aa r X i v : . [ m a t h . P R ] M a y Feller Evolution Systems: Generators andApproximation
Bj¨orn B¨ottcher ∗ May 3, 2013
Abstract
A time and space inhomogeneous Markov process is a Feller evolutionprocess, if the corresponding evolution system on the continuous functionsvanishing at infinity is strongly continuous. We discuss generators of suchsystems and show that under mild conditions on the generators a Fellerevolution can be approximated by Markov chains with L´evy increments.The result is based on the approximation of the time homogeneousspace-time process corresponding to a Feller evolution process. In par-ticular, we show that a d -dimensional Feller evolution corresponds to a d + 1-dimensional Feller process. It is remarkable that, in general, thisFeller process has a generator with discontinuous symbol. Keywords:
Markov process, evolution system, propagator, space-time process,Feller process, approximation, pseudo-differential operatorAMS Subject Classification:
Primary:
Secondary:
Markov processes with continuous time parameter are used in various applica-tions and often approximations and simulations of these processes are required.In an infinitesimal sense (see the next section) such a process is composed ofL´evy processes. Thus it is natural to try to approximate a Markov process byMarkov chains with L´evy increments. In the case of time homogeneous Markovprocesses general conditions for such an approximation were given in [4]. Weare going to extend the result to time inhomogeneous Markov processes. Themain tool is the transformation of a time inhomogeneous Markov process to atime homogeneous Markov process, which will be recalled in Section 3. Thetransformed process is usually called the corresponding space-time process andthe transformation was already used by Doob [7, p. 226] and Dynkin [8, Section4.6]. We will derive a necessary and sufficient condition for the transformedprocess to be a Feller process. Furthermore, the generator of this Feller processis analyzed. In Section 4 the approximation is presented and discussed. ∗ [email protected] , TU Dresden, Fachrichtung Mathematik, Institut f¨urMath. Stochastik, 01062 Dresden, Germany X t ) t ∈ R , an evolution system ( U ( s, t )) s,t ∈ R ,s ≤ t and a semigroup ( T ( t )) t ∈ R will beoften denoted just by X t , U ( s, t ) and T ( t ) , respectively. The Borel measurablefunctions on R d will be denoted by B ( R d ) and the continuous functions by C ( R d ). The subscripts c , b and ∞ denote functions with compact support,bounded functions and functions vanishing at ∞ , respectively; furthermore, asuperscript indicates the number of existing derivatives. The uniform norm isdenoted by k . k ∞ . Let X t be an R d valued time (and space) inhomogeneous Markov process onthe probability space (Ω , A , P ) . Then the corresponding evolution system U ( s, t ) f ( x ) := E ( f ( X t ) | X s = x ) , s ≤ t, s, t ∈ R is well defined on B b ( R n ) . The linear operators U ( s, t ) are positivity preservingand satisfy U ( s, t )1 = 1, U ( s, s ) = id and the evolution property U ( s, t ) = U ( s, r ) U ( r, t ) for s ≤ r ≤ t. Such families of operators are well studied in the literature, e.g. Yosida [17,Section XIV.4], Pazy [13, Chapter 5]. The following definitions are analogousto Gulisashvili and van Casteren [9, Section 2.3], who use the term backwardpropagator for an evolution system.Corresponding to an evolution system a family of right generators is givenby A + s f := lim h ↓ U ( s, s + h ) f − fh for each s ∈ R (1)which is defined for all f ∈ C ∞ ( R d ) such that the limit exists in a strong sense(i.e. with respect to k · k ∞ ). In this case we write f ∈ D ( A + s ) . If one weakens(1) to a pointwise limit, the corresponding operator is called extended pointwisegenerator (a notion which will be of importance in Theorem 3.3). Analogouslythe left generators are defined by A − s f := lim h ↓ U ( s − h, s ) f − fh on D ( A − s ) . The family of operators U ( s, t ) is strongly continuous, if for each