Brownian Motions on Metric Graphs with Non-Local Boundary Conditions II: Construction
aa r X i v : . [ m a t h . P R ] M a y BROWNIAN MOTIONS ON METRIC GRAPHS WITHNON-LOCAL BOUNDARY CONDITIONS II: CONSTRUCTION
FLORIAN WERNER
Abstract.
A pathwise construction of discontinuous Brownian motions onmetric graphs is given for every possible set of non-local Feller–Wentzell bound-ary conditions. This construction is achieved by locally decomposing the met-ric graphs into star graphs, establishing local solutions on these partial graphs,pasting the solutions together, introducing non-local jumps, and verifying thegenerator of the resulting process. Introduction
This article is the final part in a series of works in which we achieve a classificationand pathwise construction of Brownian motions on metric graphs. In [6], we definedBrownian motions on metric graphs in accordance with previous works of Itˆo andMcKean [1] and Kostrykin, Potthoff and Schrader [3], that is, as right continuous,strong Markov processes which behave on every edge of the graph like the standardone-dimensional Brownian motion. There, we showed that the generator A = ∆of every Brownian motion on a metric graph G satisfies at each vertex point v ∈ V a non-local Feller–Wentzell boundary condition ∀ f ∈ D ( A ) : p v f ( v ) − X l ∈L ( v ) p v,l f ′ l ( v ) + p v f ′′ ( v ) − Z G\{ v } (cid:0) f ( g ) − f ( v ) (cid:1) p v ( dg ) = 0for some constants p v ≥ p v,l ≥ l ∈ L ( v ) emanating from v , p v ≥ p v on G\{ v } , normalized by p v + X l ∈L ( v ) p v,l + p v + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) p v ( dg ) = 1 , and p v being an infinite measure if P l ∈L ( v ) p v,l + p v = 0.After having developed the necessary process transformations of concatenationsand process revivals in [8], collected the characteristic properties of Brownian mo-tions on metric graphs in [6] and constructed all Brownian motions on star graphsin [7], we are now in the position to give a complete pathwise construction of Brow-nian motions on any metric graph for every admissible set of Feller–Wentzell data,thus proving the following existence theorem: Institut f¨ur Mathematik, Universit¨at Mannheim, 68131 Mannheim, Germany
E-mail address : [email protected] . Date : September 28, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Brownian motion, non-local Feller–Wentzell boundary condition, met-ric graph, Markov process, Feller process.
Theorem 1.1.
Let G = ( V , E , I , ∂, ρ ) be a metric graph, and for every v ∈ V letconstants p v ≥ , p v,l ≥ for each l ∈ L ( v ) , p v ≥ and a measure p v on G\{ v } begiven with p v + X l ∈L ( v ) p v,l + p v + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) p v ( dg ) = 1 , and p v (cid:0) G\{ v } (cid:1) = + ∞ , if X l ∈L ( v ) p v,l + p v = 0 . Then there exists a Brownian motion X on G which is continuous inside all edges,such that its generator satisfies D ( A ) ⊆ n f ∈ C ( G ) : ∀ v ∈ V : p v f ( v ) − X l ∈L ( v ) p v,l f ′ l ( v ) + p v f ′′ ( v ) − Z G\{ v } (cid:0) f ( g ) − f ( v ) (cid:1) p v ( dg ) = 0 o . Construction Approach
The construction proceeds as follows: We will begin with Brownian motions onstar graphs which implement the corresponding “local” boundary conditions (in-cluding “small jumps”) at their respective vertices. When the process is startedon one of these star graphs and approaches (or jumps to) the vicinity of anothervertex, it is killed and revived on the relevant subgraph with the help of concatena-tion techniques. That way, we obtain a Brownian motion on a general metric graphby successive pastings of partial Brownian motions on star graphs. The accurateconstruction approach will be laid out in the following.Technically, we will not start with star graphs, but with the complete metricgraph which we then decompose into subgraphs. This approach is necessary, asthe subgraphs (that is, at some level, star graphs) must be chosen appropriatelyin order to construct the correct complete graph at the end, and the topology ofthe full graph is required for the pathwise construction and the specification of theFeller–Wentzell data.Let G = ( V , I , E , ∂, ρ ) be a metric graph having at least two vertices. We willbreak G up by decomposing the set of vertices into V = V − ⊎ V +1 and definingtwo “subgraphs” ee G j , j ∈ {− , +1 } , which possess the respective vertices V j aswell as all of the original edges (with their combinatorial structure) not incidentwith the other vertices V − j . As internal edges i which are incident with vertices ofboth subgraphs are lost, we need to replace them by new external “shadow” edges e − i , e +1 i on the respective subgraphs, see the upper graph of figure 1.By iteratively decomposing the subgraphs further up to the level of star graphs,we are able to apply our results of [7] and introduce Brownian motions on ee G − and ee G +1 with the desired boundary behavior at their vertices. In order to paste thetwo processes—and thus the two graphs—together, we need to cut out the excres-cent parts of the external “shadow” edges by removing them from the subgraphs As in the previous works, we will assume any metric graph discussed here to have no loops(see [6, Section A.2, Remark 3.1]). Furthermore, we restrict our attention to metric graphs withfinite sets of edges and vertices. A short introduction to metric graphs can be found in [6,Appendix A].
ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 3 ee G − ee G +1 v v v v v v e − i e +1 i e − i e +1 i e − i e +1 i e − i e +1 i G − G +1 v v v v v v · ·· · i i i i Figure 1.
Decomposition and gluing of metric graphs: The met-ric graph G of [6, Figure 1] is decomposed into two “subgraphs” ee G − and ee G +1 with vertices V − = { v , v , v } and V +1 = { v , v , v } .The internal edges i which are incident with vertices of both sub-graphs are replaced by new external edges e − i , e +1 i on the respec-tive subgraphs. By performing the transformations explained insection 2, subsets of these “subgraphs” are mapped to the sub-sets G − , G +1 of the graph G .and killing the partial Brownian motions whenever they hit the removed locations.The remaining parts of these external edges need to be reorientated where neces-sary (as vertices are always initial points of external edges) and then are mappedto the original internal edges in order to get proper subgraphs G − and G +1 of theoriginal graph G , see the lower graph of figure 1.The resulting Brownian motions on G − and G +1 can now be pasted togetherwith the help of the alternating copies technique established in [8, Section 3], namelyby reviving the subprocesses at the other subgraph whenever they leave the remain-ing part of one of their shadow vertices (and thus are killed).This construction approach will cause two main technical difficulties, which willprescribe the order of applied transformations: Firstly, the “global” jumps, that isjumps to other vertices or subgraphs, can only be implemented once the gluing iscomplete, as their jump destinations do not exist for the original Brownian motionson the subgraphs. They will be implemented by an instant return process with anappropriate revival measure. Moreover, the implementation of the killing portions( p v , v ∈ V ) via jumps to the cemetery must be postponed until the gluing procedureand the introduction of the global jumps is complete. The reason is that, as justmentioned, both procedures will apply the technique of identical/alternating copies, FLORIAN WERNER (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (a) adjoining “fake cemeteries” (cid:3) v (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (b) gluing the subgraphs together ∆ (c) killing on the “fake cemeteries” (cid:3) / ∆ (d) adjoining (cid:3) , reviving with an appropri-ate measure, killing on (cid:3) again Figure 2.
