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Stochastic processes on non-Archimedean spaces. III. Stochastic processes on totally disconnected topological groups

Stochastic processes on totally disconnected topological groups are investigated. In particular they are considered for diffeomorphism groups and loop groups of manifolds on non-Archimedean Banach spaces. Theorems about a quasi-invariance and a pseudo-differentiability of transition measures are proved. Transition measures are used for the construction of strongly continuous representations including irreducible of these groups. In addition stochastic processes on general Banach-Lie groups, loop monoids, loop spaces and path spaces of manifolds on Banach spaces over non-Archimedean local fields also are investigated.


Introduction.
This part is the continuation of the previous two [15,16], where stochastic processes on Banach spaces over local fields and stochastic antiderivational equations on them were investigated. This part is devoted to stochastic processes on a totally disconnected topological group which is complete, separable and ultrametrizable. In particular stochastic processes on diffeomorphism groups and loop groups of manifolds on Banach spaces over a local field are considered. These groups were defined and investigated in previous articles of the author [17,19,20,23,21]. These groups are non-locally compact and for them the Campbell-Hausdorff formula is not valid (in an open local subgroup). In this article topological groups satisfying locally the Campbell-Hausdorff formula also are considered.
Finite-dimensional Lie groups satisfy locally the Campbell-Hausdorff formula. This is guarantied, if to impose on a locally compact topological Hausdorff group G two conditions: it is a C ∞ -manifold and the following mapping (f, g) → f • g −1 from G × G into G is of class C ∞ . But for infinitedimensional G the Campbell-Hausdorff formula does not follow from these conditions. Frequently topological Hausdorff groups satisfying these two conditions also are called Lie groups, though they can not have all properties of finite-dimensional Lie groups, so that the Lie algebras for them do not play the same role as in the finite-dimensional case and therefore Lie algebras are not so helpful. If G is a Lie group and its tangent space T e G is a Banach space, then it is called a Banach-Lie group, sometimes it is undermined, that they satisfy the Campbell-Hausdorff formula locally for a Banach-Lie algebra T e G. In some papers the Lie group terminology undermines, that it is finitedimensional. It is worthwhile to call Lie groups satisfying the Campbell-Hausdorff formula locally (in an open local subgroup) by Lie groups in the narrow sense; in the contrary case to call them by Lie groups in the broad sense.
In this article also theorems about a quasi-invariance and a pseudodifferentiability of transition measures on the totally disconnected topological group G relative to the dense subgroup G ′ are proved. For measures on Banach spaces over locally compact non-Archimedean fields their quasiinvariance and pseudo-differentiability were investigated in [18] (see also [21,23] for diffeomorphism and loop groups, but for measures not related with stochastic processes). In each concrete case of G it its necessary to construct a stochastic process and G ′ . Below path spaces, loop spaces, loop monoids, loop groups and diffeomorphism groups are considered not only for finite-dimensional, but also for infinite-dimensional manifolds.
In particular, loop and diffeomorphism groups are important for the development of the representation theory of non-locally compact groups. Their representation theory has many differences with the traditional representation theory of locally compact groups and finite-dimensional Lie groups, because non-locally compact groups have not C * -algebras associated with the Haar measures and they have not underlying Lie algebras and relations between representations of groups and underlying algebras (see also [22]).
In view of the A. Weil theorem if a topological Hausdorff group G has a quasi-invariant measure relative to the entire G, then G is locally compact. Since loop groups (L M N) ξ are not locally compact, they can not have quasi-invariant measures relative to the entire group, but only relative to proper subgroups G ′ which can be chosen dense in (L M N) ξ , where an index ξ indicates on a class of smoothness. The same is true for diffeomorphism groups.
In this article classes of smoothness of the type C n by Schikhof are used. We recall shortly their definition. Let K be a local field, that is, a finite algebraic extension of the p-adic field Q p for the corresponding prime number p [32]. For b ∈ R, 0 < b < 1, we consider the following mapping: j b (ζ) := p b×ordp(ζ) ∈ Λ p for ζ = 0, j b (0) := 0, such that j b ( * ) : K → Λ p , where K ⊂ C p , C p denotes the field of complex numbers with the non-Archimedean valuation extending that of Q p , p −ordp(ζ) := |ζ| K , Λ p is a spherically complete field with a valuation group {|x| : 0 = x ∈ Λ p } = (0, ∞) ⊂ R such that C p ⊂ Λ p [7,29,30,32]. Then we denote j 1 (x) := x for each x ∈ K. Let us consider Banach spaces X and Y over K. Suppose F : U → Y is a mapping, where U ⊂ X is an open bounded subset. The mapping F is called differentiable if for each ζ ∈ K, x ∈ U and h ∈ X with x + ζh ∈ U there exists a differential such that The space of such F is denoted C(1, U → Y ). Let be partial difference quotients of order b for 0 < b < 1, x + ζh ∈ U, ζh = 0, Φ 0 F := F , where Y Λp is a Banach space obtained from Y by extension of a scalar field from K to Λ p . By induction using Formulas (1 − 4) we define partial difference quotients of orders n + 1 and n + b: and derivatives F (n) = (F (n−1) ) ′ . Then C(t, U → Y ) is a space of functions F : U → Y for which there exist bounded continuous extensionsΦ v F for each x and x + ζ i h i ∈ U and each 0 ≤ v ≤ t, such that each derivative are the integral and the fractional parts of t = n + b respectively. The norm in the Banach space C(t, U → Y ) is the following: where 0 ≤ t ∈ R, sign(y) = −1 for y < 0, sign(y) = 0 for y = 0 and sign(y) = 1 for y > 0. It is necessary to note that there are quite another groups with the same name loop groups, but they are infinite-dimensional Banach-Lie groups of mappings f : M → H into a finite-dimensional Lie group H with the pointwise group multiplication of mappings with values in H. The loop groups considered here are geometric loop groups.