v, w ∈ R , v ≤ w lim ( s,t ) → ( v,w ) s ≤ t k U ( s, t ) f − U ( v, w ) f k ∞ = 0 . (2)Note that a family of linear operators U ( s, t ) on C ∞ satisfying (2), k U ( s, t ) f k ∞ ≤k f k ∞ , U ( s, t ) f ( x ) = U ( s, r ) U ( r, t ) f ( x ) for s ≤ r ≤ t and U ( s, t ) f ≥ f ≥ Feller evolution system .We denote by ddt + ( ddt − ) the right (left) derivative. Thus the evolutionproperty leads to the following evolution equations corresponding to the process: ddt ± U ( s, t ) = U ( s, t ) A ± t , (3) dds ± U ( s, t ) = − A ± s U ( s, t ) . (4)2quation (3) + is called forward equation and (4) − is called backward equation.Note that only in the case of the backward equation it makes sense to talk aboutsolutions of the corresponding initial value problem. In the case of the forwardequation one can only consider fundamental solutions due to the interchangedorder of U and A. These equations are equivalent to the Kolmogorov equationsif the corresponding process has transition densities p ( s, x ; t, y ) . Then U ( s, t ) f ( x ) = Z f ( y ) p ( s, x ; t, y ) dy =: h f, p ( s, x ; t, . ) i L holds and thus the forward equation reads as (cid:10) f, ddt p ( s, x ; t, . ) (cid:11) L = (cid:10) f, A + t ⋆ p ( s, x ; t, . ) (cid:11) L , where A + t ⋆ is the (formal) adjoint of A + t .The operators A + s (resp. A − s ) satisfy the positive maximum principle, i.e.for f ∈ D ( A + s ) the following implication holds: If there exists x ∈ R d with f ( x ) = sup x ∈ R n f ( x ) ≥ , then A + s f ( x ) (cid:12)(cid:12) x = x ≤ . This property of A + s (resp. A − s ) is a consequence of (1) and the fact, thatfor f ∈ D ( A + s ) (resp. A − s ) attaining its positive maximum at some point x thefollowing inequality holds: U ( s, t ) f ( x ) ≤ U ( s, t ) f + ( x ) ≤ k f + k ∞ = f ( x ) , where f + := f { f ≥ } . Therefore, if the set C ∞ c ( R d ) is a subset of D ( A + s ) (resp. D ( A − s )), we knowby Courr`ege [6] (see Jacob [10, Section 4.5]) that − A + s (resp. − A − s ) on C ∞ c isa pseudo-differential operator with continuous negative definite symbol , i.e. itadmits the representation − A + s f ( x ) = (2 π ) − d Z R d e ixξ q + ( s, x, ξ ) ˆ f ( ξ ) dξ, (5)where ˆ f ( ξ ) = (2 π ) − d R R d e − ixξ f ( x ) dx denotes the Fourier transform of f and q + ( s, x, · ) is for fixed ( s, x ) a continuous negative definite function in the senseof Berg and Forst [1]. Conversely, for any operator defined via (5) the function q + is called the symbol of the operator.An explicit construction of Feller evolutions for a given symbol can be foundin [2]. Another option is to construct a Feller process with a constant drift coor-dinate and consider the process of the remaining coordinates as Feller evolution(see Theorem 3.2 and Corollary 4.2 below). For a survey of constructions ofFeller processes for a given symbol see [12].Note that for a time homogeneous evolution system (i.e. U such that U ( s, s + h ) = U ( s − h, s ) for all s, h ≥
0) the left and right generators coincide and donot depend on time. Thus T ( h ) := U ( s, s + h ) defines a Feller semigroup withgenerator A := A − s = A + s . In general, however, the left and the right generator do not coincide as thefollowing example illustrates. 3 xample 2.1.
Consider a process on R which drifts with slope α > until time s and afterwards with slope β > , α = β. Thus the process started in x ∈ R at time t is given by X t = ( x + α · ( t − t ) , t < s x + α · ( s − t ) + β · ( t − s ) , t ≥ s for t ≥ t , and for f ∈ C ∞ c ( R ) A − s f ( x ) = lim h ↓ E ( f ( X s ) | X s − h = x ) − f ( x ) h = lim h ↓ f ( x + αh ) − f ( x ) h = αf ′ ( x ) ,A + s f ( x ) = lim h ↓ E ( f ( X s + h ) | X s = x ) − f ( x ) h = lim h ↓ f ( x + βh ) − f ( x ) h = βf ′ ( x ) . Moreover the symbol corresponding to A + s as in (5) is given by q + ( s, x, ξ ) = − il ( s ) ξ with l ( s ) := ( α , s < s ,β , s ≥ s . (See Remark 3.6 for the space-time transformation of this process.) The following lemma gives some condition for the left and right generatorto coincide.