Completing the construction of Brownian motions ona metric graph: Illustrated are the steps that are performed inthe construction of the target Brownian motion on the completegraph, when starting with Brownian motions on the subgraphswhich already implement the correct reflection, stickiness and “lo-cal” jump parameters. The dotted lines indicate the range of theimplemented jump measures.which is based on reviving the process and would therefore cancel any killing effectbeforehand.The above-mentioned restrictions and interactions of these techniques lead tosome rather unwieldy “workarounds” in the upcoming complete construction. Weare giving an overview of the construction steps now, the mathematical justificationswill follow in sections 3–6.Assume that we are given a metric graph G = ( V , I , E , ∂, ρ ) and boundary weights (cid:0) p v , ( p v,l ) l ∈L ( v ) , p v , p v (cid:1) v ∈V which satisfy the conditions of Feller’s theorem [6, Theorem 1.1].As we cannot introduce the distant jumps yet, we choose for each v ∈ V adistance δ v > δ v is smaller than the lengths of all edges emanatingfrom v , and define the restricted jump measure q v := p v (cid:0) · ∩ B δ v ( v ) (cid:1) ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 5 on the ball B δ v ( v ) around v with radius δ v , and the “extended” killing parameter q v := p v + p v (cid:0) ∁ B δ v ( v ) (cid:1) . We are going to construct the complete Brownian motion with the just givenboundary weights iteratively. That is, we decompose the metric graph into twosubgraphs ee G − and ee G +1 as explained above, and assume that there exist two Brow-nian motions ee X − , ee X +1 thereon which implement the boundary conditions (cid:0) q v , ( p v,l ) l ∈ ee L j ( v ) , p v , q v (cid:1) v ∈V j , j ∈ {− , +1 } , where we set the reflection parameters for the adjoined edges to p v,e ji = p v,i .As the gluing procedure only works for processes with no additional killing ef-fects at the vertices, we further adjoin for every vertex v ∈ V an absorbing “fake”cemetery point (cid:3) v to the respective subgraph ee G j , and assimilate the killing param-eter into the jump measure by reviving the subprocesses at (cid:3) v whenever they dieat v , see figure . Then the new processes possess the boundary conditions (cid:0) , ( p v,l ) l ∈ ee L j ( v ) , p v , q v ε (cid:3) v + q v (cid:1) v ∈V j , j ∈ {− , +1 } . Next, we glue both processes together and obtain a process on the completegraph G , as illustrated in figure , with boundary conditions (cid:0) , ( p v,l ) l ∈L ( v ) , p v , q v ε (cid:3) v + q v (cid:1) v ∈V . In order to introduce the global jumps, we split the jump to (cid:3) v , with originalweight q v = p v + p v (cid:0) ∁ B δ v ( v ) (cid:1) , into killing with weight p v and non-local jumpsrelative to the measure p v (cid:0) · ∩ ∁ B δ v ( v ) (cid:1) . To this end, we need to kill the processagain: By mapping the absorbing points { (cid:3) v , v ∈ V} to the “real” cemetery ∆, seefigure , we obtain a newly killed process with boundary conditions (cid:0) q v , ( p v,l ) l ∈L ( v ) , p v , q v (cid:1) v ∈V . We adjoin another absorbing “fake” cemetery point (cid:3) and construct the next pro-cess as instant revival process with revival distribution (cid:0) p v ε (cid:3) + p v (cid:0) · ∩ ∁ B δ v ( v ) (cid:1)(cid:1) /q v .This process now implements jumps relative to the measure p v ε (cid:3) v + p v (cid:0) · ∩ ∁ B δ v ( v ) (cid:1) ,which adds to the already existing jump measure q v = p v (cid:0) · ∩ B δ v ( v ) (cid:1) . Thus, thisprocess satisfies the boundary conditions (cid:0) , ( p v,l ) l ∈L ( v ) , p v , p v ε (cid:3) + p v (cid:1) v ∈V . Finally, we transform the jumps to (cid:3) into killing by mapping (cid:3) to ∆, and obtainthe complete boundary condition (cid:0) p v , ( p v,l ) l ∈L ( v ) , p v , p v (cid:1) v ∈V . As seen above, we need to perform many process transformations in the completeconstruction, while keeping track of the resulting boundary conditions. In order tokeep our results comprehensible, we first analyze the two main components—killingon an absorbing set and introduction of jumps via the instant revival process—together with their effects on the generator separately in the next two sections.
FLORIAN WERNER Killing a Brownian Motion on an Absorbing Set
In this section, we examine how killing a Brownian motion on an absorbingset F affects the boundary conditions of its generator. It will turn out that thejump portion which originally led to F is just transformed into the killing portion,as any jump to F is now immediately triggering the killing.We implement the killing transformation by mapping the absorbing set F to ∆,that is, we consider the process ψ ( X ) for the map ψ : G → G\
F, x ψ ( x ) := ( x, x ∈ G\ F, ∆ , x ∈ F. (3.1)It has been shown in Appendix A that the transformed process ψ ( X ) is a rightprocess if X is a right process and F is an isolated and absorbing set for X .We are able to obtain the following set of necessary boundary conditions bydirectly computing the generator of the transformed process: Lemma 3.1.
