TWO-PARAMETER SEMIGROUPS , EVOLUTIONS AND THEIR APPLICATIONS TO MARKOV AND DIFFUSION FIELDS ON THE PLANE

We study two-parameter coordinate-wise C0-semigroups and their generators, as well as two-parameter evolutions and differential equations up to the second order for them. These results are applied to obtain the Hi]le-Yosida theorem for homogeneous Markov fields of the Feller type and to establish forward, backward, and mixed Ko]mogorov equations for nonhomogeneous diffusion fields on the plane.


Introduction
Let Ttl,t 2 be a two-parameter coordinate-wise C0-semigroup.The paper is organized as follows.In Section 2, we prove that Ttl't2-Tl'12t -Ttlt2,1 and establish that its generator coincides with the generator of the one-parameter semigroup T1, t.We also derive differential equations up to the second order to Ttl,t 2 and its resolvent and establish Hille-Yosida theorem for Ttl,t 2 (Remark 2).In the third section we consider two-parameter evolution operators, Tss,,tt, up to the second order.In the fourth section we study .-Markovfields on the plane with transition functions and present the Hille-Yosida theorem for .-Markovfields of the Feller type.In the fifth section the class of diffusion fields is introduced.The form of generators and relations between them are established.Forward, backward, and mixed Kolmogorov equations of the second order for the densities of diffusion fields are presented.A partial case of backward Kolmo- gorov equations was considered in [3,4].
2. Two-Parazneter Semigroups, Their Generators, and Resolvents Let X be a complex Banach space, (X) be a space of linear continuous operators from X to X, I be the identity operator on X, and D(A) be the domain of operator A. [0 +cxz) 2 with partial ordering <(< ) if -(81 82) -(t 1 t2) and < < )t, i-1,2.
Definition 1: The family {Ttl, t2, (tl, t2) G 2+ } C (X) is called a coordinate-wise two-para- meter semigroup if it satisfies the following two conditions.
Ttl is called a Co-semigroup if for any x E X, E N 2 Definition 2: A semigroup (B1) For any x G X, t 1 >0, t 2>0, (B2) For any x G X, lim T lim T ,t2x-x.t2--*0 1 t2X tl--+0 1 lim Ttl t2x x.I v t2--,0 The proof of Lemma 1 is similar to that of the classical theorem about continuity of separate- ly continuous bilinear forms [8] when we replace functionals by operators, so it is omitted.
Lemrna 2: The semigroup Ttl,t 2 is a Co-semigroup if and only if it satisfies one of the condi- lions (B1) or (B2).
Proof: The necessity is obvious.Let us prove sufficiency.Suppose, for example (B1) is satisfied.Then the one-parameter semigroups, Ttl., and T ,t 2 are continuous for any fixed t 1 and t 2. From known properties of one-parameter semigroups, for any t > 0, 1,2, there exist constants C Ci(ti) > 0 and a ai(ti) G gt such that I[ Ttl,u 11 <-C1 ealu and I I Tu, 2 11 <- for any u > 0. Now let be fixed with t > .Then, from Lemma 1, we find that Other version of the arrangement of the point with respect to are considered similarly.
n--,c 1,t2(Pn/qn)    Definition 3: 1.The generator A of Co-semigroup T is defined by whenever the limit exists.
Theorem 1" Let T T be a Co-semigroup.Then the following hold" 1) A A I A and AtJ tiA t > O, i-1,2, j 1, 2, i j.

2)
For any x G D(A), T 7 Az AT( x.

TZ Otltj
For any x G D(A2), 02T7 ATi x + t lt2A2T7 .(2) exist or do not exist simultaneously.Therefore, according to Definition 3, D(A)-D(At2,), D(A21) C D(A) and At 2 x flAX.From the same arguments applied to x e D(A) and x e D(A), we have D(A11) C D(A) C D(A) D(A), and, consequently, D(A)= D(A21) D(A).There- fore, the equalities (2) hold 2. Operators A and Ttl commute on D(A) (this follows from the corresponding properties of one-parameter semigroupsl; therefore, A A and Ty Ttlt2,1 commute on D(A)= D(A).
3. This statement can be obtained by direct calculations.

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Suppose the semigroup T is not continuous on the whole space X.In this case, let us con- sider the linear manifold, X 0 { Ix G X Ilim T. x limT/ v x x for tl, t 2 > 0}.
2) Operators Ty act from X 0 to X O.
The proof follows from equality (1) and similar results for one-parameter semigroups.