Lemma 2.2.
Fix s and select f ∈ D ( A + s ) such that there exists some δ > with f ∈ T r ∈ ( s − δ,s ] D ( A + r ) . If A + r exists uniformly for r ∈ ( s − δ, s ] , i.e. lim h ↓ sup r ∈ ( s − δ,s ] (cid:13)(cid:13)(cid:13)(cid:13) U ( r, r + h ) f − fh − A + r f (cid:13)(cid:13)(cid:13)(cid:13) ∞ = 0 (6) and r A + r is strongly continuous from the left in s , i.e. lim h ↓ k A + s − h f − A + s f k ∞ = 0 , (7) then f ∈ D ( A − s ) and A + s f = A − s f. Proof.
We have for h < δ lim h ↓ (cid:13)(cid:13)(cid:13)(cid:13) U ( s − h, s ) f − fh − A + s f (cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ lim h ↓ (cid:13)(cid:13)(cid:13)(cid:13) U ( s − h, s − h + h ) f − fh − A + s − h f (cid:13)(cid:13)(cid:13)(cid:13) ∞ + lim h ↓ k A + s − h f − A + s f k ∞ ≤ lim h ↓ sup r ∈ ( s − δ,s ] (cid:13)(cid:13)(cid:13)(cid:13) U ( r, r + h ) f − fh − A + r f (cid:13)(cid:13)(cid:13)(cid:13) ∞ + 0= 0and the result follows. (cid:4) The assumptions of Lemma 2.2 are, for example, satisfied if the generatoris a pseudo-differential operator whose symbol continuously depends on time(implying (7)) and has bounded coefficients in the sense of (17) below (implying(6)). 4
Transformation of time inhomogeneousMarkov processes
To transform an R d valued time inhomogeneous Markov process X t defined on(Ω , A , P ) into a time homogeneous Markov process e X t defined on ( e Ω , e A , e P ) wefollow [16, Section 8.5.5]. For generality in this section T will denote the timeset on which X t is defined, i.e. so far we considered T = R . But T = [0 , ∞ )would also be possible. Note that in both cases the transformed process e X t willalways be defined only on the time set [0 , ∞ ) . For X t there exists a transition function P : T × R d ×T ×B → [0 ,
1] such thatfor each s, t ∈ T , s ≤ t, x ∈ R d , B ∈ B the function P ( s, · ; t, B ) is measurable, P ( s, x ; t, · ) is a probability measure, P ( s, x ; s, B ) = B ( x ) and P ( s, X s ; t, B ) = P ( X t ∈ B | X s ) . Furthermore, since X t is a Markov process also the ChapmanKolmogorov equations P ( s, x ; t, B ) = R R d P ( r, y ; t, B ) P ( s, x ; r, dy ) hold for s ≤ r ≤ t and x, y ∈ R d . The standard way to define the transformed process is:
Transformation 3.1.