On the other hand, representation theory of non-locally compact groups is little developed apart from the case of locally compact groups. For locally compact groups theory of induced representations is well developed due to works of Frobenius, Mackey, etc. (see [10] and references therein). But for non-locally compact groups it is very little known. In particular geometric loop and diffeomoprphism groups have important applications in modern physical theories (see [23,21] and references therein).
Then measures are used for the study of associated unitary representations of dense subgroups G ′ .
2 Stochastic antiderivational equations and measures on totally disconnected topological groups.
To avoid misunderstandings we first remind our definitions from [20,23,21]. 2.1. Definitions and Notes. 1. Let X be a Banach space over a local field K. Suppose M is an analytic manifold modelled on X with an atlas At(M) consisting of disjoint clopen charts (U j , φ j ), j ∈ Λ M , Λ M ⊂ N. That is, U j and φ j (U j ) are clopen in M and X respectively, φ j : (1) c 0 (α, K) := {x = (x i : i ∈ α)|x i ∈ K, and for each ǫ > 0 the set and the standard orthonormal base (e i : i ∈ α) [29], α is considered as an ordinal due to the Kuratowski-Zorn lemma, α ≥ 1. Its cardinality is called a dimension card(α) =: dim K c 0 (α, K) over K. Then 2.1.2. Let X, Y and M be the same as in §2.1.1 for a local field K. We denote by C 0 (t, M → Y ) a completion of a subspace of cylindrical functions restrictions of which on each chart f | U l are finite K-linear combinations of functions {Qm(xm)q i | U l : i ∈ β, m} relative to the following norm: ( where multipliers J l (t, m) are defined as follows: 2.1.3.a. Let N be an analytic manifold modelled on Y with an atlas is the index of a class of smoothness, that is, for each admissible (i, j): with * empty or an index * taking value 0 respectively, are non-void clopen subsets. We denote by In view of Formulas (1 − 4) we supply it with an ultrametric Let M and N be two analytic manifolds with finite atlases, We denote by C θ 0 ((t, s), M → N) a completion of a locally K-convex for each integer γ such that 1 ≤ γ ≤ s and for each v ∈ {[t] + γn, t + γn}. 2.1.4. For infinite atlases we use the traditional procedure of inductive limits of spaces. For M with the infinite atlas, card(Λ M ) = ℵ 0 , and the Banach space Y over K we denote by C θ * (ξ, M → Y ) for ξ = t with 0 ≤ t ≤ ∞ or for ξ = (t, s) a locally K-convex space, which is the strict inductive limit where E ∈ Σ, Σ is the family of all finite subsets of Λ M directed by the inclusion E < F if E ⊂ F , U E := j∈E U j (see also §2.4 [20]). For mappings from one manifold into another f : M → N we therefore get the corresponding uniform spaces. They are denoted by C θ * (ξ, M → N). We introduce notations (1) has a refinement At ′ (M ) which is countable and its charts (Ū ′ j ,φ ′ j ) are clopen and disjoint and homeomorphic with the corresponding balls B(X, y j ,r ′ j ), where (2)φ ′ j :Ū ′ j → B(X, y ′ j ,r ′ j ) for each j ∈ Λ ′M are homeomorphisms (see [8,24]). ForM we fix such At ′ (M ).
We define topologies of groups G i (ξ,M) and locally K-convex spaces C * (ξ,M → Y ) relative to At ′ (M ), where Y is the Banach space over K. Therefore, we suppose also thatM and N are clopen subsets of the Banach spaces X and Y respectively. Up to the isomorphism of loop semigroups (see below their definition) we can suppose that s 0 = 0 ∈M and y 0 = 0 ∈ N.
For M =M \ {0} let At(M) consists of charts (U j , φ j ), j ∈ Λ M , while At ′ (M) consists of charts (U ′ j , φ ′ j ), j ∈ Λ ′ M , where due to Formulas (1, 2) we define Then we consider their subspaces of mappings preserving marked points: where for s > 0 and ξ = (t, s) in addition Condition 2.1.3.b.(4) is satisfied for each 1 ≤ γ ≤ s and for each v ∈ {[t] + nγ, t + nγ}, and the following subgroup: With the help of them we define the following equvalence relations K ξ : f K ξ g if and only if there exist sequences for each x ∈ M and lim n→∞ f n = f and lim n→∞ g n = g.
It is shown below that Ω ξ (M, N) is the monoid, which we call the loop monoid.
Suppose V is an analytic vector field on N (that is, by definition V | V i are analytic for each chart and V • ψ −1 i has the natural extension from ψ i (V i ) on the balls B(X, a i , r i )). Then by analogy with the classical case we can define the following mappingē xp y (zV ) = y + zV (y) for which by charts such that it is also disjoint and analytic and is the isomorphism. Therefore,ēxp is the locally analytic mapping,ēxp : consequently, where π N : T N → N is the natural projection, * = 0 or * = ∅ (∅ is omitted). Therefore, the following mapping is defined by the formula given below [23]). We introduce the following notation P D c (b, f (x)) for such integral by B(K, 0, 1) instead of the entire K.
2.7. Definitions. Let G be a topological Hausdorff semigroup and Naturally Definitions 2.7 have generalizations, when G is a topological manifold on which a topological group (or a semigroup) G ′ acts continuously from the left G ′ × G ∋ (g, x) → gx ∈ G.
2.8. Note. Now let us describe dense loop submonoids which are necessary for the investigation of quasi-invariant measures on the entire monoid.  3)).
