} be the solutions that are bounded in x. Then we have,

|\overline{u}(t, x) − \overline{v}(t, x)| ≤ \frac{ν}{2}∫_{0}^{t} ∫_{IR}|\overline{u}^{2}(s, y) − \overline{v}^{2}(s, y)||\frac{∂}{∂y} (σp_{t,x})(s, y)|dyds.

Hence,

sup_{x} |\overline{u}(t, x) − \overline{v}(t, x)| ≤ C_{1}cνM^{3/2}∫_{0}^{t} sup_{y} |\overline{u}(s, y) − \overline{v}(s, y)|\frac{1}{\sqrt{t − s}}ds.

By applying the Gronwall’s lemma (see the Appendix) to this inequality we get the conclusion. □

Similarly as in the previous arguments one can also prove the convergence and existence of the second derivative of \overline{u}(t, x). In fact, one has that

|\overline{u}′′_{k}(t, x) − \overline{u}′′_{k−1}(t, x)| ≤

|ν ∫_{0}^{t} ∫_{ℝ} \frac{∂}{∂y} (σ (\overline{u}_{k−1}\overline{u}′_{k−1} − \overline{u}_{k−2}\overline{u}′_{k−2})) (s, y) (\frac{∂}{∂y} + \frac{∂}{∂x}) p_{(t,x)}(s, y)dyds|

+ν ∫_{0}^{t} ∫_{ℝ} |σ (\overline{u}_{k−1}\overline{u}′_{k−1} − \overline{u}_{k−2}\overline{u}′_{k−2}) (s, y)| \frac{M exp (−\frac{|x−y|^{2}}{M(t−s)})}{\sqrt{2π}(t − s)^{1/2}}dyds.

The conclusion follows by Gronwall’s lemma.

4.4 Convergence of the sequence {u_{k}}

We have that

∫_{0}^{t} \overline{u}_{k}^{2}(s, X^{(t,x)}(s))d_{ϕ}W_{s} = ∫_{0}^{t} \overline{u}_{k}^{2}(s, X^{(t,x)}(s))d^{−}W_{s} − ∫_{0}^{t} σ\overline{u}_{k}\overline{u}′_{k}(s, X^{(t,x)}(s))ds.

Therefore Propositions 9 and 10 applied to u_{k+1} in (6) imply that as k tends to ∞,

the sequence ∫_{0}^{t}\overline{u}_{k}^{2}(s, X^{(t,x)}(s))d_{ϕ}W_{s} converges (in the L^{2}(Ω) × L_{loc}^{2}(IR) - sense) to

the limit ∫_{0}^{t}\overline{u}^{2}(s, X^{(t,x)}(s))d_{ϕ}W_{s}, uniformly in t ∈ [0, t_{0}]. Consequently we have

lim_{k→∞}u_{k}(t, x) = ∃v(t, x) (uniformly in t), where v is given by the

v(t, x) = u_{0}(X^{(t,x)}(0)) + ν ∫_{0}^{t} \overline{u}^{2}(s, X^{(t,x)}(s))d_{ϕ}W_{s}.

Since Ev(t, x) = ū(t, x) by virtue of the Proposition 11, we see that the limit v(t, x) is the unique solution of the integral equation (4).

In fact suppose that we are given two solutions to (4), say v_{1} and v_{2}, then by Proposition 11 one has that E(v_{1}(t, x)) = E(v_{2}(t, x)) = ū(t, x). Therefore if one replaces this into the equation (4) one has that v_{1}(t, x) = v_{2}(t, x) = u(t, x).