Let X be a Brownian motion on G with generator D ( A X ) ⊆ n f ∈ C ( G ) : ∀ v ∈ V : c v f ( v ) − X l ∈L ( v ) c v,l f ′ l ( v ) + c f ′′ ( v ) − Z G\{ v } (cid:0) f ( g ) − f ( v ) (cid:1) c v ( dg ) = 0 o , and F ( G be an isolated, absorbing set for X . Let Y := ψ ( X ) be the processon G\ F resulting from killing X on F , with ψ as given in equation (3.1) . Then thedomain of the generator of Y satisfies D ( A Y ) ⊆ n f ∈ C ( G\ F ) : ∀ v ∈ V\ F : (cid:0) c v + c v ( F ) (cid:1) f ( v ) − X l ∈L ( v ) c v,l f ′ l ( v ) + c f ′′ ( v ) − Z G\ ( F ∪{ v } ) (cid:0) f ( g ) − f ( v ) (cid:1) c v ( dg ) = 0 o . Proof.
For all f ∈ D ( A Y ), we have for g ∈ G\ FA X ( f ◦ ψ )( g ) = lim t ↓ E g (cid:0) f ◦ ψ ( X t ) (cid:1) − f ◦ ψ ( g ) t = lim t ↓ E g (cid:0) f ( Y t ) (cid:1) − f ( g ) t , which exists and is equal to A Y f ( g ). On the other hand, if g ∈ F , then X t ∈ F holds for all t ≥ P g -a.s., because F is absorbing for X , and it follows that A X ( f ◦ ψ )( g ) = lim t ↓ E g (cid:0) f ◦ ψ ( X t ) (cid:1) − f ◦ ψ ( g ) t = lim t ↓ E g (cid:0) f (∆) (cid:1) − f (∆) t = 0 . Thus, we have f ◦ ψ ∈ D ( A X ) for all f ∈ D ( A Y ), and A X ( f ◦ ψ ) = A Y f ∁ F inthis case. ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 7
So, if f ∈ D ( A Y ), then f ◦ ψ fulfills the boundary condition for X , that is0 = c v f (cid:0) ψ ( v ) (cid:1) − X l ∈L ( v ) c v,l f ′ l (cid:0) ψ ( v ) (cid:1) + c f ′′ (cid:0) ψ ( v ) (cid:1) − Z G\{ v } (cid:0) f (cid:0) ψ ( g ) (cid:1) − f (cid:0) ψ ( v ) (cid:1)(cid:1) c v ( dg )= c v f ( v ) − X l ∈L ( v ) c v,l f ′ l ( v ) + c f ′′ ( v ) − Z G\ ( F ∪{ v } ) (cid:0) f ( g ) − f ( v ) (cid:1) c v ( dg ) + f ( v ) c v ( F )for all v ∈ V\ F , where we used f (cid:0) ψ ( g ) (cid:1) = f (∆) = 0 for all g ∈ F . (cid:3) In general, this proof does not provide us with the Feller–Wentzell data of thekilled process, as we are only able to directly compare the Feller–Wentzell datawith the boundary data of the generator in the star graph case (cf. [6, Lemma 4.1]).Therefore, we need to derive it manually by checking its definitions given in Feller’stheorem [6, Theorem 1.2]:
Lemma 3.2.
Let X be a Brownian motion on G with Feller–Wentzell data (cid:0) c v, ∆1 , c v, ∞ , ( c v,l ) l ∈L ( v ) , c v , c v (cid:1) v ∈V , and F ( G be an isolated, absorbing set for X . Let Y := ψ ( X ) be the processon G\ F resulting from killing X on F , with ψ as given in equation (3.1) . If G\ F is a metric graph and Y is a Brownian motion on G\ F , then the Feller–Wentzelldata of Y reads (cid:0) c v, ∆1 + c v ( F ) , c v, ∞ , ( c v,l ) l ∈L ( v ) , c v , c v ( · ∩ F ∁ ) (cid:1) v ∈V\ F . Proof.