: ψ(s 0 ) = s 0 } and the equivalence relation K ξ,{k} in it for each M and N from §2.1 and §2.2. Therefore, 2.9. Note and Definition. For a commutative monoid Ω ξ (M, N) with the unity and the cancellation property (see [23]) there exists a commutative group L ξ (M, N) equal to the Grothendieck group. This group is the quotient group F/B, where F is a free Abelian group generated by Ω ξ (M, N) and B is a closed subgroup of F generated by elements is injective. We supply F with a topology inherited from the Tychonoff product topology of Ω ξ (M, N) Z , where each element z of F is N) be the loop submonoid as in §2.8 such that c > 0 and c ′ > 0. Then it generates the loop group For Λ = ω 0 we define a Banach spacẽ We introduce the following group which is called the diffeomorphism group (and the homeomorphism group for 0 ≤ t < 1), where Hom(M) is the group of continuous homeomorphisms. Each function f ∈ C 0 (t, M → X) has the following decomposition: i, n, Ord(m) = n, j} is the orthogonal basis, moreover, where β(n) < β(n + 1) for each n and there exists n 0 ∈ N with β(n) = n for each n > n 0 .
We take the following subgroup > n for each n and lim inf n→∞ s(n)/n =: ζ > 1. Then there exists the following ultrametric in G ′ : 2.12. Note. At first it is necessary to prove theorems about the quasiinvariance and the pseudo-differentiability of transition measures of stochastic processes on Banach spaces over local fileds. We consider two types of measures on c 0 (ω 0 , K). The first is the q-Gaussian measure [16]). The characteristic functional of the q-Gaussian measure is positive definite, hence µ is nonnegative (see also §2.6 [18]). The second is specified below and is the particular case of measures considered in §4.3 [15].
Let w be the real-valued nonnegative Haar measure on K with w(B(K, 0, 1)) = 1. We consider the following measure µ on c 0 (ω 0 , K) Let now on the Banach space c 0 := c 0 (ω 0 , K) there is given an operator Hence for each such S there exists the σ-additive measure µ S,S * γ,q and µ S . These measures are induced by the corresponding cylinder measures µ I,γ,q or µ I on K ℵ 0 , where I is the unit operator, since c 0 in the weak topology is isomorphic with K ℵ 0 . Here the algebra U of cylindrical subsets is generated by subsets π −1 V (A), where A is a Borel subset in K n , card(V ) = n < ℵ 0 , V ⊂ N, π V : K ℵ 0 → i∈V K i is the natural projection.
On the space C 0 0 (T, H) = C 0 0 (T, K) ⊗ H let S = S 1 ⊗ S 2 and γ = γ 1 ⊗ γ 2 , where S 1 is a linear operator on C 0 0 (T, K) and S 2 is a linear operator on H, γ 1 ∈ C 0 0 (T, K), γ 2 ∈ H such that the measure µ S,γ,q is the product of measures µ S 1 ,γ 1 ,q on C 0 0 (T, K) and µ S 2 ,γ 2 ,q on H, analogously µ S is the product of measures µ S 1 on C 0 0 (T, K) and µ S 2 on H. With the help of such measures on the space C 0 0 (T, H) the stochastic process w(t, ω) is defined as in § §4.2 and 4.3 [15] and §3.2 [16].
Theorem. Let either Conditions (i − iii) or (i − iv) be satisfied, then the transition measure P y (A) of the stochastic process ξ(t, ω, y) being the solution of Equation either 3.4(i) or 3.5.(i) [16] and depending on the parameter y ∈ V is quasi-invariant relative to each mapping U(y 2 , y; ξ(t, ω, y)) := ξ(t, ω, y 2 ) for each y and y 2 ∈ V .
In view of the preceding consideration lim n→∞ ρ(P n z, x) = ρ(z, x) for µ-almost all x ∈ C 0 0 (T, H), moreover, this convergence is uniform by z in each ball B(L, 0, c) for each finite-dimensional subspace L in X 0,s (C 0 0 (T, H)), whereP n is a projection on a subspace sp K (e 1 , ..., e n ) = K n , where {e j : j} is the orthonormal base in X 0,s (C 0 0 (T, H)). Evidently, X 0,s (C 0 0 (T, H)) is dense in C 0 0 (T, H). Stochastic antiderivational Equation 3.4.(i) [16] is the particular case of 3.5.(i). Therefore, it is sufficient to consider the latter equation. Below it is shown, that the one-parameter family of solutions ξ(t, ω, y) is of class C 1 by y ∈ V . Let X 0 (t, y) = x(y),..., (P u b+m−l ,w(u,ω) l [a m−l+b,l (u, X n (u, y), y) − a m−l+b,l (u, X n−1 (u, y), y)] where t j = σ j (t) for each j = 0, 1, 2, ..., for the shortening of the notation X n , x and a l,k are written without the argument ω, a and E are written without their variables. Then where X n ∈ C 0 0 (B R , H) for each ω, y ∈ V and for each n, K is the same constant as in §3.4, 1 ≤ g < ∞. On the other hand, Due to Condition (ii) for each ǫ > 0 and 0 On the other hand, the partial difference quotient has the continuous extensionΦ 1 (X n+1 − X n )(y; h; ζ), that is expressible throughΦ 1 of a l,k , a and E, and also through a l,k , a and E themselves, where y ∈ V , h ∈ Y , ζ ∈ K such that y + ζh ∈ V , since analogous to (X n+1 − X n ) estimates are true forΦ 1 (X n+1 − X n ). Therefore, there exists the unique solution on each B ǫ and it is of class C 1 by y ∈ V , since sup u,y max( X 1 (u, y) − X 0 (u, y) L q (Ω,H) , Φ 1 (X 1 (u, y) − X 0 (u, y)) L q (Ω,H) < ∞ and lim l→∞ c l C = 0 for each C > 0, hence there exists lim n→∞ X n (t, y) = X(t, y) = ξ(t, ω, y)| Bǫ , where C := M sup u∈Bǫ,y∈V max( X 1 (u, y) − X 0 (u, y) q L q (Ω,H) , Φ 1 (X 1 (u, y) − X 0 (u, y)) q L q (Ω,H) ≤ (c + 1)K < ∞, here B ǫ is an arbitrary ball of radius ǫ in B R , t ∈ B ǫ . Therefore, ξ(t, ω, y) ∈ C 1 (V, L q (Ω, F, λ; C 0 (B R , H)).