Thus we have shown the existence and uniqueness of the local solution u(t, x), 0 ≤ t ≤ t_{0} ≤ T. The proof of the Theorem 4 is completed when we show that the discussion can be extended to the case t ∈ [t_{0}, T]. But this is an immediate consequence of the flow property below;

u(t, x) = u(t_{0}, X^{(t,x)}(t_{0})) + ν ∫_{t_{0}}^{t} \overline{u}^{2}(s, X^{(t,x)}(s))d_{ϕ}W_{s}, t ≥ t_{0}. □

In order to carry this step one has to note that u(t_{0}, x) is independent of σ{W_{s} − W_{t_{0}}, s ≥ t_{0}} and that t_{0} < 1 is independent of all constants as stated in Lemma 6. Also the constants found in Lemmas 7 and 8 do not explode in a finite number of iterations therefore the extension of the previous arguments follow.

4.5 Verification of Equation (5)

It is trivial that the initial condition is satisfied as ū(0, x) = Eu_{0}(X_{0}^{(0,x)}) = u_{0}(x). Repeating the same arguments as we have done previously, one proves that \overline{u}(t, x) is two times differentiable in space. That is, one proves that the sequence \overline{u}′′_{k} converges uniformly and that its limit is \overline{u}′′(t, x). From (13) and (3) we have that \overline{u}(t, x) is differentiable in time and

\frac{∂\overline{u}}{∂t}(t, x) = \frac{1}{2}σ(t, x)^{2} ∫_{IR} u_{0}(y)\frac{∂^{2}p_{t,x}}{∂x^{2}} (0, y)dy + b(t, x) ∫_{IR} u_{0}(y)\frac{∂p_{t,x}}{∂x} (0, y)dy

−\frac{νσ(t, x)^{2}}{2} ∫_{0}^{t} ds ∫_{IR} σ\overline{u} \overline{u}′(s, y)\frac{∂^{2}p_{t,x}}{∂x^{2}} (s, y)dy

−νb(t, x)∫_{0}^{t} ds ∫_{IR} σ\overline{u} \overline{u}′(s, y)\frac{∂p_{t,x}}{∂x} (s, y)dy

−νσ(t, x)\overline{u} \overline{u}′(t, x)

= \frac{σ(t, x)^{2}}{2} \frac{∂^{2}\overline{u}}{∂x^{2}}(t, x) + b(t, x)\frac{∂\overline{u}}{∂x}(t, x) − νσ(t, x)\frac{∂\overline{u}^{2}}{∂x} (t, x).
□

5 Concluding Remarks

Here we have obtained a new stochastic representation formula for the solution of the Burgers equation. This is called a particle equation because its possible approximation process will require the use of a particle method to approximate the expectation in the equation. We hope that this representation will allow numerical simulation of the process u and the study of other properties related with this process for small values of ν. We have assumed here that the coefficients a and b are bounded but one could derive the above results with linear growth conditions doing appropiate changes in the arguments given for the proofs. In comparison with the stochastic representation obtained in [1] and [2] we do not require any integrability properties of u_{0}. In particular, one may have that ∫_{IR} |u_{0}(x)|dx = ∞ and still the stochastic representation in Theorem 4 is valid.

6 Acknowledgements

The authors express their thanks to the referee for a careful reading of the article. The second author was partially supported by grants BFM 2000-807 and BFM 2000-0598. This research was done while the second author was visiting the Laboratory of Applied Mathematics and Stochastics at Kanazawa University.

References

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[2] Bossy, M. and Talay D.: Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation. Annals of Applied Probability, 6:818-861, 1996.

[3] Friedman, A.: Partial differential equations of parabolic type. 1964. Prentice-Hall.

[4] Ikeda, N. and Watanabe, S.:Stochastic Differential Equations and Diffusion Processes. Amsterdam Oxford New York: North-Holland/Kodansha 1989.

