ON WEAK SOLUTIONS OF RANDOM DIFFERENTIAL INCLUSIONS

In the paper we study the existence of solutions of the random differential inclusion 
x˙t∈G(t,xt)     P.1,t∈[0,T]-a.e.x0=dμ, 
where G is a given set-valued mapping value in the space Kn of all nonempty, 
compact and convex subsets of the space ℝn, and μ is some probability measure 
on the Borel σ-algebra in ℝn. Under certain restrictions imposed on F and μ, we 
obtain weak solutions of problem (I), where the initial condition requires that the 
solution of (I) has a given distribution at time t=0.

lar, random cases were considered in [3], [5], [7].This work deals with the inclusion with a purely stochastic initial condition.First, we recall several notions and results needed in the sequel.Let Kc(S be the space of all nonempty compact and convex subsets of a metric space S equipped with the Hausdorff metric H (see e.g., [1], [4])" H(A,B)-max([I(A,B),(B,A));A,B E gc(s), where H(A,B)-sup inf p(a b).By [IA [[ we denote the distance H(A,O).For S being a aEAbEB separable Banach space, (Kc(S),H) is a polish metric space.
Let I-[0, T], T > 0. For a given multifunction G:I--.Kc(S by DHG(to) we denote its h both equal to the same set DHF(Io) Kc(S).
1This work was supported by KBN grant No. 332069203.
For S-R n and K n-Kc(Rn), we denote by C I -C(I,Kn) the space of all H-continuous set- valued mappings.In C I we consider a metric p of uniform convergence p(F, G): sup H(X(t), Y(t)), for X, Y C T. 0<t<T Then C1 is a polish metric space.
Let (f2, , P) be a given complete probability space.We recall now the notion of a multival- ued stochastic process.The family of set-valued mappings X-(Xt) > 0 is said to be a multival- ued stochastic process if for every t > 0, the mapping Xt: f2K n is me-surable, i.e., Xt(U): {w: Xt(w gl U :/: 0} , for every open set U C_ n (see e.g., [1,4]).It can be noted that U can be also chosen as closed or Borel subset.We restrict our interest to the case when 0 _< t < T, T > 0.
If the mapping t-,Xt(w is continuous (H-continuous) with probability on (P.1), then we say that the process X has continuous "paths."Let us notice that the set-valued stochastic process X can be though as a random element X: ft--C I. Indeed, it follows immediately from [3] and from the fact that the topology of the uniform convergence and the compact-open topology in C I are the same.
Definition 1: A probability measure # (on C i) is a distribution of the set-valued process X (Xt) o < < T if one has #(A) P((A)) for every Borel subset A from C I.
A distribution of X will be denoted by pX.Definition 2: A set-valued mapping F: I x Kn--K n is said to be an integrably bounded of the Caratheodory type if: T 1) there exists a measurable function m:I-,R+ such that fm(t)dt<oc and I! F(t, A)II <_ re(t) )is H-continuous t-a.e. 3) F(. ,A) is a measurable multifunction for every A Kn.
Let us consider now the multivalued random differential equation: DHX F(t, Xt) P.1, t e [0, T]-a.e. (II) Xo d where the initial condition requires that the set-valued solution process X (Xt) E I has a given distribution # at the time t =0.By a weak solution of (II) we understand a system (,,P(Xt)tE I) where (Xt)te I is a set-valued process on some probability space (,,P)such that (II) isomer.
Theorem 1: Let F: I x Kn---K n be an integrably bounded set-valued function of the Caratheo- dory type and let # be an arbitrary probability measure on the space Kn.Then there exists a weak solution of (II).