Let X t be as above. • New state space : T × R d with elements e x := ( s, x ) , s ∈ T , x ∈ R d . On this space we consider the σ -algebra e B consisting of all sets B ⊂ T × R d such that for all s ∈ T the cuts B s := { x : ( s, x ) ∈ B } are elements ofthe Borel σ -algebra on R d . • New sample space : e Ω :=
T × Ω with elements e ω := ( s, ω ) , s ∈ T ,ω ∈ Ω , and the σ -algebra e A := { A ⊂ e Ω : A s ∈ A , ∀ s ∈ T } where A s := { ω : ( s, ω ) ∈ A } . • Space-time process : e X t ( e ω ) = e X t ( s, ω ) := ( s + t, X s + t ( ω )) , t ∈ [0 , ∞ ) with the probability measure defined for A ∈ e A and e x ∈ T × R d by e P e x ( A ) = e P ( A | e X = ( s, x )) := P ( A s | X s = x ) , i.e. the transition probabilities are given by e P ( e X t ∈ B | e X = e x ) = e P ( e X t ∈ B | e X = ( s, x )) = P ( X s + t ∈ B s + t | X s = x ) where B ∈ e B , e x ∈ T × R d , and thus the transition function is defined by e P ( t, e x, B ) := P ( s, x ; s + t, B s + t ) . In the transformation the change of the probability space might seem coun-terintuitive, since the process is extended by adding a deterministic drift ina further dimension but no further randomness is introduced. Neverthelessit is canonical, if one recalls the construction of Markov processes using Kol-mogorov’s theorem.To see that the new process is a Markov process denote by F t its filtrationand note that for each t ∈ [0 , ∞ ), e x ∈ T × R d , B ∈ e B the function e x e P ( t, e x, B )is measurable, since for e x = ( s, x ) the function is measurable in x and the5 -Algebra corresponding to s is the power set of T . Furthermore, e P ( t, e x, . )is a probability measure, e P (0 , e x, B ) = B ( e x ) holds and for all t, r ∈ T and e x, e y ∈ T × R d the Chapman Kolmogorov equation Z e P ( r, e y, B ) e P ( t, e x, d e y ) = Z R d P ( s + t, y ; s + t + r, B s + t + r ) P ( s, x ; s + t, dy )= P ( s, x ; s + t + r, B s + t + r )= e P ( t + r, e x, B )holds. Hence the process is a Markov process if and only if e P e x ( e X t + h ∈ B |F t ) = e P ( h, e X t , B )i.e. for all A ∈ F t e P e x ( A ∩ { e X t + h ∈ B } ) = Z A e P ( h, e X t ( e ω ) , B ) e P e x ( d e ω ) . This equality holds since the transformation given above and the Markov prop-erty of X t yield Z A e P ( h, e X t ( e ω ) , B ) e P e x ( d e ω )= Z A s P ( s + t, X s + t ( ω ); s + t + h, B s + t + h ) P ( dω | X s = x )= P ( A s ∩ { X s + t + h ∈ B s + t + h }| X s = x )= e P e x ( A ∩ { e X t + h ∈ B } ) . Thus e X t is a Markov process and therefore there exists a correspondingsemigroup T ( t ) on f ∈ B b ( T × R d ) given by T ( t )( e x ) = e E ( f ( e X t ) | e X = e x ) . (8)The following theorem provides a necessary and sufficient condition for T ( t )to be a Feller semigroup, i.e. a strongly continuous positivity preserving con-traction semigroup on C ∞ . Theorem 3.2.
Let X t be a Markov process with corresponding evolution system U ( s, t ) . Furthermore, let e X t be the time homogeneous transformation (as definedabove) of X t , and T ( t ) be the semigroup associated with e X t as in (8) . Then thefollowing statements are equivalent:i) ( U ( s, t )) s,t ∈T ,s ≤ t is a Feller evolution system on C ∞ ( R d ) ,ii) ( T ( t )) t ≥ is a Feller semigroup on C ∞ ( T × R d ) . Proof.
