Markov dynamics on the cone of discrete Radon measures
aa r X i v : . [ m a t h - ph ] J a n Markov dynamics on the coneof discrete Radon measures
Dmitri Finkelshtein ∗ , Yuri Kondratiev † , and Peter Kuchling ‡ Department of Mathematics, Swansea University, Bay Campus, FabianWay, Swansea SA1 8EN, U.K. Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 110 131, 33501Bielefeld, GermanyFebruary 1, 2021
Configuration spaces form an important and actively developing area in the infinite di-mensional analysis. The spaces not only contain rich mathematical structures which re-quire non-trivial combination of continuous and combinatoric analysis, they also providea natural mathematical framework for the applications to mathematical physics, biology,ecology, and beyond.Spaces of discrete Radon measures (DRM) may be considered as generalizations ofconfiguration spaces. Main peculiarity of a DRM is that its support is typically not aconfiguration (i.e. not a locally finite set). The latter changes drastically the techniquesfor the study of the spaces of DRM.Spaces of DRM have various motivations coming from mathematics and applications.In particular, random DRM appear in the context of the Skorokhod theorem [17] in thetheory of processes with independent increments. Next, in the representation theoryof current groups, the role of measures on spaces of DRM was clarified in fundamentalworks by Gelfand, Graev, and Vershik; see [15] for the development of this approach.Additionally, DRM gives a useful technical equipment in the study of several models inmathematical physics, biology, and ecology.In the present paper, we start with a brief overview of the known facts about thespaces of DRM (Section 2). In [10], the concept of Plato subspaces of the spaces ofmarked configurations was introduced. Using this, one can define topological, differentialand functional structures on spaces of DRM, as well as transfer the harmonic analysisconsidered in [11] to the spaces of DRM. This allows us to extend the study of non-equilibrium dynamics, see e.g. [8, 12, 13], to the spaces of DRM. ∗ [email protected] . † [email protected] . ‡ [email protected] . Cone of discrete Radon measures
Let X be a locally compact Polish space, and let B c ( X ) denote the family of all Borelsets from X with a compact closure. The cone of nonnegative discrete Radonmeasures on X is defined as follows: K ( X ) := n η = X i s i δ x i ∈ M ( X ) (cid:12)(cid:12)(cid:12) s i ∈ (0 , ∞ ) , x i ∈ X o . By convention, the zero measure η = 0 is included in K ( X ). The support of an η ∈ K ( X )is given by τ ( η ) := (cid:8) x ∈ X : 0 < η ( { x } ) =: s x ( η ) (cid:9) , and τ (0) := ∅ . If η is fixed and x ∈ τ ( η ), we write s x := s x ( η ). Therefore, η (Λ) = X x ∈ τ ( η ) ∩ Λ s x < ∞ , Λ ∈ B c ( X ) , η ∈ K ( X ) . We stress that, in general, the number of points | τ ( η ) | in the support of a measure η ∈ K ( X ) may be infnite. Let henceforce | · | denote the number of elements of a set.For η, ξ ∈ K ( X ) we write ξ ⊂ η if τ ( ξ ) ⊂ τ ( η ) and s x ( ξ ) = s x ( η ) for all x ∈ τ ( ξ ).If, additionally, | τ ( ξ ) | < ∞ , we write ξ ⋐ η .We fix the vague topology on M ( X ), which is the coarsest topology such that themappings η
7→ h f, η i := X x ∈ τ ( η ) s x f ( x ) . are continuous for all continuous functions f : X → R with compact support. We en-dow K ( X ) with the corresponding subspace topology, and also let B ( K ( X )) denote thecorresponding Borel σ -algebra. Configuration spaces
Let Y be a locally compact Polish space. The space of locally finite configurations over Y is defined as follows:Γ( Y ) = (cid:8) γ ⊂ Y : | γ ∩ ∆ | < ∞ for all compact ∆ ⊂ Y (cid:9) . Then Γ( Y ) is naturally embedded into the space of Radon measures Γ( Y ) ⊂ M ( Y ); weendow it with the vague topology defined on M ( Y ). Let B (Γ( Y )) be the correspondingBorel σ -algebra. 2e denote R ∗ + := (0 , ∞ ) and consider Y = R ∗ + × X . Let Γ p ( R ∗ + × X ) ⊂ Γ( R ∗ + × X )denote the set of all pinpointing configurations ; the latter means that γ ∈ Γ p ( R ∗ + × X )iff ( s , x ) , ( s , x ) ∈ γ for an x ∈ X implies s = s .For a pinpointing configuration γ ∈ Γ p ( Y ), we introduce the local mass of a pre-compact set Λ ∈ B c ( X ): γ (Λ) = Z R ∗ + × X s Λ ( x ) dγ ( s, x ) = X ( s,x ) ∈ γ s Λ ( x ) ∈ [0 , ∞ ] . Finally, we define the space of pinpointing configurations with finite local mass:Π( R ∗ + × X ) := (cid:8) γ ∈ Γ p ( R ∗ + × X ) : γ (Λ) < ∞ for all Λ ∈ B c ( X ) (cid:9) . We endow Π( R ∗ + × X ) with the subspace topology coming from Γ( R ∗ + × X ), and one canconsider the corresponding (trace) Borel σ -algebra.The mapping R : Π( R ∗ + × X ) → K ( X ) given by γ = X ( s,x ) ∈ γ δ ( s,x ) γ := X ( s,x ) ∈ γ sδ x (2.1)provides a natural bijection. It can be shown that both R and R − are measurable withrespect to the Borel σ -algebras constructed above, i.e. B (Π( R ∗ + × X )) and B ( K ( X )) are σ -isomorphic, see [9]. Discrete measures with finite support