From Proposition 3.11 [16] it follows, that the multiplicative operator functional T (t, v; ω; y) is of class C 1 by the parameter y ∈ V such that ξ(t, ω, y) = T (t, v; ω; y)ξ(v, ω, y) for each t and v ∈ T .
Due to the existence and uniqueness of the solution ξ(t, ω, y) for each y ∈ V , there exists the operator U(y 2 , y; ξ(t, ω, y)) := ξ(t, ω, y 2 ), that may be nonlinear by ξ. The variation of the family of solutions {ξ(t, ω, y) : y} corresponds to the differential D y ξ(t, ω, y). Since ξ(t, ω, y) is of class C 1 by y, then U(y 2 , y; ξ(t, ω, y) is of class C 1 by y and y 2 . The operator U(y 2 , y; * ) has the inverse, since U(y, y 2 ; U(y 2 , y; ξ(t, ω, y))) = ξ(t, ω, y) for each y 2 and y ∈ V , t ∈ T and ω ∈ Ω. Therefore, U −1 (y 2 , y; * ) is also of class C 1 by y 2 and y. In view of Conditions (iii, iv) and ξ(t, ω, y 2 ) − ξ(t, ω, y) ∈ X 0,s (H). On the other hand, either µ S,γ,q or µ S is quasi-invariant relative to shifts z ∈ X 0,q (C 0 0 (T, H)) and S = S 1 ⊗S 2 , consequently, the transition measure P y is quasi-invariant relative to shifts z ∈ X 0,s (H). In view of Conditions (ii−iv) ∂U(y 2 , y; η)/∂η −I ∈ L r (H) for each y 2 and y ∈ V , where η ∈ {ξ(t, ω, y) : y}, either r = q or r = 0 respectively. Since µ S (C 0 0 (T, H)) = 1, then P y (H) = 1, hence U(y 2 , y; * ) is defined P y -almost everywhere on H for each y 2 and y ∈ V . Therefore, there exists n such that for each j > n the mappings In view of Theorem 3.28 [18] for each y 2 and y ∈ V the transition measures P y 2 and P y are equivalent.
2.14. Theorem. Let Conditions 2.13.(i − iv) be satisfied and let φ be a C 1 -diffeomorphism of a subset V clopen in K onto the unit ball B(K, 0, 1).

Then
(1) the transition measure P y corresponding to µ S,γ,q is pseudo-differentiable by the parameter y = φ(z) of order b ∈ C for each Re(b) ≥ 0, where z ∈ V ; (2) P y corresponding to µ S with h k such that k δ k < ∞, where δ k := sup x∈B(K,0,1) |P D c (b, h k (x))|, is pseudo-differentiable by the parameter y = φ(z) of order b for each b ≥ 0, moreover, P y is pseudo-differentiable for each b ∈ C, when each f k is locally constant, that is, h k = 0 for each k ∈ N.
Proof. Up to a constant multiplier the operator P D c (b, h(x)) of §2.7 coincides with the pseudo-differential operator D b (h(x)Ch B(K,0,1) (x)) from §9 [31], where Ch A is the characteristic function of the subset A in K. If ψ ∈ L 2 (K, w, C) and b > 0, then due to the Cauchy-Schwarz-Bunyakovskii inequality there exists K\B(K, 1 ((x, y))w(dx) is the Fourier transform [31,29] (see also §3.6 [15]). In view of Theorem 7.4 [31] the Fourier transform f → F [f ] is the bijective continuous isomorphism of L 2 (K, w, C) onto itself such that f (x) = lim r→∞ B(K,0,r) F [f ](y)χ 1 (−(y, x))w(dy) and (f, If f is a locally constant function as in §2.13, then P D c (b, f ) exists for each b ∈ C. On the other hand, Let g be a continuously differentiable function g : R → R such that . Analogous two statements are true for the operator P D c instead of P D.
In view of Equation 9.(1.5) [31] D α D β ψ = D β D α ψ = D α+β ψ for each α = −1, β = −1 and α + β = −1 for each ψ ∈ D ′ such that there exists D α ψ, D β ψ and D α+β ψ, where D ′ is the topologically dual space to the space D of locally constant functions φ : K → R. On the other hand, D is dense in D ′ in the weak topology (see §6 [31]). Evidently, L 2 ∩ D is dense in L 2 (K, w, R) also. The characteristic functional of the Gaussian measure belongs to D ′ and is locally constant on K \ {0}. Due to § §7.2 and 7.3 the Fourier transform is the linear topological isomorphism of D on D and of D ′ on D ′ . Then µ g S,γ,q (dx)/w(dx) ∈ L 1 (K, w, R) ∩ D ′ for each g ∈ C 0 0 (T, H) * . In view of Theorem 4.3 [18] and using the Kakutani theorem as in §2. 13 we get the statements of this theorem, since the quasi-invariance factor P y (dx)/P u (dx) is pseudo-differentiable as the function by y of order b for each fixed u ∈ B(K, 0, 1).
2.15. Theorem. Let G be either a loop group or a diffeomorphism group defined as in § §2.9 and 2.11, then there exists a stochastic process ξ(t, ω) on G which induces a quasi-invariant transition measure P on G relative to G ′ and P is pseudo-differentiable of order b for each b ∈ C such that Re(b) ≥ 0 relative to G ′ , where a dense subgroup G ′ is given in § §2.10 and 2.11.