Weak Solutions of Random Differential Inclusions
As an application of Theorem 1, we show the existence of a weak solution of the random differential inclusion k G(t, xt) P.1, t [0, T]-a.e. () The weak solution of (I) is understood similarly as above, where # is now a given probability mea- sure on Let T o denote the family of nonempty open subsets of Rn, and let C-{Cv; V E if0}, where C V {K Kn: K 3 V :/: 0}.Then we have that %n_ g(C) (see e.g.Proposition 3.1 [4]), where %n is a Borel g-field induced by the metric space (Kn, H).
Lemma 1: The following hold true" i) The property i) is obvious.Let V1, V2,... G 0 be such that A n-C V for n-1, 2, To establish ii), let us observe that [.
Let us suppose that iii) does not hold.Then for some k >_ 1, V k V k + 1" Hence there exists a point xGV k such that xV k+l.But then {x}ECvk and {x}CYk+l contradicts to Cyk C_ Cyk + .
To obtain our main result we need the following lemma: Lemma 2: If is a probability measure on the Borel g-algebra (Rn), then there exists a probability measure fi on the space g n such that "fi(Cv) #(V), Y G o" Proof: Let C be the family generating Borel r-field %n.We define a set-function u on C by u(Cy) #(Y).Let us observe that u is well-defined.Indeed, if Cv1 CV2 and #(Yl) /z(V2) then V 1 -7(= V 2. Hence V I\V 2 0 or V2\V 1 .Without loss of generality we may assume the first case.Then there exists x V 2 such that x V 1.But then {x} CV2 and {x} Cv1 which contradicts with an equality Cv1-Cv2.Similarly, it can be shown that if the sets Cv1 and Cv2 are disjoint, then the sets V1, V 2 have the same property too.Hence we get u(C yl tA Cv2 u(Cyl + u(Cy2 for disjoint Cv1 and Cv2.From Lemma 1 we conclude that,, if Cv1 C_ Cv2 C_..., then oo oo U Cv eCandu(U Cv )-limu(Cy ).Finally let us observe that u is g-subadditive.Next we define another set Moreover, u(Kn) 1.
function as follows: (A)" -inf{u(D)" A C D,D C}, A C Kn.
Standard calculations show that " is an outer measure on Kn.Thus from the Caratheodory Theorem, is a probability measure on the g-field of -measurable subsets in Kn.Setting - [%n, we obtain a desired probability measure.
We now present the following existence theorem.
Theorem 2: Let us suppose that G" I x n--,Kn is an integrably bounded multifunction of the Caratheodory type.Then for any probability measure p on Rn, there exists a weak solution of problem (I).
Proof." Lemma 2 yields the existence of a probability measure on the metric space (K n, H) with the property: fi(Cy)-#(Y), Y o" Let F:I x KnK n be a multifunction defined by F(t,A)--bG(t,A), for A g n.Hence from Lemma 1.1 [9], the set-valued mapping F is integrably bounded of the Caratheodory type too.Consequently, by Theorem 1, there exists a probability space (gt,Y,P) and the set-valued stochastic process X-(Xt) 0 <t< T (on it) with continuous "paths" and with values in K n which is a weak solution of the equation DHX F(t, Xt) P.l,t E [0, T]-a.e.Xo d From Kuratowski and Ryll-Nardzewski Selection Theorem [4] we can choose :fn as a measurable selection of X 0. Then by Theorem 4 [5] (see also [3]), there exists a stochastic process x (zt) o < < T as a selection of X that is a solution (in strong sense) of the random differential inclusion't G(t, xt) P.1, t [0, T]-a.e.To complete the proof, it is sufficient to show that z 0 d #; Let us notice that {W:Xo(W V} {w: (w) E V} C {w: X 0 VI Y # q}}, Y 0" Because of X0 a-and "fi(Cv) #(Y) we have Px(v) < #(V). (.) Using regularity properties of probability measures (on a separable metric space) (see e.g., Th. 1.2 [8]), we have that PX(B)-inf{pX(V): B C V, V 0} and #(B)= inf{#(V):B C V,V 0} for every Borel subset B of n.
we get PX(B)< It(B).But P x and It are probability measures. equgl.
Hence from inequality (,) Therefore they have to be w)= {(w)} for w e f.