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Theorem 2: The linear operator A is a generator of a coordinate-wise Co-semigroup if and only if it is a generator of a one-parameter Co-semigroup.
V1 lemark 2: It follows from Theorem 2 that the conditions of the well-known Hille-Yosida theorem are necessary and sufficient for the closed operator A with D(A) X to generate coordin- ate-wise semigroup.
Remark 3: The statement similar to Theorem 2 for an n-parameter coordinate-wise semigroup is true and would have the same proof.
It is well known that in the one-parameter case, the Laplace transform of semigroup is a resolvent of its generator, defined in the appropriate half-plane of C. Analogously, in the case of the multiplicative semigroup Ttl,t2, given by equations, Ttl,t 2 Tl(tl)72(t2) and Tl(tl)T2(t2)-T2(t2)Tl(tl) where T i(ti) for i-1,2 is a one-parameter semigroup, the two-dimensional Laplace transform of T(tl, t2) is decomposed into a product of one-dimensional transforms and is the product of resol- vents of semigroup generators.There are no such simple relations for coordinate-wise semigroups.
In this vein, we can obtain only the following result.
Theorem 3: Let {T, @ N2+} be a contractive coordinate-wise semigroup (this assumption is made for the sake of simplicity), and let Lz, w-Lz, w(f) and L z Lz(g be two-and one-dimen- sional Laplace transform of functions f and g, respectively.Then, the following hold.1) For any z, w > O, Lz, w(Ttl,t2)-Ll,zw-Ll(R(zw)/t2) where R is a resolvent of the generator of the semigroup T1, t. 2) For any x E D(A2), A202Lz, w   OzOw + AL , zwL , ,
3) Finally, from the equality L(u) L1, u with u-zw, we obtain that We call any operator, Tss,,tt, in this family a two-parameter evolution operator (or simply an evolution).
Definition 4: The family of evolutions is said to be continuous if, for any 0 _ Further, we consider only continuous families of evolutions.Let us denote (u,a) + (u, u + a), (u, a) (u a, u), a>O; ss',(t,k) -t- (s(Ot Definition 5: 1) The elements of the family of operators {Ast + }, defined as A + +/- lim ,l_---r[:lTt,kx, s, x" h, k-On considered on the sets where corresponding limits exist, are called generators of evolutions.
2) The elements of the family of operators {A 1' A2, +/- ss't } defined as stt Al-t-t s,t, x" lim A1T + A' + h, , t, ' and x: lim AT +/- ss k, s, , x considered on the sets where corresponding limits exist, are called i-generators (i 1,2) of evolutions.If A i' + =A i'-or A + + =A +-=A-+ =A--, then we denote the common value as A or A respectively.
Definition 6: Right and left derivatives of evolutions are defined as O+T ss tt' lim(Tss, + h, tt' Tss' tt') and ss', tt' lim(T
In similar ways, one can define right and left derivatives of other families of operators, depending on s, s', t and t'.
Sufficient conditions for (9) can be formulated in a similar way.mark 5: Let A ' + A i' -, 1,2 and let families 1 of operators {Astt, (s, t, t') e N } and {As,t(s s', t) e N } be continuously differentiable in (t, t') and (s, s') respectively on the set s, s t, t, Then one can write equalities 1) through 4) of Lemma 6 in the form: c) There exists the limit, limA;,; (51T,t,t')T', tt,z.
Remark 6: The following conditions are sufficient for (El) d).
Proofi We prove only the statement 1) of Theorem 4.
The following statements are proved analogously to the proof of Theorem 4.
Theorem 5: 1) Let the following conditions hold.
c) There exists C > O such that [I Tus,tt l] < C while u G [s-5, s] for some S > O.
Then ss tt