We consider the subcone of all discrete nonnegative Radon measures with finite support: K ( X ) := (cid:8) η ∈ K ( X ) : | τ ( η ) | < ∞ (cid:9) = ∞ G n =0 K ( n )0 ( X ) , where K ( n )0 ( X ) := { η ∈ K ( X ) : | τ ( η ) | = n } , n ∈ N ; K (0)0 ( X ) := { } . The mapping R , given by (2.1), provides provides a bijection between K ( X ) and theset Γ , p ( R ∗ + × X ) of pinpointing finite configurations on R ∗ + × X . We define the Borel σ -algebra on K ( X ) as the smallest σ -algebra which makes this mapping R measurable.Any measurable function G : K ( X ) → R can be identified with the sequence ofsymmetric functions on ( R ∗ + × X ) n , n ∈ N , through the equalities: G ( n ) ( s , x , . . . , s n , x n ) := G (cid:16) n X i =1 s i δ x i (cid:17) , n X i =1 s i δ x i ∈ K ( n )0 ( X ) , n ∈ N . We set also G (0) := G (0) ∈ R .A set A ⊂ K ( X ) is called bounded if there exist Λ ∈ B c ( X ), N ∈ N , and a segment I := [ a, b ] ⊂ R ∗ + such that, for all η ∈ A , τ ( η ) ⊂ Λ , | τ ( η ) | ≤ N, s x ∈ I for all x ∈ τ ( η ) . The family of all bounded measurable subsets of K ( X ) is denoted by B b ( K ( X )). Ameasure ρ on K ( X ) is called locally finite if ρ ( A ) < ∞ for all A ∈ B b ( K ( X )).3n example of a locally finite measure on K ( X ) is the Lebesgue–Poisson measure λ ν ⊗ σ with the intensity measure ν ⊗ σ , where ν and σ are non-atomic Radon measureson R ∗ + and X , respectively, and ν has a finite first moment: Z R ∗ + sν ( ds ) < ∞ . The Lebesgue–Poisson measure λ ν ⊗ σ is then defined through the equality Z K ( X ) G ( η ) λ ν ⊗ σ ( dη ) == G (0) + ∞ X n =1 n ! Z ( R ∗ + × R d ) n G ( n ) ( s , x , . . . , s n , x n ) ν ( ds ) . . . ν ( ds n ) σ ( dx ) . . . σ ( dx n ) , which should hold for any G : K ( X ) → R + .We also consider a special case of the measure ν = ν θ , where ν θ ( ds ) = θs e − s ds (2.2)for some θ >
0. For a fixed non-atomic Radon measure σ on X , we then denote λ θ := λ ν θ ⊗ σ . A function G : K ( X ) → R is said to be a bounded function with bounded support if | G ( η ) | ≤ C A ( η ), η ∈ K ( X ), for some C > A ∈ B b ( K ( X )). The set of all boundedfunctions on K ( X ) with bounded support is denoted by B bs ( K ( X )). Clearly, for anylocally finite measure ρ on K ( X ), Z K ( X ) | G ( η ) | ρ ( dη ) < ∞ , G ∈ B bs ( K ( X )) . Note that B bs ( K ( X )) is dense in L ( K ( X ) , λ ν ⊗ σ ), where ν and σ are as above.We will need the following identity. Lemma 2.1 (Minlos lemma) . Let λ ν ⊗ σ be defined as the above.1. Let G : K ( X ) → R , H : ( K ( X )) → R . Then Z K ( X ) Z K ( X ) G ( ξ + ξ ) H ( ξ , ξ ) λ ν ⊗ σ ( dξ ) λ ν ⊗ σ ( dξ )= Z K ( X ) G ( η ) X ξ ⊂ η H ( ξ, η − ξ ) λ ν ⊗ σ ( dη ) .
2. Let H : K ( X ) × R ∗ + × R d → R . Then Z K ( X ) X x ∈ τ ( η ) H ( η, s x , x ) λ ν ⊗ σ ( dη )= Z K ( X ) Z R ∗ + × R d H ( η + sδ x , s, x ) ν ( ds ) σ ( dx ) λ ν ⊗ σ ( dη ) , , provided, at least one side of the equality exists. armonic analysis on the cone For any G ∈ B bs ( K ( X )), we define KG : K ( X ) → R by (cf. [11])( KG )( η ) := X ξ ⋐ η G ( ξ ) . (2.3) Proposition 2.2 (see [6, 16]) . For any G ∈ B bs ( K ( X )) , there exist C > , Λ ∈ B c ( X ) , N ∈ N , and a segment I = [ a, b ] ⊂ R ∗ + such that, for each η ∈ K ( X ) , ( KG )( η ) = ( KG ) (cid:16) X x ∈ τ ( η ) ∩ Λ I ( s x ) s x (cid:17) , (cid:12)(cid:12) ( KG )( η ) (cid:12)(cid:12) ≤ C (cid:0) | τ ( η ) ∩ Λ | (cid:1) N . Note that (2.3) can be also defined pointwise on a wider class of functions (see [6, 16]for details). In particular, for the Lebesgue-Poisson exponents e λ ( f, η ) := Y y ∈ τ ( η ) f ( s y , y ) , η ∈ K ( X ) , e λ ( f,
0) := 1 , one has that Ke λ ( f, η ) = Y y ∈ τ ( η ) (1 + f ( s y , y )) , η ∈ K ( X ) , provided that e.g. | f ( s, y ) | ≤ C s Λ ( y ) for ( s, y ) ∈ R ∗ + × X , where C >
0, Λ ∈ B c ( X ).Note also that, for any f ∈ L ( R ∗ + × X, dνdσ ), Z K ( X ) e λ ( f, η ) λ ν ⊗ σ ( dη ) = exp Z R ∗ + × X f ( s, x ) ν ( ds ) σ ( dx ) ! . (2.4)For measurable G , G : K ( X ) → R , we define their ⋆ -convolution as follows:( G ⋆ G )( η ) = X ξ + ξ + ξ = η : τ ( ξ i ) ∩ τ ( ξ j )= ∅ G ( ξ + ξ ) G ( ξ + ξ ) . Then, for any G , G ∈ B bs ( K ( X )), K ( G ⋆ G ) = KG · KG . Let µ be a probability measure on the space ( K ( X ) , B ( K ( X ))) such that Z K ( X ) | η (Λ) | N µ ( dη ) < ∞ for any Λ ∈ B c ( X ) and N ∈ N . Then µ is said to have finite local moments of allorders. The space of all such measures is denoted by M ( K ( X )). In particular, K ( B bs ( K ( X ))) ⊂ L ( K ( X ) , µ ) , µ ∈ M ( K ( X )) . The corresponding correlation measure ρ µ on ( K ( X ) , B ( K ( X ))) is then defined bythe relation ρ µ ( A ) := Z K ( X ) ( K A )( η ) µ ( dη ) , A ∈ B b ( K ( X )) . roposition 2.3 (see [6, 16]) . Let µ ∈ M ( K ( X )) . Then1. The corresponding correlation measure ρ µ is locally finite.2. For any G ∈ L ( K ( X ) , ρ µ ) , the sum in (2.3) converges µ -almost surely, and3. KG ∈ L ( K ( X ) , µ ) with Z K ( X ) KG ( η ) µ ( dη ) = Z K ( X ) G ( η ) ρ µ ( dη ) , (2.5) k KG k L ( µ ) ≤ k G k L ( ρ µ ) . Let ν, σ be as above. Consider the Poisson measure π ν ⊗ σ on Γ( R ∗ + × X ) with theintensity measure ν ⊗ σ on R ∗ + × X , then π ν ⊗ σ (Π( R ∗ + × X )) = 1 (see [6, 16] for details).Hence, we may view π ν ⊗ σ as a probability measure on Π( R ∗ + × X ), and consider the cor-responding push-forward measure on K ( X ) under the mapping R . This measure belongsto M ( K ( X )), and the corresponding correlation measure is just λ ν ⊗ σ .In the special case ν = ν θ given through (2.2), θ >
0, the corresponding push-forwardmeasure on K ( X ) is called the Gamma measure G θ with the intensity θ > µ ∈ M ( K ( X )) and ρ µ be the corresponding correlation measure. A function k µ : K ( X ) → R is called the correlation function of µ if it is the density of the correlationmeasure with respect to the Lebesgue-Poisson measure λ ν ⊗ σ , i.e. if ρ ( dη ) = k µ ( η ) λ ν ⊗ σ ( dη ) . For sufficient conditions for the existence of the correlation function, see [6, 16].