Note that C ∞ ( T × R d ) = { f ∈ C ( T × R d ) : lim | e x |→∞ f ( e x ) = 0 } . f ∈ C ∞ ( T × R d ) and define g s ( x ) := f ( s, x ) = f ( e x ) for all e x = ( s, x ) ∈T × R d then the semigroup has the representation T ( t ) f ( e x ) = e E ( f ( f X t ) | f X = e x )= e E ( f ( f X t ) | f X = ( s, x ))= E ( f ( s + t, X s + t ) | X s = x )= E ( g s + t ( X s + t ) | X s = x )= U ( s, s + t ) g s + t ( x ) . (9)i) ⇒ ii): Clearly T ( t ) is a positivity preserving contraction semigroup on B b .Thus it remains to show that1. T ( t ) maps C ∞ ( T × R d ) into C ∞ ( T × R d ),2. T ( t ) is strongly continuous on C ∞ ( T × R d ).First note, that since U ( s, t ) is strongly continuous, it is also locally uniformstrongly continuous, i.e. for each compact K ⊂ T lim ( s,t ) → ( v,w ) s ≤ t k U ( s, t ) g − U ( v, w ) g k ∞ = 0 uniformly for v, w ∈ K, v ≤ w. (10)The first step of proving 1. is to show that e x T ( t ) f ( e x ) is continuous. Let e x = ( s, x ) , e y = ( r, y ) ∈ T × R d with e y fixed. Then | T ( t ) f ( e x ) − T ( t ) f ( e y ) | = | U ( s, s + t ) g s + t ( x ) − U ( r, r + t ) g r + t ( y ) |≤ | U ( s, s + t )( g s + t − g r + t )( x )) | + | U ( s, s + t ) g r + t ( x ) − U ( r, r + t ) g r + t ( x ) | + | U ( r, r + t ) g r + t ( x ) − U ( r, r + t ) g r + t ( y ) | holds, where the first term can be estimated by k g s + t ( . ) − g r + t ( . ) k ∞ using thecontraction property of U ( s, t ) and the second term converges by the local uni-form strong continuity (10). Thus each of these terms is smaller than ε for | s − r | < δ for some δ >
0. Furthermore, r and t being fixed the function x U ( r, r + t ) g r + t ( x ) is continuous. Hence also the last term gets smaller than ε for | x − y | < δ for some δ >
0. Thus taking | e x − e y | < δ ∧ δ yields thecontinuity.The next step is to show that T ( t ) f ( e x ) | e x |→∞ −−−−→ . Let ε > . It holds that | T ( t ) f ( e x ) | = | U ( s, s + t ) g s + t ( x ) | ≤ sup x | g s + t ( x ) | = sup x | f ( s + t, x ) | , (11)and note that | e x | = | s | + | x | , i.e. for | e x | → ∞ at least one of | s | and | x | islarge.Since f ∈ C ∞ ( T × R d ) there exists R = R ( t ) such that | f ( s + t, x ) | < ε uniformly in x for | s | > R . Otherwise, if | s | ≤ R , let h t ( x ) := sup | s | ≤ R | g s + t ( x ) | and note that h t ∈ C ∞ ( R d ) . Thus | T ( t ) f ( e x ) | = | U ( s, s + t ) g s + t ( x ) | ≤ U ( s, s + t ) h t ( x ) .
7y the uniformity of (10) and the Heine-Borel theorem the set { s ∈ T : | s | ≤ R } can be covered by equally sized balls with centres in some finite set R ⊂{ s ∈ T : | s | ≤ R } such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup | s | ≤ R U ( s, s + t ) h t − max r ∈R U ( r, r + t ) h t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ ≤ ε . Since R is finite and U ( s, t ) is an evolution system on C ∞ there exists an R such that for | x | > R max r ∈R U ( r, r + t ) h t ( x ) < ε . Hence for | e x | > R := √ R ∨ R either | s | > R and thus (11) implies the resultor | s | ≤ R and | x | > R and therefore | T ( t ) f ( e x ) | ≤ sup | s | ≤ R U ( s, s + t ) h t ( x ) < ε. To show 2. note that | T ( t ) f ( e x ) − f ( e x ) | = | U ( s, s + t ) g s + t ( x ) − g s ( x ) |≤ | U ( s, s + t ) g s + t ( x ) − U ( s, s + t ) g s ( x ) | + | U ( s, s + t ) g s ( x ) − g s ( x ) |≤ k g s + t − g s k ∞ + | U ( s, s + t ) g s ( x ) − g s ( x ) |≤ (cid:20) sup s,x (cid:12)(cid:12) f ( s + t, x ) − f ( s, x ) (cid:12)(cid:12)(cid:21) + | U ( s, s + t ) g s ( x ) − g s ( x ) | . holds, where the first term converges to 0 as t → f. For the second term fix ε >
0. Then there exists (analogous to (11) andthe reasoning thereafter) an