Markov Fields and Semigroups
Let (f,F,P) be a complete probability space; let (E,g) be a measurable space; let X- {X)-, e N2} be a stochastic field with the values in E that is constant on the set (N2\R2+) U {[0, c)x{0}}U{{0}x[0,c)}.Put F7 =(r{x,g <}VN, F V F-r and F =FIVF 2 t.>0 t2 where N is the class of P-zero sets of F. -Definition 7: The field X is called an ,-Markov field if for any g _< t and B E g P{X e B/F) P{X e B/Xs, Xslt2, Xtls2 } [5, 11].
Definition 9: X is called an ,-Markov field with transition function P, if for any m >_ 1 and 1, with Bij e for i-1, m and for j 1, n, with (si, tj) G N2+, we have Pin ":N " (x -j fI I (under the assumption that the right-hand sides do not depend on x).In this case, the collection (p10, p20, pit, p2s) is called an ,-transition function on (E, g).
s 1 s, tf(x) and s, tl, tf( s, 1 tf(x), where Ts, tf(x) s, We denote their common value as T s, tf(x ).__Then T s, tf(x )__ is a coordinate-wise contractive semi- group on the space B(E) of bounded measurable functions f'E--,R.Further, we consider only homogeneous fields.
Definition 10: Transition function P(s,t;x,B) is said to be continuous in probability (Pcontinuous), if for any > 0 vlimt0 P (s, t,x, U(x)) 1, where U(x) is any C-neighborhood of x. ,-Markov field with a P-continuous transition function will be called a P-continuous field.
The index-will be omitted.
Let us denote CB(E C B(E) as the space of continuous bounded functions on E.
Lemma 7: The following conditions are equivalent. (F1) The field X is P-continuous.
(F3) lim T u vf(x) f(x) for any f G CB(E and x G E.
Definition 11: Transition function P(s t,x,B) is said to be Feller, if for any N 2 T 7 (CB) C C B. The corresponding ,-Markov ned win be called a Feller field. (Note that if E is a compact set, then CB(E C(E), where C(E)is the space of continuous functions.) Theorem 6: 1) Let X be a P-continuous field.Then T-i Tl,tlt2 Ttlt2,1 on CB(E). 2) Let E be a compact set and X be a P-continuous Feller field.Then Ti is a Co-semigroup on C(E).

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The Hille-Yosida theorem for Feller fields on compact sets is similar to the one-parameter case.
Theorem 7: An operator A with domain D(A) that is dense in C(E) generates a P-contin- uous Feller field on the compact set E if and only if the following conditions are satisfied.
Proof: If A generates a P-continuous Feller field, then from Theorem 1, A is a generator of a Feller one-parameter semigroup, T Tl,t, and necsity follows.If assum..ption (G) is satisfied, then from Theorem 1, there exists a semigroup Tt, tO, such that Tt(C(E))C C(E) and Ttf---,f as t--0 for any f C(E) [2, p. 167].Let Ts, Tst.Then Ts, t(C(E)) C C(E).Since T s, tf(x), for any s,t and x fixed, is a linear functional on C(E), then there exists a measure P(s, t, x, B) on such that Ts, tf(x) f f(y)P(s, t, x, B).Moreover, P(s, t, x, E) 1 and P is a transition function by the semigroup property of T s, t.Now, as with the proof of Lemma 7, choose f(y) c > 0 with y E\U(x), f(x) 0 and f C(E).Then P(s,t,x, Ue(x)) <_ ct-lTs, tf(x) c-lTl,stf(x)--c-lf(x) 0 as s V t0, i.e., the transition function is P-continuous.The construction of an .-Markovfield with transi- tion function P, under the assumptions of its Feller property and P-continuity, is realized in [6].
Definition 12: An ,-Markov field with transition function, t', B} P(s, t s', t', x, B): P{(s, t) (s', x, x, x, with (s, t), (s', t') e 2+, (s, t) _ (s', t'), x e n, and B e %(n), is called a diffusion field, if the following conditions are true for any > 0 uniformly in x K where K is any compact set, K C n.
where c I(x,y)[ <_ n e for Y-x < 6, into the following estimations sup I(hk)-I(T The proof follows from ( 13)and ( 14).
zwLz, w x-w e dz z e + z zw, w .
-Parameter Evolution Operators and Their Generators Let us consider the family of operators, t'{Tss,,tt,O_s_s',O_t_t'}C(X s', EU+, satisfying the following conditions.
lim (As, k, t' s, t, Two-Parameter Semigroups, Evolutions and Their Applications 289 Remark 4: The following conditions are sufficient for (8).
It follows from [11] that any ,-Markov field with transition function is a Markov field.
the following conditions are satisfied.(K3)a) A i-i on the set T for i= O, 1.{ o o o o} c(4+ b) The set p,-s' at" at-7-s --' C x and satisfies (C, )-condition. 7 ).
2) Let the following conditions hold.