Statistical dynamics
We are going to describe evolutions of measures µ µ t in the space M ( K ( X ))through a (formal) Markov generator L . We assume that L is defined on a linear set D ⊂ K ( B bs ( K ( X ))) such that LF ∈ L ( K ( X ) , µ t ), t ≥
0, for all F ∈ D . Then we definethe evolution of measures through the equality ddt Z K ( X ) F ( η ) µ t ( dη ) = Z K ( X ) ( LF )( η ) µ t ( dη ) , (2.6)for all t ≥ F ∈ D (recall that, by Proposition 2.3, F ∈ L ( K ( X ) , µ t ) for t ≥ F = KG , G ∈ B bs ( K ( X )), and defining b LG through the identity K b LG = LKG, G ∈ B bs ( K ( X )) , one can rewrite (2.6), by using(2.5), as follows: ddt Z K ( X ) G ( η ) ρ µ t ( dη ) = Z K ( X ) ( b LG )( η ) ρ µ t ( dη ) (2.7)for all t > G ∈ B bs ( K ( X )). Here b LG = K − LKG, G ∈ B bs ( K ( X )) , K − F )( η ) := X ξ ⊂ η ( − | τ ( η ) |−| τ ( ξ ) | F ( ξ ) , η ∈ K ( X ) . We will restrict our attention to the dynamics of correlation measures which havecorrelation functions: ρ µ t ( dη ) = k t ( η ) dλ ν ⊗ σ ( η ), assuming k be given. Then (2.7) can berewritten as follows: ddt Z K ( X ) G ( η ) k t ( η ) dλ ν ⊗ σ ( η ) = Z K ( X ) ( b LG )( η ) k t ( η ) dλ ν ⊗ σ ( η ) (2.8)for all t > G ∈ B bs ( K ( X )).Let L △ denote the dual operator to b L , i.e. Z K ( X ) ( b LG )( η ) k ( η ) dλ ν ⊗ σ ( η ) = Z K ( X ) G ( η )( L △ k )( η ) dλ ν ⊗ σ ( η ) (2.9)for all G, k : K ( X ) → R , such that both sides of the latter equality are finite. Then onecan rewrite (2.8) as follows ddt Z K ( X ) G ( η ) k t ( η ) dλ ν ⊗ σ ( η ) = Z K ( X ) G ( η )( L △ k t )( η ) dλ ν ⊗ σ ( η ) (2.10)for all t > G ∈ B bs ( K ( X )). The latter weak-type equation defines hence the evolutionof the correlation functions generated by the Markov operator L . We can consider alsoits strong form: ∂∂t k t ( η ) = L △ k t ( η ) , t > . (2.11)considered on a suitable class of correlation functions. Let ν and σ be non-atomic Radon measures on R ∗ + and X , respectively. We define( LF )( η ) = X x ∈ τ ( η ) m ( s x )[ F ( η − s x δ x ) − F ( η )]+ X x ∈ τ ( η ) Z R ∗ + × X q ( s x , s ) a ( x − y )[ F ( η + sδ y ) − F ( η )] ν ( ds ) σ ( dy )for F ∈ K ( B bs ( K ( X ))), cf. [13]. Here a : X → [0 , ∞ ), m : R ∗ + → [0 , ∞ ), q : R ∗ + × R ∗ + → [0 , ∞ ) are such that a ( − x ) = a ( x ) , x ∈ X, a ∈ L ( X, dσ ) ∩ L ∞ ( X, dσ ) , m ∈ L ∞ ( R ∗ + , dν ) ,q ∈ L ∞ ( R ∗ + × R ∗ + , dν dν ) , Z q ( s ′ , · ) ν ( ds ′ ) ∈ L ∞ ( R ∗ + , dν ) . (3.1) Proposition 3.1.
For any G ∈ B bs ( K ( X )) , b LG := K − LKG satisfies ( b LG )( η ) = − X x ∈ τ ( η ) m ( s x ) G ( η ) (3.2)+ X x ∈ τ ( η ) Z R ∗ + × X q ( s x , s ) a ( x − y )[ G ( η − s x δ x + sδ y ) + G ( η + sδ y )] ν ( ds ) σ ( dy ) . roof. Firstly, we note that, for any G ∈ B bs ( K ( X )) and F := KG , F ( η − s x δ x ) − F ( η ) = − K ( G ( · + s x δ x ))( η − s x δ x ) F ( η + s x δ x ) − F ( η ) = K ( G ( · + s x δ x ))( η ) . Then X x ∈ τ ( η ) s x [ F ( η − s x δ x ) − F ( η )] = − X x ∈ τ ( η ) s x K ( G ( · + s x δ x ))( η − s x δ x )= − X x ∈ τ ( η ) s x X ξ ⋐ η − s x δ x G ( ξ + s x δ x ) = − X ξ ⋐ η X x ∈ τ ( ξ ) s x G ( ξ );and X x ∈ τ ( η ) Z R ∗ + × X q ( s x , s ) a ( x − y )[ F ( η + sδ y ) − F ( η )] ν ( ds ) σ ( dy )= X x ∈ τ ( η ) Z R ∗ + × X q ( s x , s ) a ( x − y ) K ( G ( · + sδ y ))( η ) ν ( ds ) σ ( dy )= X x ∈ τ ( η ) X ξ ⋐ η Z R ∗ + × X q ( s x , s ) a ( x − y ) G ( ξ + sδ y ) ν ( ds ) σ ( dy )= X x ∈ τ ( η ) X ξ ⋐ η − s x δ x Z R ∗ + × X q ( s x , s ) a ( x − y ) G ( ξ + sδ y ) ν ( ds ) σ ( dy )+ X x ∈ τ ( η ) X ξ ⋐ η − s x δ x Z R ∗ + × X q ( s x , s ) a ( x − y ) G ( ξ + s x δ x + sδ y ) ν ( ds ) σ ( dy )= X ξ ⋐ η X x ∈ τ ( ξ ) Z R ∗ + × X q ( s x , s ) a ( x − y ) G ( ξ − s x δ x + sδ y ) ν ( ds ) σ ( dy )+ X ξ ⋐ η X x ∈ τ ( ξ ) Z R ∗ + × X q ( s x , s ) a ( x − y ) G ( ξ + sδ y ) ν ( ds ) σ ( dy ) , that proves the statement.Let, for fixed ν and σ , X n := L ∞ (cid:0) ( R ∗ + × X ) n , ( ν ⊗ σ ) ⊗ n (cid:1) , n ∈ N . Let k · k n denote the norm in X n .Let L ∞ ( K ( X )) denote the set of all functions k : K ( X ) → R such that k ( n ) ∈ X n foreach n ∈ N . Note that, for all G ∈ B bs ( K ( X )) and k ∈ L ∞ ( K ( X )), Z K ( X ) | G ( η ) k ( η ) | λ ν ⊗ σ ( dη ) < ∞ . Proposition 3.2.