We are using the notations of [6, Theorem 1.2], and indicate the corre-sponding process in the superscript of the variables. Fix v ∈ V\ F . The processes’exit behaviors totally coincide, except if X exits from a small neighborhood of v by jumping into F (then Y jumps to ∆). Thus, E v ( τ Xε ) = E v ( τ Yε ) holds for allsufficiently small ε >
0, and the exit distributions read P v (cid:0) Y τ Yε ∈ dg ∩ ( G\ F ) (cid:1) = P v (cid:0) X τ Xε ∈ dg ∩ ( G\ F ) (cid:1) , P v (cid:0) Y τ Yε = ∆ (cid:1) = P v (cid:0) X τ Xε ∈ { ∆ } ∪ F (cid:1) . Therefore, we have ν Y,vε = ν X,vε (cid:0) · ∩ ( G\ F ) (cid:1) and, as d ( v, f ) = + ∞ for all f ∈ F , Z F (cid:0) − e − d ( v,g ) (cid:1) ν X,vε ( dg ) = ν X,vε ( F ) = P v ( X τ Xε ∈ F ) E v ( τ Xε ) . It follows that K Y,vε = 1 + P v ( Y τ Yε = ∆) E v ( τ Yε ) + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) ν Y,vε ( dg )= 1 + P v ( X τ Xε = ∆) E v ( τ Xε ) + Z ( G\{ v } ) ∪ F (cid:0) − e − d ( v,g ) (cid:1) ν X,vε ( dg )= K X,vε . FLORIAN WERNER As F is isolated, we get µ Y,v = µ X,v (cid:0) · ∩ (cid:0)
G\{ v }\ F (cid:1)(cid:1) , and conclude that c Y,v, ∆1 = lim ε ↓ (cid:16) P v ( X τ Xε = ∆) E v ( τ Xε ) K X,vε + P v ( X τ Xε ∈ F ) E v ( τ Xε ) K X,vε (cid:17) = c X,v, ∆1 + µ X,v ( F )= c X,v, ∆1 + c X,v ( F ) , as well as c Y,v, ∞ = c X,v, ∞ , c Y,v,l = c X,v,l for each l ∈ L ( v ), c Y,v = c X,v , and c Y,v = c X,v (cid:0) · ∩ ( G\ F ) (cid:1) . (cid:3) Remark . We will apply Lemma 3.2 in the following context: Let X be a Brow-nian motion on G and F be an isolated and absorbing set for X , such that for itsfirst entry time H F := inf { t ≥ X t ∈ F } and H X as given in [6, Definition 2.1], H X < H F P g -a.s.holds true for all g ∈ ∁ F .It then follows from Theorem A.1 that the killed process Y = ψ ( X ) is a rightprocess, and therefore strongly Markovian. If G\ F is a metric graph, then, as H Y = H X and Y t = X t for all t ≤ H X < H F , the properties of [6, Theorem 2.5]follow for Y from the respective ones of X . Thus, Y is a Brownian motion on G\ F ,and Lemma 3.2 can be applied in order to deduce the Feller–Wentzell data of Y .In particular, the condition above is satisfied if F can only be reached from ∁ F via jumps from vertices, which, as F is isolated and thus has positive distancefrom any vertex v ∈ V\ F , cannot happen immediately due to the normality of theprocess. 4. Introduction of Non-Local Jumps
We will introduce the “global” jumps, namely jumps to other subgraphs, withthe help of the technique of instant revivals as established in [8, Theorem 1.7].In order to prepare this approach, we examine the effect of this method on theFeller–Wentzell data. Similar results were already attained in the examinationsconcerning Brownian motions on star graphs (see [6, Lemma 4.2, Lemma 4.3]).The next lemma shows that, as expected, the killing weight will be transformed toan additional jump portion with distribution given by the revival kernel. It alsoclarifies that this technique can only be used for the implementation of finite jumpmeasures.
Lemma 4.1.
Let X be a Brownian motion on G with Feller–Wentzell data (cid:0) c v, ∆1 , c v, ∞ , ( c v,l ) l ∈L ( v ) , c v , c v (cid:1) v ∈V , lifetime ζ X , and exit times τ Xε := inf (cid:8) t ≥ d ( X t , X ) > ε (cid:9) for ε > . If c v, ∆1 > ,consider the instant revival process Y , constructed from X with the revival kernel k ( v, · ) = κ v , v ∈ V , for some probability measure κ v on G , and k ( g, · ) = ε g for all g / ∈ V . Suppose thatfor every v ∈ V there exists δ > such that(i) κ v (cid:0) B δ ( v ) (cid:1) = 0 , and(ii) for all ε < δ , X τ Xε ∈ B δ ( v ) holds P Xv -a.s. on { τ Xε < ζ X } . ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 9
Then Y is a Brownian motion on G . For all v ∈ V , the generator A Y of Y satisfiesfor every f ∈ D ( A Y ) c v, ∞ f ( v ) − X l ∈L ( v ) c v,l f ′ l ( v ) + c v Af ( v ) − Z G\{ v } (cid:0) f ( g ) − f ( v ) (cid:1) ( c v + c v, ∆1 κ v )( dg ) = 0 . If additionally d ( v, x ) = + ∞ holds for every x ∈ supp κ v , then the Feller–Wentzelldata of Y at v reads (cid:0) , c v, ∞ , ( c v,l ) l ∈L ( v ) , c v , c v + c v, ∆1 κ v (cid:1) . Proof.
