SMOOTHNESS OF DISTRIBUTION FOR

SOLUTIONS OF SDE’S WITH INTERACTION^{∗}

Pilipenko A.Yu.

Abstract. Consider the following system of stochastic differential equation with interaction

(1)  \begin{aligned}dx(u, t) = & ∫ b_{0}(x(u, t), x(v, t))μ_{0}(dv)dt+ \\ & + ∑_{k=1}^{m} b_{k}(x(u, t), x(v, t))μ_{k}(dv)dω_{k}(t), t ∈ [0, T] \end{aligned}

x(u, t) = φ(u), u ∈ ℝ^{d}.

The conditions that guarantee the smoothness of x(u, t) distribution are studied.

Consider the equation (1), where b_{k} : ℝ^{2d} → ℝ^{d}, k = \overline{1, m}, φ : ℝ^{d} → ℝ^{d}, μ_{k} are probability measures on ℝ^{d}, ω_{1}(t), …, ω_{m}(t) are independent one-dimensional Wiener processes. The aim of this article is to find some conditions that ensure smoothness of x(u, t) distribution for fixed u ∈ ℝ^{d}, t > 0.

The process x(u, t), u ∈ ℝ can be considered as an random element in some infinitedimensional space (C(ℝ, ℝ) roughly). So our investigation can be studied as a part of the following problem. Let ξ(t) be a solution of SDE

dξ(t) = A(ξ(t))dt + B(ξ(t))dW (t)

in some multidimensional space E. Let π be a projection in E to some finite dimensional subspace of E. The problem is to prove the smoothness of πξ(t) distribution w.r.t. the Lebesgue measure on πE. There are some results that ensure differentiability of the ξ(t) distribution w.r.t. the directions of some infinitedimensional subspace H ⊂ E (cf. [1,2]). This property is more strong than differentiability of πξ(t). But reasoning of [1,2] cannot be applied in a case of the equation (1). Really, if we have only finite number of Wiener processes then (infinitedimensional) Malliavin’s matrix cannot be invertible.

Let us introduce the following notations. Set

(2)  (Bψ)(u) = ∥B_{k}ψ(u)∥_{k=\overline{1,m}} = ∥ ∫_{ℝ^{d}} b_{k}(ψ(u), ψ(v))μ_{k}(dv)∥_{k=\overline{1,m}}

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 60H07, 60G30.

Key words and phrases. Malliavin Calculus, Partially Hypoellipticity Problem, SDE with interaction.

∗ This article was supported (in part) by Ministry of Education and Science of Ukraine, project N 01.07/103.

for each measurable function ψ : ℝ^{d} → ℝ^{d}.

For fixed u ∈ ℝ^{d}, ψ a map Bψ(u) will be considered as a linear operator from ℝ^{d} to ℝ^{m} or as d × m matrix.

The main result of the article is the following theorem

Theorem 1. Assume that

1) for each k = 0, …, m a function b_{k} : ℝ^{2d} → ℝ^{d} is infinitely differentiable and bounded with all its derivatives;

2) for each k = 0, …, m a probability measure μ_{k} has all positive moments;

3) a function φ : ℝ^{d} → ℝ^{d} is infinitely differentiable and sup_{x} ∥∇^{k}φ(x)∥ < ∞ for each k ∈ ℕ.

Suppose that for some orthoprojection π : ℝ^{d} → ℝ^{d} and a point u ∈ ℝ^{d} the following condition is satisfied: ∃c > 0 ∀ψ : ℝ^{d} → ℝ^{d}

πBψ(u)(Bψ(u))^{∗}π^{∗} ≥ cπ.

Then the distribution of πx(u, t) in πℝ^{d} has C^{∞}-density with respect to the Lebesgue measure in πℝ^{d}.

Remark 1. The result of a Theorem 1 can easily be generalized to a case of multiple interaction, i.e. if we consider terms

∫ b_{k}(x(u, t), x(v_{1}, t), …, x(v_{n}, t))μ_{k}(dv_{1}, …, dv_{n})

instead of ∫ b_{k}(x(u, t), x(v, t)μ_{k}(dv). This can be done by extension of a space.

Remark 2. The existence and uniqueness of solution (1) is proved in [3]. Besides there exists a modification of x(u, t) which is continuous in (u, t). Namely this modification is considered further.

Remark 3. There is one difference between research of smoothness of a solution of SDE or its projection. At first case it is natural to demand some non-degenerativity of coefficients in the starting point. But in the second case it is natural to require a non-degenerativity of coefficients in the terminal point (cf.[4]). The same effect appears in our situation also.

We need the following well-known theorem of the Malliavin calculus.