For any k ∈ L ∞ ( K ( X )) , the mapping ( L △ k )( η ) = − X x ∈ τ ( η ) m ( s x ) k ( η )+ X y ∈ τ ( η ) Z R ∗ + × X q ( s, s y ) a ( x − y ) k ( η − s y δ y + sδ x ) ν ( ds ) σ ( dx )+ X y ∈ τ ( η ) X x ∈ τ ( η ) \{ y } q ( s x , s y ) a ( x − y ) k ( η − s y δ y )8 s well-defined and, for any G ∈ B bs ( K ( X )) , Z K ( X ) G ( η )( L △ k )( η ) λ ν ⊗ σ ( dη ) = Z K ( X ) ( b LG )( η ) k ( η ) λ ν ⊗ σ ( dη ) . Proof.
The result is a straightforward application of the Minlos lemma.
Theorem 3.3.
Let (3.1) hold. We define, for s > , κ ( s ) := Z X a ( x ) σ ( dx ) · Z R ∗ + q ( s ′ , s ) ν ( ds ′ ) , r ( s ) := κ ( s ) − m ( s ) , and set R := ess sup s> r ( s ) ∈ R . Let ≤ k ∈ L ∞ ( K ( X )) .1. There exists a unique point-wise solution to the initial value problem (2.11) ; more-over, ≤ k t ∈ L ∞ ( K ( X )) .2. Suppose that, for some C > , k k ( n )0 k n ≤ C n n ! , n ∈ N . Then, for all t > , n ∈ N k k ( n ) t k n ≤ e tR ( C + t ) n n ! if R < ,e tnR ( C + t ) n n ! if R ≥ .
3. Denote µ = k m k L ∞ ( R ∗ + ,dν ) . Suppose that there exists B ⊂ R ∗ + × X such that α : = min n ess inf ( s,x ) ∈ B k (1)0 ( s, x ) , ess inf ( s ,x ) , ( s ,x ) ∈ B q ( s , s ) a ( x − x ) o > β : = α · ( ν ⊗ σ )( B ) < µ. Denote also T n := n − P j =1 1 j for n ≥ ; T := 0 ; b x ( n ) := ( s , x , . . . , s n , x n ) . Then k ( n ) t ( b x ( n ) ) ≥ α n e ( β − µ ) nt n ! for b x ( n ) ∈ B n , t ≥ T n . Proof.
1) Consider a convolution-type operator on X n , n ∈ N : for 1 ≤ i ≤ n ,( A i k ( n ) )( s , x , . . . , s n , x n ) := Z R ∗ + × X q ( s, s i ) a ( x − x i ) k ( n ) i ( s, x ) ν ( ds ) σ ( dx ) , where k ( n ) i ( s, x ) := k ( n ) ( s , x , . . . , s i − , x i − , s, x, s i +1 , x i +1 , . . . , s n , x n ) . (3.3)We define, for k ( n ) ∈ X n , n ∈ N , 1 ≤ i ≤ n , b x ( n ) := ( s , x , . . . , s n , x n )( B i k ( n ) )( b x ( n ) ) := m ( s i ) k ( n ) ( b x ( n ) ) , ( C i k ( n ) )( b x ( n ) ) := κ ( s i ) k ( n ) ( b x ( n ) ) , ( V i k ( n ) )( b x ( n ) ) := r ( s i ) k ( n ) ( b x ( n ) ) . M i := A i − C i is the jump generator w.r.t. i -th variable: i.e. for fixed s j , x j ,1 ≤ j ≤ n , j = i ,( M i k ( n ) i )( s i , x i ) = Z R ∗ + × X q ( s, s i ) a ( x − x i ) (cid:0) k ( n ) i ( s, x ) − k ( n ) i ( s i , x i ) (cid:1) ν ( ds ) σ ( dx ) , where k ( n ) i is given through (3.3).We set also A ( n ) := n X i =1 A i ; B ( n ) := n X i =1 B i ; V ( n ) := n X i =1 V i ; M ( n ) := n X i =1 M i . Finally, we consider mappings from X n − to X n , n ≥ W i k ( n − )( s , x , . . . , s n , x n ) := (cid:16)X j = i q ( s j , s i ) a ( x j − x i ) (cid:17) × k ( n − ( s , x , . . . , s i − , x i − , s i +1 , x i +1 , . . . , s n , x n )for 1 ≤ i ≤ n , and W ( n ) := n X i =1 W i , n ≥ . We set also W := W (1) := 0.It is straightforward to see that, under assumptions (3.1), operators A i , B i , V i , M i , andhence A ( n ) , B ( n ) , V ( n ) , M ( n ) , are bounded linear operators on X n ; and also W i and W ( n ) are linear bounded operators from X n − to X n .The initial value problem (2.11) can be hence rewritten as follows ∂∂t k ( n ) t = A ( n ) k ( n ) t − B ( n ) k ( n ) t + W ( n ) k ( n − = M ( n ) k ( n ) t + V ( n ) k ( n ) t + W ( n ) k ( n − t ; n ∈ N k ( n )0 ∈ X n . Since W (1) k (0) t = 0 and all operators are bounded, the latter system can be solved recur-sively: k ( n ) t ( s , x , . . . , s n , x n ) = e t ( M ( n ) + V ( n ) ) k ( n )0 ( s , x , . . . , s n , x n )+ Z t e ( t − τ )( M ( n ) + V ( n ) ) ( W ( n ) k ( n − τ )( s , x , . . . , s n , x n ) dτ (3.4)Let X + n denote the cone of all non-negative (a.e.) functions in X n , n ∈ N . By theTrotter–Lie formula for bounded operators, e tM ( n ) = lim m →∞ (cid:0) e tm A ( n ) e − tm C ( n ) (cid:1) m . By the very definition, A ( n ) : X + n → X + n , hence, e tA ( n ) = ∞ X j =0 tj ! ( A ( n ) ) j : X + n → X + n , t ≥ . e − tC ( n ) is just a multiplication operator by a non-negative function, hence, it pre-serves