R > x sup | s | >R | U ( s, s + t ) g s ( x ) − g s ( x ) | < ε t <
1. Furthermore, r U ( s, s + t ) g r ( x ) − g r ( x ) isequicontinuous since | U ( s, t + s ) g r ( x ) − g r ( x ) − U ( s, t + s ) g q ( x ) + g q ( x ) | ≤ k g r − g q k ∞ and therefore we find, as above, a finite set R ⊂ { r ∈ T : | r | ≤ R } such that forall x ∈ R d sup | r |≤ R | U ( s, s + t ) g r − g r ( x ) | ≤ max r ∈R | U ( s, s + t ) g r ( x ) − g r ( x ) | + ε . Since R is finite and U ( s, t ) satisfies (10) there exists a δ > | s |≤ R max r ∈R | U ( s, s + t ) g r ( x ) − g r ( x ) | < ε t < δ. Putting the above together yields for t < δ ∧ x sup s | U ( s, s + t ) g s ( x ) − g s ( x ) | ≤ sup x sup | s |≤ R | U ( s, s + t ) g s ( x ) − g s ( x ) | + ε ≤ sup | s |≤ R max r ∈R | U ( s, s + t ) g r ( x ) − g r ( x ) | + 2 ε < ε. T ( t ) is strongly continuous.ii) ⇒ i): X t is a Markov process, hence U ( s, t ) is a positivity preserving contrac-tion evolution system on B b . Equation (9) with f ( e x ) := g ( x ) , g ∈ C ∞ ( R d ) readsas T ( t ) f ( e x ) = U ( s, s + t ) g ( x )and thus U ( s, s + t ) : C ∞ ( R d ) → C ∞ ( R d ) . The function e x T ( t ) f ( e x ) is uniformly continuous and thus s T ( t ) f ( s, x )is equicontinuous (w.r.t. x ∈ R d ). Finally | U ( s, t ) g ( x ) − U ( v, w ) g ( x ) | = | T ( t − s ) f ( s, x ) − T ( w − v ) f ( v, x ) |≤ | T ( t − s ) f ( s, x ) − T ( w − v ) f ( s, x ) | + | T ( w − v ) f ( s, x ) − T ( w − v ) f ( v, x ) |≤ sup e x ∈T × R d | T ( t − s ) f ( e x ) − T ( w − v ) f ( e x ) | + sup x ∈ R d | T ( w − v ) f ( s, x ) − T ( w − v ) f ( v, x ) | yields the strong continuity. (cid:4) Now it is straightforward to calculate the generator of T ( t ): Theorem 3.3.
Let X t be a Feller evolution with evolution system U ( s, t ) andright generators A + s . Furthermore, let e X t be its time homogeneous transforma-tion with associated semigroup T ( t ) as in (8) . Then the (extended pointwise)generator L of T ( t ) is given for all f ∈ C ∞ ( T × R d ) satisfying • f ( ., x ) ∈ C ( T ) for all x ∈ R d , • f ( s, . ) ∈ D ( A + s ) for all s ∈ T by Lf ( e x ) = ∂∂s f ( s, x ) + A + s g s ( x ) where e x = ( s, x ) and g s ( x ) = f ( s, x ) . (12) Remark 3.4.
Note that (12) does not imply that the given f is in the domainof the generator L . For this one would have to ensure that Lf is in C ∞ ( T × R d ) .See Remark 3.6 for further discussion and Lemma 3.7 for a sufficient condition. Proof of Theorem 3.3.
Using (9) yields T ( t ) f ( e x ) − f ( e x ) t = E ( f ( s + t, X s + t ) | X s = x ) − E ( f ( s, X s + t ) | X s = x ) t + U ( s, s + t ) g s ( x ) − g s ( x ) t where the second term converges for t → A + g s ( x ) . For the first term set f (1 , ( s, x ) := ∂∂s f ( s, x ) and note that E (cid:18) f ( s + t, X s + t ) − f ( s, X s + t ) t (cid:12)(cid:12)(cid:12)(cid:12) X s = x (cid:19) = E (cid:18) t Z s + ts f (1 , ( r, X s + t ) dr (cid:12)(cid:12)(cid:12)(cid:12) X s = x (cid:19) . (cid:12)(cid:12)(cid:12)(cid:12) E (cid:18) t Z s + ts f (1 , ( r, X s + t ) dr (cid:12)(cid:12)(cid:12)(cid:12) X s = x (cid:19) − E ( f (1 , ( s, X s + t ) | X s = x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup y sup r ∈ [ s,s + t ] | f ( r, y ) − f ( s, y ) | vanishes for t → f is uniformly continuous. Finally, defining the function h s ( x ) := f (1 , ( s, x ) and using strong continuity yields E ( f (1 , ( s, X s + t ) | X s = x ) = U ( s, s + t ) h s ( x ) t → −−−→ h s ( x ) = ∂∂s f ( s, x ) , which proves the statement. (cid:4) Furthermore if C ∞ c is in the domain of all right generators, then the operator L has a representation as pseudo-differential operator: Corollary 3.5.