[5] Mitrinovic,D.S., Pecaric,J.E. Fink,A.M.; Inequalities Involving Functions and Their Integrals and Derivatives, 1991, Kluwer Academic Publishers

[6] Ogawa,S.: On a class of SPDEs called the Brownian particle equation - Model for nonlinear diffusions, Monte Carlo Methods and Applications, vol.7, No.1-2, pp.321–328, VSP Publisher, Netherland, 2001

[7] Ogawa,S.: ”On the Brownian particle equations and the noncausal stochastic calculus”, Rendiconti Academia Nazionale delle Scienze detta dei XL, 119, XXV, pp.125–139, 2001

[8] Ogawa,S.: ”On the numerical simulation of nonlinear SDEs and Burgers like processes”, Proceedings of The Workshop on Turbulent Diffusion and Related Problems in Stochastic Numerics (at ISM in Oct 1996) (ISM Joint Research Report 99), pp.140-159, 1997

[9] Ogawa,S.: ”Topics in the theory of noncausal stochastic calculus”, in Diffusion Processes and Related Problems in Analysis, vol.1, (edt. Mark Pinsky), Birkhäuser Boston Inc., pp.411-420, 1990

[10] Ogawa,S.: ”The stochastic integral of noncausal type as an extension of the symmetric integrals”, Japan J. Appl.Math., vol.2, No.1, pp.229–240, 1985

[11] Ogawa,S.: ”Sur le produit direct du bruit blanc par lui-même”, C.R.Acad.Sci., t.288, Série A, pp.359-362, 1979 [12] Ogawa,S.: ”A partial differential equation with the white noise as a coefficient”,

Z.W.verw. Geb., 28, pp.53–71, 1973

[13] Ogawa,S.: ”On a Riemann definition of the stochastic integral, I (and II)”, Proc.Japan Acad., 46, pp.153–156 (and 157– 161), 1970

7 Appendix

Here we prove the following assertion

lim_{k→∞}\sqrt{k + 1} · B(\frac{k + 1}{2}, \frac{1}{2}) = \sqrt{2π}

First, using the definition of the beta function we have that

B(\frac{k + 1}{2}, \frac{1}{2}) = \frac{Γ(\frac{1}{2})Γ(\frac{k+1}{2})}{Γ(\frac{k}{2} + 1)}

One studies the above quantities in the cases k even and odd. We will do one case leaving the other for the reader. For k = 2n, n ∈ N, we have by properties of the gamma function that Γ(\frac{k}{2} + 1) = \frac{k}{2}! and that

Γ(\frac{k + 1}{2}) = \frac{k − 1}{2}Γ(\frac{k − 1}{2})
= \frac{(k − 1)!!}{2^{k/2}}Γ(\frac{1}{2})
= \frac{k!}{2^{k} (\frac{k}{2})!}Γ(\frac{1}{2})

Therefore we have that

B(\frac{k + 1}{2}, \frac{1}{2}) = \frac{Γ(\frac{1}{2})^{2}_{\frac{k!}{2^{k}(\frac{k}{2})!}}}{(\frac{k}{2})!}

Using Stirling’s approximation for the factorial we have

B(\frac{k + 1}{2}, \frac{1}{2}) ∼ \frac{πk^{k+1/2}}{\sqrt{2π}2^{k} (\frac{k}{2})^{k+1}}

∼ \begin{matrix}√ & \overline{2π} \\ √ & \overline{k} \end{matrix}

In the other case k = 2n + 1 one proceeds similarly finding the same limit. □

The Gronwall inequality we used here is of some particular type due to the fact that the kernel function is degenerate. For this reason we give a brief account of this inequality Lemma 12 Ley y(t) be a real, non-negative function such that it satisfies

y(t) ≤ A_{1} + A_{2} ∫_{0}^{t} \frac{y(s)}{\sqrt{t − s}}ds, t ∈ [0, T],

for two positive constants A_{1} and A_{2}. Then

y(t) ≤ A_{1} exp(2A_{2}\sqrt{t}).

The idea of the proof is to apply 6.Theorem 1 in Mitrinović et. al., page 358 for the kernel

k(t, s) = \frac{1}{\sqrt{t − s + ϵ}}

obtain the Gronwall inequality that will depend on ϵ and then take limits as ϵ goes to 0.□