X + n as well. As a result, e tM ( n ) : X + n → X + n . Using again the Trotter–Lie formulafor e t ( M ( n ) + V ( n ) ) = lim m →∞ (cid:0) e tm M ( n ) e tm V ( n ) (cid:1) m , (3.5)we conclude by the same arguments that it also preserves X + n . Since W ( n ) : X + n − → X + n ,we get recursively from (3.4) that k ( n )0 ∈ X + n , n ∈ N , implies k ( n ) t ∈ X + n , n ∈ N , t > M ( n ) e tM ( n ) e tM ( n ) preserves X + n , we have, forany f n ∈ X + n , which hence satisfies the inequality 0 ≤ f n ≤ k f n k n , that 0 ≤ e tM ( n ) f n ≤ e tM ( n ) k f n k n = k f n k n , and thus k e tM ( n ) f n k n ≤ k f n k n , f n ∈ X + n . Since e tV ( n ) is a multiplication operator, k e tV ( n ) k = e t ess sup V ( n ) ( b x ( n ) ) ≤ e tnR . Therefore, by (3.5), k e t ( M ( n ) + V ( n ) ) f n k n ≤ e tnR k f n k n , f n ∈ X + n . Then, by (3.4), k k ( n ) t k n ≤ e tnR k k ( n )0 k n + Z t e ( t − τ ) nR k W ( n ) k ( n − τ k n dτ ≤ e tnR k k ( n )0 k n + n ( n − Z t e ( t − τ ) nR k k ( n − τ k n − dτ. For n = 1, it reads as k k (1) t k ≤ e tR k k (1)0 k ≤ Ce tR ≤ ( C + t ) n e tR . For n ≥
2, consider two cases separately.Let
R <
0. Then, assuming that k k ( n − τ k n − ≤ e τR ( C + τ ) n − ( n − , τ ≥ , and using the inequality e ( t − τ ) nR ≤ e ( t − τ ) R , τ ∈ [0 , t ], R <
0, we get k k ( n ) t k n ≤ e tnR C n n ! + n !( n − Z t e ( t − τ ) nR e τR ( C + τ ) n − dτ ≤ e tR C n n ! + n !( n − Z t e ( t − τ ) R e τR ( C + τ ) n − dτ = e tR C n n ! + n !( n − e tR Z t ( C + τ ) n − dτ = e tR C n n ! + n !( n − e tR ( C + t ) n − C n n ≤ ( C + t ) n n ! e tR . Let now R ≥
0. Then, assuming that k k ( n − τ k n − ≤ e τ ( n − R ( C + τ ) n − ( n − , τ ≥ ,
11e get k k ( n ) t k n ≤ e tnR C n n ! + n !( n − B n − Z t e ( t − τ ) nR e τ ( n − R ( C + τ ) n − dτ = e tnR C n n ! + e tnR n !( n − Z t e − τR ( C + τ ) n − dτ ≤ e tnR C n n ! + e tnR n !( n −
1) ( C + t ) n − C n n ≤ ( C + t ) n e tnR n ! .
3) We rewrite (3.4) in the form k ( n ) t ( s , x , . . . , s n , x n ) = e t ( A ( n ) − B ( n ) ) k ( n )0 ( s , x , . . . , s n , x n )+ Z t e ( t − τ )( A ( n ) − B ( n ) ) ( W ( n ) k ( n − τ )( s , x , . . . , s n , x n ) dτ. (3.6)By the Trotter–Lie formula, e t ( A ( n ) − B ( n ) ) = lim m →∞ (cid:0) e tm A ( n ) e − tm B ( n ) (cid:1) m . For any f n ∈ X + n , n ∈ N , b x ( n ) := ( s , x , . . . , s n , x n ), we get, using the notation (3.3),( A ( n ) f n )( b x ( n ) ) = n X i =1 Z R ∗ + × X q ( s, s i ) a ( x − x i ) f n ( s, x ) ν ( ds ) σ ( dx ) ≥ α n X i =1 Z B f n ( s, x ) ν ( ds ) σ ( dx ) . Therefore, if b n > f n ( b x ( n ) ) ≥ b n , b x ( n ) ∈ B n , (3.7)then ( A ( n ) f n )( b x ( n ) ) ≥ nb n β, b x ( n ) ∈ B n , where β := α ( ν ⊗ σ )( B ). Iterating, one gets for each j ∈ N ,(( A ( n ) ) j f n )( b x ( n ) ) ≥ n j b n β j , b x ( n ) ∈ B n , and hence, for any τ > e τA ( n ) f n )( b x ( n ) ) ≥ b n e nβτ , b x ( n ) ∈ B n . Let µ = k m k L ∞ ( R ∗ + ,dν ) . Then (3.7) implies( e − τB ( n ) f n )( b x ( n ) ) ≥ e − µnτ f n ( b x ( n ) ) ≥ e − µnτ b n , b x ( n ) ∈ B n . Therefore, ( e τA ( n ) e − τB ( n ) f n )( b x ( n ) ) ≥ e ( β − µ ) nτ b n , b x ( n ) ∈ B n , and hence (( e tm A ( n ) e − tm B ( n ) ) m f n )( b x ( n ) ) ≥ e ( β − µ ) nt b n , b x ( n ) ∈ B n . n = 1. Then, for any ( s, x ) ∈ B and t ≥ k (1) t ( s, x ) = e t ( A (1) − B (1) ) k (1)0 ( s, x ) ≥ αe ( β − µ ) t . Let now n ≥
2. Suppose that, for all τ ≥ T n − , k ( n − τ ( b x ( n − ) ≥ α n − e ( n − β − µ ) τ ( n − , b x ( n − ∈ B n − . Then ( W ( n ) k ( n − τ )( b x ( n ) ) ≥ α n n ( n − e ( n − β − µ ) τ ( n − , b x ( n ) ∈ B n , and therefore, by (3.6), for n ≥ t ≥ T n , and b x ( n ) ∈ B n , k ( n ) t ( b x ( n ) ) ≥ α n Z t e ( β − µ ) n ( t − τ ) n ( n − e ( n − β − µ ) τ ( n − dτ ≥ α n n ! e ( β − µ ) nt ( n − Z tT n − e − ( β − µ ) τ dτ ≥ α n n ! e ( β − µ ) nt ( n − t − T n − ) ≥ α n n ! e ( β − µ ) nt ( n − T n − T n − ) = αn ! e ( β − µ ) nt . The statement is fully proved.