Proof. These topological groups also have structures of C ∞ -manifolds, which are infinite-dimensional over the local field K, but they do not satisfy the Campbell-Hausdorff formula in any open local subgroup [21,23]. Their manifold structures and actions of G ′ on G will be sufficient for the construction of desired measures. These separable Polish groups have embeddings as clopen subsets into the corresponding tangent Banach spaces Y ′ and Y in accordance with [24] and § §2.1-2.11, where Y ′ is the dense subspace of Y . As usually T G = x∈G T x G and T x G = (x, Y ).
Let G be a complete separable relative to its metric ρ C ∞ -manifold on a Banach space Y over K such that it has an embedding into Y as the clopen subset. Let τ G : T G → G be a tangent bundle on G. It is trivial, since T G = G ×Y for the considered here case. Let θ : Z G → G be a trivial bundle on G with the fibre Z such that Z G = Z × G, then L 1,2 (θ, τ G ) be an operator bundle with a fibre L 1,2 (Z, Y ) (see §2.13 [15]). Let Π := τ G ⊕ L 1,2 (θ, τ G ) be a Whitney sum of bundles τ and L 1,2 (θ, τ G ).
Since G is clopen in Y , the valuation group of K is discrete in (0, ∞), then it has a clopen disjoint covering by balls B(Y, x j , r j ). That is, the atlas At(G) of G has a refinement At ′ (G) being a disjoint atlas.
On Y consider the measure µ S,γ,q or µ S as in §2.12. Then in view of Theorems 4.3 [15] and 2.2 [16] there exists the stochastic process w(t, ω) corresponding to µ S,γ,q or µ S (see also Definitions 4.2 [15] and 3.2 [16]). Suppose that f and h k for each k ∈ N defining the measure µ S satisfy the Conditions of §2.12 and of Theorem 2.14.
Now let G be a loop or a diffeomorphism group of the corresponding manifolds over the field K. Consider for G a field U with a principal part (a η , E η ), where a η ∈ T η G and E η ∈ L 1,2 (H, T η G) and ker(E η ) = {0}, θ : H G → G is a trivial bundle with a Banach fiber H and H G := G×H, L 1,2 (θ, τ η ) is an operator bundle with a fibre L 1,2 (H, T η G) such that (a η , E η ) satisfies Conditions of Theorem 3.4 [16]. For Equation 3.5.(i) [16] we take additionally (a l,k ) η for each l, k satisfying conditions of Theorem 3.5 [16]. To satisfy conditions of quasi-invariance and pseudo-differentiability of transition measures theorems we choose a η , E η and (a k,l ) η of class C 1 and satisfying Conditions 2.13.(iii,iv) by y := η ∈ G ′ =: V for each k, l.
We can take initially µ I,s,q or µ I a cylindrical measure on a Banach space X ′ such that T η G ′ ⊂ X ′ ⊂ T η G. If A η is the L q -operator or the L 1 -operator with ker(A η ) = {0}, then A η gives the σ-additive measure µ Aη,A * η z,q or µ Aη in the completion X ′ 1,η of X ′ with respect to the norm x 1 := A η x (see also §2.12).
There exists the solution ξ(t, ω, η) = ξ η (t, ω) of stochastic antiderivational Equation 3.4.(i) or 3.5.(i) [16]. When the embedding θ of T η G ′ into T η G is θ = θ 1 θ 2 with θ 1 and θ 2 of class L q for µ S,γ,q or of class L 1 for µ S , then there exists A η such that µ Aη,Aηz,q or µ Aη is the quasi-invariant and pseudodifferentiable of order b measure on T η G relative to shifts on vectors from T η G ′ (see Theorems 2.13 and 2.14). Henceforth we impose such demand on A η for each η ∈ G ′ . Consider left shifts L h : G → G such that L h η := h • η. Let us take a e ∈ T e G ′ , A e ∈ L 1,q (T e G ′ , T e G) or A e ∈ L 1,1 (T e G ′ , T e G) respectively, (a k,l ) η ∈ L k+l ((T e G) ⊗(k+l) ; T e G) for each k and each l, where H, T eG ′ and T e G in their own norm uniformities are isomorphic with c 0 (ω 0 , K). Then we put a x = (DL x )a e and A x = (DL x ) • A e for each x ∈ G, hence a x ∈ T e G and In view of the choice of G ′ in G each partial difference quotientΦ n L h (X 1 , ..., X n ; ζ 1 , ..., ζ n ) is of class C 0 and D n L h is of class L n+1,s (T G ′ ⊗n ×G ′ , T G) for each vector fields X 1 , ..., X n on G ′ , ζ 1 , .., ζ n ∈ K with ζ j p 2 (X j ) + h ∈ G ′ and h ∈ G ′ , since for each 0 ≤ l ∈ Z the embedding of T l G ′ into T l G is the product of two operators of the L q -class or the embedding is of the L 1 -class, where T 0 G := G, X = (x, X x ) ∈ T x G, x ∈ G ′ , X x ∈ Y ′ , p 1 (X) = x, p 2 (X) = X x . Take a dense subgroup G ′ from §2.10 or §2.11 correspondingly and consider left shifts L h for h ∈ G ′ .