Let T = R . If A + s is defined on the test functions C ∞ c ( R d ) and thus can be represented (cf. (5) ) as pseudo-differential operator with symbol q + ( s, x, ξ ) then L given by (12) is on C ∞ c ( T × R d ) a pseudo-differential operatorand its symbol is given by q L (˜ x, ˜ ξ ) := − iσ + q + ( s, x, ξ ) (13) with e x = ( s, x ) and e ξ = ( σ, ξ ) , s, σ ∈ R , x, ξ ∈ R d . Proof.
By linearity the symbol of the sum of the operators is the sum of thesymbols and for s with covariable σ the operator dds corresponds to − iσ . (cid:4) Note that for T = [0 , ∞ ) one would have to take care of boundary terms.Alternatively, in the setting of Corollary 3.5, a time inhomogeneous Markovprocess X t only defined for positive times, i.e. T = [0 , ∞ ) can be extended onto T = R by setting for s < q + ( s, x, ξ ) := q + (0 , x, ξ ) . (14) Remark 3.6.
One subtlety of Corollary 3.5 is that it states that the generatorof the time homogeneous process can be defined on C ∞ c ( T × R d ) , but it doesnot state that C ∞ c ( T × R d ) is a subset of the domain of the generator of thisprocess. To understand this, consider a Feller process on R whose sample pathare deterministic with slope α > below level s and slope β > above level s , α = β . Its generator is Lf ( x ) = l ( x ) f ′ ( x ) with l ( x ) := ( α , x < s β , x ≥ s (15) for all f ∈ C ∞ ( R ) with f ( s ) = 0 , where the last restriction is due to therequirement that L : D ( L ) → C ∞ . Thus C ∞ c ( R ) is not a subset of the domain,nevertheless (15) is well defined for f ∈ C ∞ c ( R ) and L has a representation aspseudo-differential operator with symbol − il ( x ) ξ. Finally note that the above discussion also applies to the process introducedin Example 2.1 whose transformed process has the symbol − iσ − il ( s ) ξ where ( s, x ) has covariable ( σ, ξ ) .
10e close this section with a result which ensures that C ∞ c is in the domainof the transformed process. Lemma 3.7.
In the setting of Theorem 3.3 let C ∞ c ( R d ) ⊂ D ( A + s ) for all s , anddenote by q + ( s, x, ξ ) the symbol of A + s . If s q + ( s, x, ξ ) is continuous for all x, ξ then C ∞ c ( T × R d ) ⊂ D ( L ) . Proof.
Let f ∈ C ∞ c ( T × R d ) then Lf is in C ∞ ( T × R d ) by dominated con-vergence, and (12) implies that C ∞ c ( T × R d ) is a subset of the domain of thepointwise generator. Finally the result follows, since for Feller semigroups thepointwise generator coincides with the generator ([15, Lemma 31.7], see also [14,Lemma III.6.7]). (cid:4) Now we are going to show that a process X t with symbol q + given by (5) can– under the assumptions of the following lemma – be approximated by Markovchains with time steps of size n . For each n ∈ N the approximating Markovchain ( Z n ( k )) k ∈ N is defined by Z n (0) := X and transition kernels (from x attime k into dy at time k + 1) ν kn ,x, n ( dy ) where Z R d e iyξ ν s,x, n ( dy ) = e ixξ − n q + ( s,x,ξ ) . (16) Lemma 4.1.