We modify the contact model by adding a competition term, see e.g. [2,3,5,7]. The modelis given by the following operator for F ∈ K ( B bs ( K ( X ))):( LF )( η ) = X x ∈ τ ( η ) m ( s x )[ F ( η − s x δ x ) − F ( η )]+ X x ∈ τ ( η ) X y ∈ τ ( η ) \{ x } q − ( s x , s y ) a − ( x − y ) [ F ( η − s x δ x ) − F ( η )]+ X x ∈ τ ( η ) Z R ∗ + × X q + ( s x , s ) a + ( x − y )[ F ( η + sδ y ) − F ( η )] ν ( ds ) σ ( dy ) . Here m : R ∗ + → R + is the mortality rate function, 0 ≤ a ∈ ± ∈ L ( X, dσ ) ∩ L ∞ ( X, dσ )are spatial dispersion and competition kernels, such that a ± ( − x ) = a ± ( x ), x ∈ X ; and q ± : R ∗ + × R ∗ + → R + are symmetric functions. We denote κ ± := Z X a ± ( x ) σ ( dx ) > . Proposition 4.1.
For any G ∈ B bs ( K ( X )) , b LG := K − LKG = b L G + b L G + b L G + b L G, here ( b L G )( η ) := − X x ∈ τ ( η ) m ( s x ) G ( η ) − X x ∈ τ ( η ) X y ∈ τ ( η ) \{ x } q − ( s x , s y ) a − ( x − y ) G ( η ) , ( b L G )( η ) := − X x ∈ τ ( η ) X y ∈ τ ( η ) \{ x } q − ( s x , s y ) a − ( x − y ) G ( η − s x δ x )( b L G )( η ) := X x ∈ τ ( η ) Z R ∗ + × X q + ( s x , s ) a + ( x − y ) G ( η − s x δ x + sδ y ) ν ( ds ) σ ( dy )( b L G )( η ) := X x ∈ τ ( η ) Z R ∗ + × X q + ( s x , s ) a + ( x − y ) G ( η + sδ y ) ν ( ds ) σ ( dy ) . For
C > α ∈ R , we set C ( η ) := C | τ ( η ) | e α P y ∈ τ ( η ) s y , η ∈ K ( X ) , and define the space L α,C := L ( K ( X ) , C ( η ) dλ θ ( η )) . (4.1)We denote its norm by k · k α,C .We define D ( η ) := X x ∈ τ ( η ) m ( s x ) + X x ∈ τ ( η ) X y ∈ τ ( η ) \{ x } q − ( s x , s y ) a − ( x − y ) , η ∈ K ( X ) , and consider also the linear set D := { G ∈ L α,C : DG ∈ L α,C } . Theorem 4.2.
Let
C > and α ∈ R . Suppose that there exist β > , such that Z R ∗ + q − ( s, τ ) e ατ ν ( dτ ) ≤ βm ( s ) , s > , (4.2) Z R ∗ + q + ( s, τ ) e ατ ν ( dτ ) ≤ βe αs m ( s ) , s > , (4.3) q + ( s, τ ) a + ( x ) ≤ βe ατ q − ( s, τ ) a − ( x ) , s, τ > , x ∈ X, (4.4) κ + + κ − C + 1 C < β . (4.5) Then ( b L, D ) is the generator of an analytic semigroup T ( t ) , t ≥ , in L α,C .Proof. Firstly, using the same arguments as in [8, Lemma 3.3], we can show that ( b L , D )is the generator of an analytic contraction semigroup in L α,C .Next, we recall (see e.g. [4]) that, for a Banach space Z , a linear operator ( B, D ( B ))is (relatively) A -bounded w.r.t. a linear operator ( A, D ( A )), if D ( A ) ⊆ D ( B ) and if thereexist constants a, b ∈ R + such that k Bx k ≤ a k Ax k + b k x k (4.6)for all xD ( A ). The A -bound of B is a := inf { a ≥ ∃ b ∈ R + such that (4.6) holds } A being the generator of an analytic semigroup, ( A + B, D ( A )) generates an analyticsemigroup for every A -bounded operator B having A -bound a < .We are going to show now that, under assumptions above, the operator b L + b L + b L is b L -bounded. Indeed, for each G ∈ D , k b L G k α,C ≤ Z K ( X ) X x ∈ τ ( η ) X y ∈ τ ( η − s x δ x ) q − ( s x , s y ) a − ( x − y ) | G ( η − s y δ y ) | C ( η ) λ ( dη ) (2.1) = Z K ( X ) Z R ∗ + × X X x ∈ τ ( η ) q − ( s x , s ) a − ( x − y ) | G ( η ) | C ( η + sδ y ) ν ( ds ) σ ( dy ) λ ( dη )= κ − C Z K ( X ) | G ( η ) | C ( η ) X x ∈ τ ( η ) Z R ∗ + q − ( s x , s ) e αs ν ( ds ) λ ( dη ) ≤ βκ − C Z K ( X ) X x ∈ τ ( η ) m ( s x ) | G ( η ) | C ( η ) λ ( dη ) ≤ βκ − C k b L G k α,C , where we used (4.2). Next, by (4.3), we have, for each G ∈ D , k b L G k α,C ≤ Z K ( X ) X x ∈ τ ( η ) Z R ∗ + × X q + ( s x , s ) a + ( x − y ) | G ( η − s x δ x + sδ y ) | ν ( ds ) σ ( dy ) C ( η ) λ ( dη ) (2.1) = κ + Z K ( X ) X y ∈ τ ( η ) Z R ∗ + × X q + ( s, s y ) a + ( x − y ) | G ( η ) | C ( η − s y δ y + sδ x ) ν ( ds ) σ ( dx ) λ ( dη )= Z K ( X ) | G ( η ) | C ( η ) X y ∈ τ ( η ) Z R ∗ + × X q + ( s, s y ) e − αs y e αs a + ( x − y ) ν ( ds ) σ ( dx ) λ ( dη ) ≤ κ + β Z K ( X ) | G ( η ) | C ( η ) X y ∈ τ ( η ) e − αs y e αs y m ( s y ) λ ( dη ) ≤ κ + β k b L G k α,C . Finally, by (4.4), we get, for any G ∈ D , k b L G k α,C ≤ Z K ( X ) X x ∈ τ ( η ) Z R ∗ + × X q + ( s x , s ) a + ( x − y ) | G ( η + s y δ y ) | C ( η ) ν ( ds ) σ ( dy ) λ ( dη ) (2.1) = Z K ( X ) X y ∈ τ ( η ) X x ∈ τ ( η ) \{ y } q + ( s x , s y ) a + ( x − y ) | G ( η ) | C ( η − s y δ y ) λ ( dη ) ≤ β Z K ( X ) X y ∈ τ ( η ) X x ∈ τ ( η ) \{ y } q − ( s x , s y ) a − ( x − y ) e αs y C − e − αs y C ( η ) λ ( dη ) ≤ βC k b L G k α,C . Combining the estimates with the assumption (4.5), one gets the statement.15 roposition 4.3.