The considered here groups G are separable, hence the minimal σ-algebra generated by cylindrical subalgebras f −1 (B n ), n=1,2,..., coincides with the σalgebra B of Borel subsets of G, where f : G → K n are continuous functions, B n is the Borel σ-algebra of K n . Moreover, G is the topological Radon space (see Theorem I.1.2 and Proposition I.1.7 [5]). Let P (t 0 , ψ, t, W ) := P ({ω : ξ(t 0 , ω) = ψ, ξ(t, ω) ∈ W }) be the transition probability of the stochastic process ξ for t ∈ T , which is defined on a σ-algebra B of Borel subsets in G, W ∈ B, since each measure µ Aη,A * η z,q is defined on the σ-algebra of Borel subsets of T η G (see above). On the other hand, T (t, τ, ω)gx = gT (t, τ, ω)x is the stochastic evolution family of operators for each τ = t ∈ T . There exists the transition measure P (t 0 , ψ, t, W ) such that it is a σ-additive quasi-invariant pseudo-differentiable of order b relative to the action of G ′ by the left shifts L h on µ measure on G, for example, t 0 = 0 and ψ = e with the fixed t 0 ∈ T (see Definitions 2.6 and 2.7).
2.16. Note. In §2.15 G ′ is on the Banach space Y ′ and G on the Banach space Y over K such that G ′ and G are complete relative to their uniformities In view of §2.15 the transition measure P is quasi-invariant and pseudo-differentiable of order b relative to the 1-parameter group ρ.
This approach is also applicable to the case of two Polish manifolds G ′ and G of class C ∞ on Y ′ and Y over K. The quasi-invariance and pseudodifferentiability of the measure P on G relative to the 1-parameter group ρ (by the definition) means such properties of P relative to the Evidently, considering different (a, E) and {a k,l : k, l} we see that there exist c = card(R) nonequivalent stochastic (in particular, Wiener) processes on G and c orthogonal quasi-invariant pseudo-differentiable of order b ∈ C with Re(b) > 0 measures on G relative to G ′ .
If M is compact, then in the case of the diffeomorphism group its dense subgroup G ′ can be chosen such that G ′ ⊃ Dif f (t ′ , M) for dim K M = n ∈ N and t ′ = t + s for 0 ≤ t ∈ R, s > nv, v = dim Qp (K). Analogously can be considered the manifold M ⊂ B(K n , 0, r) and the group G := Dif f (an r , M) of analytic diffeomorphisms f : M → M having analytic extensions on B(K, 0, r) with the corresponding norm topology, where r > 0 and r < ∞. Then there exists the stochastic process ξ on T e G such that it generates the transition measure P on T e G, its restriction on the clopen subset G embedded into T e G produces the quasi-invariant and pseudo-differentiable of each order b ∈ C with Re(b) ≥ 0 measure P | G relative to the dense subgroup G ′ := Dif f (an R , M) for R > r > 0, since the embedding T e G ′ into T e G is of class L 1 (see also §2.17).
2.17. Theorem. Let G be a separable Banach-Lie group over a local field K. Then there exists a probability quasi-invariant and pseudo-differentiable of each order b ∈ C with Re(b) > 0 transition measure P on G relative to a dense subgroup G ′ such that P is associated with a non-Archimedean stochastic process.
Proof. We consider two cases: (I) G satisfies locally the Campbell-Hausdorff formula; (II) G does not satisfy it in any neighbourhood of e in G. The first case permits to describe G ′ more concretely. There exists the embedding of G into T e G as the clopen subset, since G is the Polish group. The second case can be considered quite analogously to §2.15, where the dense subgroup G ′ can be characterized by the condition that the embedding of T e G ′ into T e G is θ = θ 1 θ 2 with θ 1 and θ 2 of class L q or θ of class L 1 , where s = q or s = 1 for stochastic processes associated either with µ S,γ,q or µ S respectively.
It remains to consider the first case. For G there exists a Banach-Lie algebra g and the exponential mapping exp : V → U, where V is a neigbourhood of 0 in g and U is a neighbourhood of e in G such that exp( The radii of convergence of the exponential and Hausdorff series corresponding to log(exp(X).exp(Y )) are positive such that for each 0 < R < p 1/(1−p) to a ball B(g, 0, R) there corresponds a clopen subgroup G 1 supplied with the Hausdorff function (see §II.6 and §II.8 [2]). Therefore, the exponential mapping exp supplies G with the structure of the analytic manifold over K. By At(G) = {(U j , φ j ) : j ∈ N} is denoted the analytic atlas of N, that is φ j : U j → V j are diffeomorphisms of U j onto V j , where U j and V j are clopen in G and in g respectively, connecting mappings φ j • φ −1 i are analytic on φ i (U i ∩ U j ) ⊂ g. Therefore, the exponential mapping provides G with the covariant derivation ∇ and a bilinear tensor Γ such that ∇ X Y = L ∇ X Y − L T (X, Y )/2, where the left-invariant derivation on G is defined by L ∇ XỸ = 0 for an arbitrary left-invariant vector field Y and all vector fields X on G, a vector fieldỸ is called left-invariant if T L gỸ (h) =Ỹ (gh), L g h := gh for each g, h ∈ G, T L g is the tangent mapping of L g , ∇ X Y u = DY u .X u + Γ u (X u , Y u ). For such ∇ the torsion tensor is zero (see §1.7 [14], [12] and §14.7 [31]). It defines the rigid analytic geometry and the corresponding atlas on G. Nevertheless At(G) has the refinement At ′ (G) such that charts of At ′ (G) compose the disjoint covering of G.