Theorem 2 [5]. Let (X, μ, H) be an abstract Wiener space, F = (F_{1}, …, F_{d}) : X → ℝ^{d}. Assume that

1) F_{1}, …, F_{d} lie in ∩_{p,k≥1}W_{p,k};

2) the Malliavin matrix (DF_{i}, DF_{j})_{i,j=1}^{d} is invertible a.e.;

3) (det ∥(DF_{i}, DF_{j})∥)^{−1} ∈ ∩_{p≥1}L_{p}.

Then the distribution of F has C^{∞}-density in ℝ^{d}.

Further, we consider X = C_{0}([0, T], ℝ^{m}), X is a space of continuous ℝ^{m}-valued functions starting at zero. Let μ be the distribution of Wiener process (ω_{1}(t), …, ω_{m}(t)) in X, H = L_{2}([0, T], ℝ^{m}). A Hilbert space H is imbedded to X by the inclusion map

H ∋ h ↦ ∫_{0}^{·} h(s)ds ∈ X.

SMOOTHNESS OF DISTRIBUTION FOR SOLUTIONS OF SDE’S WITH INTERACTION 3

Lemma 1. The random elements x(u, t) belongs to ∩_{p,k≥1}W_{p,k}(ℝ^{d}) and their stochastic derivatives

y_{s}(u, t) := D_{s}x(u, t)

satisfy the following SDE with interaction (3)

\begin{matrix}y_{s}(u, t) = B(x(·, s))(u)+ \\ ∫_{s}^{t} ∫_{ℝ^{d}}((b_{0})′_{1}(x(u, z), x(v, z))y_{s}(u, z) + (b_{0})′_{2}(x(u, z), x(v, z))y_{s}(v, z))μ_{0}(dv)dz+ \\ + ∑_{k=1}^{m} ∫_{s}^{t} ∫_{ℝ^{d}}((b_{k})′_{1}(x(u, z), x(v, z))y_{s}(u, z) + (b_{k})′_{2}(x(u, z), x(v, z))y_{s}(v, z))μ_{k}(dv)dw_{k}(z) \end{matrix}

where B is defined in (2), s ≤ t.

Besides ∀p ≥ 1 ∃C_{p} > 0 ∀s_{1}, s_{2} ∈ [0, T] ∀t ∈ [0, T], t ≥ s_{1} ∨ s_{2} ∀u ∈ ℝ^{d} :

(4)  E∥y_{s_{1}}(u, t)∥^{p} ≤ C_{p};
(5)  E∥y_{s_{1}}(u, t) − y_{s_{2}}(u, t)∥^{p} ≤ C_{p}|t_{1} − t_{2}|^{p/2}.

The proof of the Lemma 1 is quite standard. So let us only outline the main steps.

Step 1. ∀p ≥ 1 ∃C_{p} ∀s_{1}, s_{2} ∈ [0, T] ∀u_{1}, u_{2} ∈ ℝ^{d} :

(6)  E∥x(u, s_{1})∥^{p} ≤ C_{p},
(7)  E∥x(u_{1}, s_{1}) − x(u_{2}, s_{2})∥^{p} ≤ C_{p}(|s_{1} − s_{2}|^{p/2} + ∥u_{1} − u_{2}∥^{p/2}).

To prove this the technique of [6] for stochastic flows is used again with estimations of [3].

Step 2. There exists a solution of (3) that satisfies (4), (5).

The proof can be done using (6), (7), analogously to a proof of Step 1.

Step 3. Let φ_{n}(t) = \frac{k}{2^{n}} , where t ∈ [\frac{k}{2^{n}}; \frac{k+1}{2^{n}}) . Denote by x_{n}(u, t) the solution of

\begin{aligned}dx_{n}(u, t) & = B_{0}x_{n}(·, φ_{n}(t))(u)dt + ∑_{k=1}^{n}(B_{k}x_{n}(·, φ_{n}(t)))(u)dw_{k}(t) \\ x_{n}(u, 0) & = φ(u), \end{aligned}

where B_{k} are defined in (2).

Analogously to the case of diffusion equations (without interaction) [5] it can be proved that

1) x_{n}(u, t) lies in ∩_{p,k≥1}W_{p,k}(ℝ^{d});

2) for all u, t a sequence x_{n}(u, t) converges to x(u, t) in all L_{p}, p ≥ 1 uniformly in

u ∈ ℝ^{d};

3) for all m ∈ ℕ s_{1}, …, s_{n} ∈ [0, t], u ∈ ℝ^{d} there exists a limit of D_{s_{1}} … D_{s_{m}}x_{n}(u, t)

in each L_{p}, p ≥ 1 uniformly in s_{1}, …, s_{n}, t, u.

4) D_{s}x_{n}(u, t) converges to y_{s}(u, t) which is defined in (3).