For any k ∈ L ∞ ( K ( X )) , the mapping ( L △ k )( η ) = − D ( η ) k ( η ) − Z R ∗ + × X X x ∈ τ ( η ) q − ( s x , s ) a − ( x − y ) k ( η + sδ y ) ν ( ds ) σ ( dy )+ X y ∈ τ ( η ) Z R ∗ + × X q + ( s, s y ) a + ( x − y ) k ( η − s y δ y + sδ x ) ν ( ds ) σ ( dx )+ X y ∈ τ ( η ) X x ∈ τ ( η ) \{ y } q + ( s x , s y ) a + ( x − y ) k ( η − s y δ y ) is well-defined and, for any G ∈ B bs ( K ( X )) , (2.9) holds. Hence, one can consider the dual semigroup T ∗ ( t ) in the dual space to the space L α,C ,which is isomorphic to K α,C = L ∞ (cid:0) K ( X ) , C ( η ) λ ν ⊗ σ ( dη ) (cid:1) . (4.7)This semigroup is ∗ -weakly differentiable (with respect to the duality (2.9)), and k t = T ∗ ( t ) k solves the weak equation (2.10) (see [8] for details). Note that, for some A, B > k T ∗ ( t ) k = k T ( t ) k ≤ Ae Bt , t > . Therefore, for λ ν ⊗ σ -a.a. η ∈ K ( X ), | k t ( η ) | ≤ Ae Bt C | τ ( η ) | e α P x ∈ τ ( η ) s x . (4.8)As we can see, comparing with the result (3.3), the strong mortality and competition ratesprevent factorial growth of correlation functions in n . A further analysis of the classicalsolution to the strong equation (2.11) can be done by using the sun-dual semigrouptechniques, see [8] for details. We consider now the Glauber-type dynamics. The corresponding analogue on the con-figuration spaces was studied in many papers, see e.g. [1, 8, 12, 14]. The generator of theGlauber dynamics is obtained from the Gibbs measure on the cone, which was constructedin [9] as follows. Let X = R d and consider a pair potential φ : X × X → R which satisfies the following two properties: • there exists R > φ ( x, y ) = 0 if | x − y | > R (where | · | denotes the Euclidean norm on R d );16 there exists δ > | x − y |≤ δ φ ( x, y ) > b d c d sup x,y (cid:12)(cid:12) − φ ( x, y ) ∨ (cid:12)(cid:12) , where b d is the volume of a unit ball in R d and c = c d,δ,R := √ d (1 + R/δ ) (see [9]for details).Fix also a θ >
0. It was shown in [9] that there exists a tempered Gibbs measure µ on K ( R d ) which fulfills the Dobrushin-Lanford-Ruelle equations Z K ( R d ) π ∆ ( B | η ) µ ( dη ) = µ ( B ) for any ∆ ∈ B c ( R d ) , where π ∆ is the so-called local specification constructed by φ and θ (see [9] for theprecise definitions and further details). Heuristically, µ ( dη ) = 1 Z exp (cid:16) − X x,y ∈ τ ( η ) s x s y φ ( x, y ) (cid:17) G θ ( dη ) , where Z is a normalizing factor. Proposition 5.1 (Georgii–Nguyen–Zessin identity, [9, Theorem 6.4]) . Let µ be a temperedGibbs measure on K ( X ) . Then, for any measurable function F : R d × K ( R d ) → R + , Z K ( R d ) Z R d F ( x, η ) η ( dx ) µ ( dη )= Z K ( R d ) Z R ∗ + × R d F ( x, η + sδ x ) e − Φ(( s,x ); η ) sν θ ( ds ) σ ( dx ) µ ( dη ) , (5.1) where, for η := ( s y , y ) y ∈ τ ( η ) ∈ K ( R d ) , Φ (( s, x ); η ) := 2 s X y ∈ τ ( η ) s y φ ( x, y ) . (5.2)We consider, for the fixed φ and θ , the following (pre-)Dirichlet form: E ( F, G ) := Z K ( X ) Z X D − x F ( η ) D − x G ( η ) η ( dx ) G θ ( dη ) , for F, G ∈ K ( B bs ( K ( X ))). Proposition 5.2.
Let
F, G ∈ K ( B bs ( K ( X ))) and ( LF )( η ) := X x ∈ τ ( η ) s x [ F ( η − s x δ x ) − F ( η )]+ Z R ∗ + × X [ F ( η + s x δ x ) − F ( η )] e − Φ(( s,x ); η )) sν θ ( ds ) σ ( dx ) where Φ is defined by (5.2) . Then E ( F, G ) = − Z K ( X ) ( LF )( η ) G ( η ) G θ ( dη ) . roof. By (5.1), we have E ( F, G ) = Z K ( X ) Z R ∗ + × X D − x F ( η ) D − x G ( η ) η ( dx ) G θ ( dη )= 12 Z K ( X ) Z R ∗ + × X D − x F ( η )( G ( η − s x δ x ) − G ( η )) η ( dx ) G θ ( dη )= Z K ( X ) Z R ∗ + × X D − x F ( η )( G ( η − s x δ x ) η ( dx ) G θ ( dη ) − Z K ( X ) Z R ∗ + × X D − x F ( η ) G ( η ) η ( dx ) G θ ( dη ) (5.1) = 12 Z K ( X ) Z R ∗ + × X D − x F ( η + s x δ x ) G ( η ) e − Φ(( s,x ); η ) sν θ ( ds ) σ ( dx ) G θ ( dη ) − Z K ( X ) Z R ∗ + × X D − x F ( η ) G ( η ) η ( dx ) G θ ( dη )= − Z K ( X ) Z R ∗ + × X ( F ( η + s x δ x ) − F ( η )) G ( η ) e − Φ(( s,x ); η ) sν θ ( ds ) σ ( dx ) G θ ( dη ) − Z K ( X ) X x ∈ τ ( η ) s x ( F ( η − s x δ x ) − F ( η )) G ( η ) G θ ( dη )= − Z K ( X ) ( LF )( η ) G ( η ) G θ ( dη ) . We denote S ( η ) := X x ∈ τ ( η ) s x , η ∈ K ( X ) , and also, for the fixed ( s, x ) ∈ R ∗ + × X , we set f s,x ( τ, y ) := e − sτφ ( x,y ) − , ( τ, y ) ∈ R ∗ + × X. Proposition 5.3.