Let a x be a an analytic vector field and A x be an anlytic operator field on G such that A x is an injective compact operator of class L s for each x ∈ G, since g is of separable type over a spherically complete field K and hence isomorphic with c 0 (ω 0 , K) (see Chapter 5 in [29]), where s = q or s = 1. Let w x (t, ω) be a non-Archimedean stochastic (or, in particular, Wiener) process in T x G such that a x t + A x w x (t) ∈ T x G, since the space C 0 0 is isomorphic with c 0 . For a ball B R := B(K, 0, R) in K for 0 < R < ∞ let B(K, t j , r) be a disjoint paving for sufficiently small 0 < r < ∞ for which ξ q x,k = ξ q x (t k ) for k = 0, 1, ..., n, ξ q x (0) = x, q denotes the partition of B R into B(K, t j , r). Then there exists the process ξ = lim q ξ q (t) which is by our definition a solution of the following stochastic equation: x) such that f (t, ξ) := ln ξ(t,ω) ξ(t, ω) satisfies the condition of Theorem 4.8 [15] on the corresponding domain W , where (t, x) ∈ W ⊂ T × H. In view of Theorem 4.8 [15] after coordinate mapping of a chart (U, φ) this equation takes the following form on g: (ii) φ(ξ(t, ω)) = φ(ξ(t 0 , ω)) + (P u a φ ξ(u) )| u=t + (P w φ ξ(u) (u,ω) E)| u=t , denotes Γ for the chart (U, φ), ψ corresponds to another chart (V, ψ) such that U ∩ V = ∅, f, g and h are vector fields, since [∂(ψ • φ −1 )/∂t] = 0, that is Corollary 4.7 [15] is applicable instead of Theorem 4.8 [15] because f corresponds to (ψ • φ −1 ) (see §1.5 [14] and [3]). Since a x and A x are analytic, then a and E satisfy conditions of Theorem 3.4 [16]. The function Γ is analytic on the corresponding domain. On the other hand, g is isomorphic with c 0 (ω 0 , K) as the Banach space. If Z is the center of g, then ad : g/Z → gl(c 0 (ω 0 , K)) is the injective representation, where gl(c 0 ) denotes the general linear algebra on c 0 , ad(x)y := [x, y] for each x, y ∈ g. Since Z is commutative it aslo has injective representation in gl(c 0 ), consequently, g has embedding into gl(c 0 (ω 0 , K), since c 0 ⊕ c 0 is isomorphic with c 0 . Therefore, each x ∈ g can be written in the form x = i,j x i,j X i,j , where {X i,j : i, j ∈ N} is the orthonormal basis of g as the Banach space, x i,j ∈ K, lim i+j→∞ x i,j = 0, consequently, g has an embedding into L 0 (c 0 (ω 0 , K). Then Γ can be written in local coordinates x s(i,j) := x i,j , where s : N 2 → N is a bijection for which lim i+j→∞ s(i, j) = ∞, X i,j =: q s(i,j) , since (x n ) j,k = l 1 ,...l n−1 x j,l 1 x l 1 ,l 2 ...x l n−1 ,k X j,k , when X j,k = (δ a,j δ b,k : a, b ∈ N), ψ • φ −1 (x) = s q s a s m x m with a s m ∈ K and lim s+|m|+Ord(m)→∞ a s m = 0, since exp(x) has a radius of convergence 0 <r = p −1 for char(K) = 0 (see Theorem 25.6 [30]), where m = (m 1 , ..., m k ), k = Ord(m), 0 ≤ m 1 ∈ Z,..., 0 ≤ m k−1 ∈ Z, 0 < m k ∈ Z, 0 ≤ k ∈ Z. Evidently, there exists 0 < r < ∞ such that the series for ψ • φ −1 converges in B(c 0 , 0, r) for V ∩ U = ∅. Hence each a m−l,l := (∂ m−2 Γ φ φ(x) /∂ m−2 x)/m! for m ≥ 2 and a 1,0 = a 0,1 = (∂φ/∂x) satisfies Condition 3.5.(ii) [16]. Due to Theorem 3.5 [16] there exists the unique solution of Equation 3.5.(i). Consider G ′ corresponding to g ′ such that the embedding θ of g ′ into g is of class L 1 for µ S or θ = θ 1 θ 2 , where θ 1 and θ 2 are of class L q for µ S,γ,q .
Let T ∈ L s (g) or T = T 1 T 2 with T 1 and T 2 ∈ L s (g), where s = 1 or s = q. Consider h 1 := T (g), h 2 := sp K ([h 1 , g] ∪ h 1 ) and by induction h n+1 := sp K ([h n , g] ∪ h n ), then h n+1 ⊃ h n and h n is the subalgebra in g for each n ∈ N. In view of Proposition 2.12.2 the space L s (g) is the ideal in L(g). Therefore, h := n h n is the ideal in g due to the anticommutativity and the Jacobi identity. Since K is spherically complete there exists h n+1 ⊖h n =: t n+1 for each n ∈ N and t 1 := h 1 such that t n is the K-linear subspace of g ( [29]). By c 0 (g, {t n : n}) =: y is denoted the completion in g of vectors z such that z = n z n with z n ∈ t n for each n and lim n→∞ z n = 0. Evidently y is the proper ideal in g such that h ⊂ y, since g is infinite dimensional over K. Then the embedding θ of y into g is either of class L 1 or θ = θ 1 θ 2 such that θ 1 and θ 2 belong to L q .
Due to this let a and A be such that [a x , y] ⊂ y and [A x , ad(y)] ⊂ ad(y) for each x ∈ G, where ad(x)g := [x, g] for each x, g ∈ g, that is, ad(x) ∈ L(g). If g ∈ y ∩ V , then exp(ad(g)) − I is either of the class L 1 or the product of two operators each of which is of the class L q .