Let ( X t ) t ∈ R be a Markov process on R d with corresponding Fellerevolution system U ( s, t ) . Assume that C ∞ c ( R d ) is an operator core for the cor-responding family of right generators A + s , i.e.for all s the closure of A + s (cid:12)(cid:12) C ∞ c is A + s . Furthermore, assume that the symbol − q + ( s, x, ξ ) of A + s (cid:12)(cid:12) C ∞ c satisfies | q + ( s, x, ξ ) | ≤ c (1 + | ξ | ) for all s, x, ξ, (17) s q + ( s, x, ξ ) is continuous for all x, ξ. (18) Then the test functions C ∞ c ( R d +1 ) are an operator core for the generator L (given in Theorem 3.3) of the corresponding Feller process e X t and for the symbol − e q ( e x, e ξ ) of L (cid:12)(cid:12) C ∞ c ( R d +1 ) exists a c > such that | e q ( e x, e ξ ) | ≤ c (1 + | e ξ | ) for all e x, e ξ. (19) Proof.
By Corollary 3.5 e q ( e x, e ξ ) = − iσ + q + ( s, x, ξ )11olds and thus (17) implies (19). Furthermore, the core property follows byLemma 3.7 and linearity, since L = dds + A + s and these operators act on different components. (cid:4) Under the conditions of the Lemma 4.1 the process e X satisfies the assump-tions of the approximation theorem in [4]. Thus, the process e X t is approximatedby the Markov chains ( e Y n ([ tk ])) k ∈ N on R × R d with e Y n (0) := ( r, X ) and tran-sition kernel µ e x, n ( d e y ) where Z R × R d e i e y e ξ µ e x, n ( d e y ) = e i e x e ξ − n e q ( e x, e ξ ) . Since the last d coordinates of e X t started in (0 , x ) coincide with X t startedin x we only need to check if we can simplify the above expression for thesecoordinates. Using e x = ( s, x ) , e y = ( r, y ) and e ξ = ( σ, ξ ) we get e q ( e x, e ξ ) = − iσ + q ( s, x, ξ ) , (20) e i e x e ξ − n e q ( e x, e ξ ) = e i ( s + n ) τ + ixξ − q ( s,x,ξ ) , (21) µ e x, n ( d e y ) = δ s + n ( dr ) × ν s,x, n ( dy ) . (22)Thus the processes e X can be approximated by the Markov chain defined by(16). An approximations of this type is easily implemented for simulations, see[3] for an implementation of the time homogeneous case.Finally we restate, using the transformation introduced in Section 3, a resultby Chernoff ([5], see also [11, Theorem 2.5]) which shows that the approximationgiven above could also be used to construct Feller evolutions directly for a givenfamily of probability measures ν s,x, n as in (16). The construction is formulatedin terms of the operators V ( s, h ) g ( x ) := Z R d g ( y ) ν s,x,h ( dy ) s ∈ R , h ≥ g ∈ C ∞ ( R d ) . Corollary 4.2. If ( V ( s, h )) s ∈ R ,h ≥ is a family of strongly continuous linearcontractions on ( C ∞ ( R d ) , k · k ∞ ) which satisfies the following properties:i) V s, = id, ii) the strong derivatives ddh V ( s, h ) (cid:12)(cid:12)(cid:12)(cid:12) h =0 = V ( s, ′ are densely defined,iii) lim n →∞ (cid:13)(cid:13)(cid:13) V ( s, tn ) V ( s + tn , tn ) V ( s + tn , tn ) · · · V ( s + ( n − tn , tn ) g − U ( s, t ) g (cid:13)(cid:13)(cid:13) ∞ = 0 for all g ∈ C ∞ ( R d ) . Then ( U ( s, t )) s,t ∈ R ,s ≤ t is a Feller evolution system on C ∞ ( R d ) and its gener-ators A + s extend V ( s, ′ . Moreover the convergence in iii) is uniform for t, s from compact intervals. eferences [1] C. Berg and G. Forst. Potential Theory on Locally Compact AbelianGroups . Springer, Berlin, 1975.[2] B. B¨ottcher. Construction of time inhomogeneous Markov processes viaevolution equations using pseudo-differential operators.
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