Combining results of Steps 1 – 3 we get the proof of the Lemma 1.

The application of the following proposition to the Theorem 2 and Lemma 1 completes a proof of the Theorem 1.

Lemma 2. Assume that a random vector ξ = (ξ_{1}, …, ξ_{j}) ∈ ∩_{p,k≥1}W_{p,k}(ℝ^{j}) satisfies the following conditions

1. ∀α > 1∃p = p(α) > 0∃c = c(α) > 0∀s, t ∈ [0, T] :

(8)  E∥D_{s}ξ − D_{t}ξ∥^{p} ≤ c|t − s|^{α};

2. ∃t_{0} ∈ [0, T]∃A > 0 :

(D_{t_{0}}ξ)(D_{t_{0}}ξ)^{∗} ≥ Aid_{ℝ^{j}} a.s.,
where id_{ℝ^{j}} is identity matrix in ℝ^{j}.

Then the distribution of ξ in ℝ^{j} has C^{∞}-density.

Proof. By a Theorem 2 we have only to verify that

(9)  det (∫_{0}^{T} D_{s}ξ(D_{s}ξ)^{∗}ds) ∈ ∩_{p≥1}L_{p}.

The inequality (8) implies the existence of continuous in s version of D_{s}ξ. Moreover [7]: ∀α > 1 ∃k = k(α) ∀ε > 0 :

E sup_{|s−t_{0}|<ε} ∥D_{s}ξ − D_{t_{0}ξ}∥^{p(α)} ≤ Kε^{α}.

Observe that

\begin{aligned}D_{s}ξ(D_{s}ξ)^{∗} & ≥ \frac{1}{2}D_{t_{0}}ξ(D_{t_{0}ξ})^{∗} − (D_{s}ξ − D_{t_{0}}ξ)(D_{s}ξ − D_{t_{0}}ξ)^{∗} ≥ \\ & ≥ \frac{A}{2}id_{ℝ^{j}} − (D_{s}ξ − D_{t_{0}ξ})(D_{s}ξ − D_{t_{0}ξ})^{∗}. \end{aligned}

So for each α > 0 :

P (det ∥Dξ(Dξ)^{∗}∥ < (\frac{A}{4})^{j} ε) ≤

≤ P (∃s ∈ [0, T], |s − t_{0}| < ε : D_{s}ξ(D_{s}ξ)^{∗} ≥ \frac{A}{4}id_{ℝ^{j}}) ≤

≤ P ( sup_{|s−t_{0}|<ε} ∥D_{s}ξ − D_{t_{0}}ξ∥ > \frac{A^{1/2}}{2} ) ≤

≤ \frac{E sup_{|s−t_{0}|<ε} ∥D_{s}ξ − D_{t_{0}}ξ∥^{p(α)}}{(A^{1/2}/2)^{p(α)}} ≤ const ·ε^{α}.

Thus

P (det D_{ξ}(Dξ)^{∗} < ε) = o(ε^{α}), ε → 0+

for each α > 0 and therefore (9) is satisfied.

Lemma 2 is proved.

By (3) we have

D_{t}πx(u, t) = πB(x(·, t))(u).

So all conditions of the Lemma 2 are satisfied with ξ = πx(u, t), t_{0} = t, j = dim πℝ^{d}, ℝ^{j} = πℝ^{d}, p(α) = 2α (see (5)).

Theorem 1 is proved.

References

1. Bell D.R., Degenerate stochastic differential equations and hypoellipticity, Pitman Monographs and Surveys in Pure and Applied Mathematics 79 (1995).

2. Belopolskaya Ya.I., Daletsky Yu.L., Smoothness of transition probabilities of Markov processes described by stochastic differential equations in Hilbert space, Reports of Ukrainian Academy of Sciences 9 (1994), 40-45.

3. A.A.Dorogovtsev, P.Kotelenez, Smooth stationary solutions of quasilinear stochastic partial differential equations: 1.Finite mass, Probability theory and its applications (to appear) (2002).

4. Takeuchi A., The Malliavin calculus for SDE with jumps and the partially hypoelliptic problem, Osaka journal of mathematics(to appear) (2002).

5. Watanabe S., Lectures on stochastic differential equations and the Malliavin calculus, TATA Inst. of Fundamental Research, 1984.

6. H.Kunita, Stochastic flows and stochastic differential equations, vol. 24, Cambridge studies in advanced mathematics, 1990.

7. Buldygin V.V., Kozachenko Yu.V., Metric characterization of random variables and random processes, vol. 188, Amer. Math. Soc, Providence, 1998.

Institute of Mathematics,National Academy of Science of Ukraine, Tereshchenkivska 3, 252601, Kiev, Ukraine

E-mail address: apilip@imath.kiev.ua