There exists a countable family (g j , W j ) : j ∈ N) of elements g j ∈ G \ W for each j > 1 and clopen subsets e ∈ W j ⊂ W such that g 1 = e, W 1 = W and {g j W j : j} is a locally finite covering of G, since G is separable and ultrametrizable (see §5.3 [8]). If P is a quasi-invariant and pseudodifferentiable of order b measure on a clopen subgroup W relative to a dense subgroup W ′ , then P (S) := ( j P ((g −1 j S) ∩ W j )2 −j )( j P (W j )2 −j ) −1 for each Borel subset S in G is quasi-invariant and pseudo-differentiable of order b measure on G relative to the dense subgroup G ′ := j g j (W j ∩ W ′ ). The group G is totally disconnected and is left-invariantly ultrametrizable (see §8 and Theorem 5.5 [13] and §6.2 [8]), consequently, in each neighbourhood of e there exists a clopen subgroup in G. Then conditions of Theorems 2.13 and 2.14 are satisfied. Therefore, analogously to §2.15 there are S, γ and the stochastic process corresponding to µ S,γ,q or µ S such that the transition measure P is quasi-invariant and pseudo-differentiable relative to G ′ .
2.18. Theorem 2.15 gives the subgroup G ′ concretely for the given group G, but Theorem 2.17 describes concretely G ′ only for the case of G satisfying the Campbell-Hausdorff formula. For a Banach-Lie group not satisfying locally the Campbell-Hausdorff formula Theorem 2.17 gives only the existence of G ′ .
These transition measures P =: ν on G induce strongly continuous unitary regular representations of G ′ given by the following formula: T ν h f (g) := (ν h (dg)/ν(dg)) 1/2 f (h −1 g) for f ∈ L 2 (G, ν, C) =: H, T ν h ∈ U(H), U(H) denotes the unitary group of the Hilbert space H. For the strong continuity of T ν h the continuity of the mapping G ′ ∋ h → ρ ν (h, g) ∈ L 1 (G, ν, C) and that ν is the Borel measure are sufficient, where g ∈ G, since G is the Polish space and hence the Radon space (see Theorem I.1.2 [5]). On the other hand, the continuity of ρ ν (h, g) by h from the Polish group G ′ into L 1 (G, ν, C) follows from ρ ν (h, g) ∈ L 1 (G, ν, C) for each h ∈ G ′ and that G ′ is the topological subgroup of G (see [11]).
Then analogously to §2.15 there can be constructed quasi-invariant and pseudo-differentiable measures on the manifold M relative to the action of the diffeomorphism group G M such that G ′ ⊂ G M . Then Poisson measures on configuration spaces associated with either G or M can be constructed [26]. There exists the stochastic process corresponding to µ S,γ,q with the certain choice of a, E and a k,l such that the regular representation is irreducible, for the stochastic process corresponding to µ S it can be taken the family of {f k : k} and a, E and a k,l such that the regular representation is irreducible.
More generally it is possible to consider instead of the group G a Polish topological space X on which G ′ acts jointly continuously: φ : (G ′ × X) ∋ Proof. In view of Lemma I.2.17 [23] it is sufficient to consider the case of M with the finite atlas At ′ (M). The rest of the proof is quite analogous to that of Theorem 2.15 using the definitions of the quasi-invariance and the pseudo-differentiability for semigroups from §2.7.
3.2. Definition and Note. In view of §2.5 each space N ξ has the additive group structure, when N = B(Y, 0, R), 0 < R ≤ ∞. Therefore, the factorization by the equivalence relation K ξ × id produce the monoid of paths C θ 0 (ξ,M → N)/(K ξ ×id) =: S ξ (M, N) in which compositions are defined not for all elements, where y 1 idy 2 if and only if y 1 = y 2 ∈ N.
There exists a composition f 1 f 2 = (g 1 g 2 , y) if and only if y 1 = y 2 = y, where f i = (g i , y i ), g i ∈ Ω ξ (M, N) and y i ∈ N ξ , i ∈ {1, 2}. The latter semigroup has elements e y such that f = e y • f = f • e y for each f , when their composition is defined, where y ∈ N ξ , f = (g, y), g ∈ Ω ξ (M, N), e y = (e, y). If N ξ is a monoid, then S ξ (M, N) can be supplied with the structure of a direct product of two monoids. Therefore, P ξ (M, N) := L ξ (M, N) × N ξ is called the path group.
3.3. Theorem. On the path group G = P ξ (M, N) from §3.2, when N = B(Y, 0, R) and N ξ is supplied with the additive group structure, and each b 0 ∈ C with Re(b 0 ) ≥ 0 there exists a stochastic process η(t, ω) for which a transition measure P is quasi-invariant and pseudo-differentiable of each order b ∈ C with Re(b) ≥ Re(b 0 ) relative to a dense subgroup G ′ .
Proof. Since P ξ (M, N) = L ξ (M, N) × N ξ , it is sufficient to construct two stochastic processes on L ξ (M, N) and N ξ and to consider transition measures for them. In view of Theorems 2.15 and 2.17 the desired processes and transition measures for them exist. Then there exists a stochastic process η(t, ω) on G for which the transition measure P is quasi-invariant and pseudo-differentiable of each order b ∈ C with Re(b) ≥ Re(b 0 ) on G relative to a dense O-subgroup G ′ .
Proof. The uniform space C 0 0 (ξ, M → N) has the embedding as the clopen subset into C 0 0 (ξ, M → Y ). Here we can take a ∈ T G ′ and A ∈ L 1,s (θ, τ ) without relations with DL h , where s = q or s = 1 respectively. Then repeating the major parts of the proof of §2.15 without L h and so more simply, but using actions of vectors fields of T G ′ by ρ X on G we get the statement of this theorem, since (D X ρ X )Y and [(∇ X ) n (D X ρ X )]Y are products of two operators of of class L n+2,q ((T G ′ ) n+2 , T G) and also of class L n+2,1 ((T G ′ ) n+2 , T G) for each C ∞ -vector fields X and Y on G ′ and each n ∈ N. In view of §2.16 there exists a stochastic process η(t, ω) for which the transition measure P is quasi-invariant and pseudo-differentiable relative to each 1-parameter diffeomorphism group of G ′ associated with a U G × U Y ′ -C ∞ -vector field on G ′ .