For any G ∈ B bs ( K ( X )) , b LG := K − LKG satisfies ( b LG )( η ) = − S ( η ) G ( η )+ Z R ∗ + × X s X ξ ⊂ η G ( ξ + sδ x ) e − Φ(( s,x ) ,ξ ) e λ ( f s,x , η − ξ ) ν θ ( ds ) σ ( dx ) . Proof.
The proof can be done in the same way as e.g. in [8], see also the proof of (3.2).We consider again the space (4.1) with
C > α ∈ (0 , D := { G ∈ L α,C | S ( η ) G ( η ) ∈ L α,C } . Theorem 5.4.
Let
C > , α ∈ (0 , , θ > , and φ ( x, y ) ≥ , x, y ∈ X, (5.3) be such that θ · sup x ∈ X Z X φ ( x, y ) σ ( dy ) ≤ α (1 − α )2 C . (5.4)
Then the operator ( b L, D ) generates an analytic semigroup in the space L α,C . roof. Firstly, we consider the operator( b L G )( η ) = − S ( η ) G ( η ) , η ∈ K ( X )in L α,C with its maximal domain D . It can be shown identically to the proof of [8,Lemma 3.3], that (cid:0) b L , D (cid:1) is a generator of a contraction analytic semigroup in L α,C .Next, we define b L := b L − b L , i.e., for G ∈ B bs ( K ( X )),( b L G )( η ) = Z R ∗ + × X s X ξ ⊂ η G ( ξ + sδ x ) e − Φ(( s,x ) ,ξ ) e λ ( f s,x , η − ξ ) ν θ ( ds ) σ ( dx ) . We are going to show now that, under (5.3), operator b L is b L -bounded. Indeed, k b L G k α,C (5.3) ≤ Z K ( X ) X ξ ⊂ η Z R ∗ + × X s | G ( ξ + sδ x ) | e λ ( | f s,x | , η − ξ ) C ( η ) ν θ ( ds ) σ ( dx ) λ θ ( dη ) (2.1) = Z K ( X ) Z K ( X ) Z R ∗ + × X s | G ( ξ + sδ x ) | e λ ( | f s,x | , ξ ) C ( ξ + ξ ) ν θ ( ds ) σ ( dx ) λ θ ( dξ ) λ θ ( dξ ) (2.1) = Z K ( X ) Z K ( X ) X x ∈ τ ( ξ ) s x | G ( ξ ) | e λ ( | f s x ,x | , ξ ) C ( ξ − s x δ x + ξ ) λ θ ( dξ ) λ θ ( dξ ) ≤ Z K ( X ) | G ( ξ ) | C ( ξ ) X x ∈ τ ( ξ ) Z K ( X ) s x e λ ( | f s x ,x | , ξ ) C ( ξ ) C ( s x δ x ) − λ θ ( dξ ) λ θ ( dξ )= C − Z K ( X ) | G ( ξ ) | C ( ξ ) X x ∈ τ ( ξ ) s x e − αs x Z K ( X ) e λ ( | f s x ,x | Ce αs · , ξ ) λ θ ( dξ ) λ θ ( dξ )= C − Z K ( X ) | G ( ξ ) | C ( ξ ) X x ∈ τ ( ξ ) s x e − αs x exp Z R ∗ + × X | f s x ,x ( s, y ) | Ce αs ν θ ( ds ) σ ( dy ) ! λ θ ( dξ )where we have used (2.4); next, since, under (5.3), | f s,x ( τ, y ) | ≤ sτ φ ( x, y ), we mayestimate ≤ C − Z K ( X ) | G ( ξ ) | C ( ξ ) X x ∈ τ ( ξ ) s x e − αs x exp Z R ∗ + × X s x sφ ( x, y ) Ce αs ν θ ( ds ) σ ( dx ) ! λ θ ( dξ ) (2.2) = C − Z K ( X ) | G ( ξ ) | X x ∈ τ ( ξ ) s x exp " Cθ Z R ∗ + × X φ ( x, y ) e ( α − s dsσ ( dx ) − α ! s x C ( ξ ) λ θ ( dξ ) ≤ C k b L G k , for α ∈ (0 , ≥ Cθ Z R ∗ + × X φ ( x, y ) e ( α − s dsσ ( dx ) − α = 2 Cθ Z X φ ( x, y ) σ ( dx ) Z R ∗ + e ( α − s ds − α = 2 C − α θ Z R d φ ( x, y ) σ ( dx ) − α, that holds under (5.4). Therefore, b L has b L -bound C < that yields the statement.19y using the Minlos identity (2.1), we immediately get the following result: Proposition 5.5.
For any k ∈ L ∞ ( K ( X )) , the mapping (cid:0) L △ k (cid:1) ( η ) = − S ( η ) k ( η )+ X x ∈ τ ( η ) s x e − Φ(( s,x ) ,η − s x δ x ) Z K ( X ) e λ ( f s x ,x , ξ ) k ( η + ξ − s x δ x ) λ θ ( dξ ) is well-defined and, for any G ∈ B bs ( K ( X )) , (2.9) holds. Again, one can consider the dual semigroup T ∗ ( t ) in the space (isomorphic to) K α,C given by (4.7), so that k t = T ∗ ( t ) k solves the weak equation (2.10), and (4.8) holds.Further analysis of the sun-dual semigroup T ⊙ ( t ) (which provides a solution to (2.10) ona subspace of K α,C ) can be done in the same way as in [8]. Acknowledgements
P.K. was supported by the DFG through the IRTG 2235 “Searching for the regular in theirregular: Analysis of singular and random systems”.
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