By [8, Theorem 1.7], Y is a right process and thus strongly Markovian.As Y t = X t holds a.s. for all t ≤ H X = H Y , Y is a Brownian motion on G .Fix v ∈ V . We are going to examine the components evolving in the generator ofthe process Y and compare them to the respective ones of X . The components inFeller’s theorem [6, Theorem 1.2] for the process X at the vertex v will be named c X , ν Xε , K Xε , etc., instead of c v , ν vε , K vε . The proof will be based on the followingtwo main principles: • Due to assumption (i), the processes Y and X are equivalent in a neigh-borhood of v , more precisely: There exists δ > δ in assumption (i) and the minimal length of all edges incident with v )such that ∀ ε ≤ δ : E Yv ( τ Yε ) = E Xv ( τ Xε ) , and for all n ∈ N , f , . . . , f n ∈ b B ( G ), 0 ≤ t < . . . < t n , P Yv (cid:0) f ( Y t ) · · · f ( Y t n ); t n < τ Yδ (cid:1) = P Xv (cid:0) f ( X t ) · · · f ( X t n ); t n < τ Xδ (cid:1) . In particular, we have for all ε < δ , A ∈ B ( G ): P Yv (cid:0) Y τ Yε ∈ A | Y τ Yε ∈ B δ ( v ) (cid:1) = P Xv (cid:0) X τ Xε ∈ A | X τ Xε ∈ B δ ( v ) (cid:1) . • Due to assumption (ii), the process X only has jumps from v into B δ ( v ) orto ∆, that is, ∀ ε < δ : P Xv (cid:0) X τ Xε ∈ B δ ( v ) ∪ { ∆ } (cid:1) = 1 . Therefore, Y only can jump into ∁ B δ ( v ) if the underlying process X is killedand revived again, which yields P Yv (cid:0) Y τ Yε ∈ ∁ B δ ( v ) (cid:1) = P Xv (cid:0) X τ Xε = ∆ (cid:1) , and the jump distribution is given by the reviving kernel P Yv (cid:0) Y τ Yε ∈ A | Y τ Yε ∈ ∁ B δ ( v ) (cid:1) = κ v ( A ) , A ∈ B ( G ) . Furthermore, the revived process Y is not able to die at all, yielding P Yv (cid:0) Y τ Yε = ∆ (cid:1) = 0 . Let f ∈ D ( A Y ) and fix v ∈ V . The vertex v cannot be a trap for Y , as otherwise v would either be a trap for X , which is impossible by c v, ∆1 >
0, or Y would berevived at v when X dies there, which contradicts assumption (i). Thus, Dynkin’sformula yields Af ( v ) = lim ε ↓ E Yv (cid:0) f ( Y τ Yε ) (cid:1) − f ( v ) E Yv ( τ Yε ) . We are going to reiterate the steps in the proof of Feller’s theorem [6, Theo-rem 1.2] for the process Y , but we will be using the normalization factor K Xε of X instead of K Yε . This will not pose any problems because K Xε ≥ K Yε holds true,which is seen as follows: With the scaled exit distributions from ∁ B ε ( v ) ν Yε ( A ) = P Yv (cid:0) Y τ Yε ∈ A (cid:1) E Yv ( τ Yε ) , ν Xε ( A ) = P Xv (cid:0) X τ Xε ∈ A (cid:1) E Xv ( τ Xε ) , A ∈ B (cid:0) G\{ v } (cid:1) , for Y and X , assumption (i) asserts that for all sufficiently small ε > K Xε = 1 + P Xv (cid:0) X τ Xε = ∆ (cid:1) E Xv ( τ Xε ) + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) ν Xε ( dg )= 1 + P Yv (cid:0) Y τ Yε ∈ ∁ B δ ( v ) (cid:1) E Yv ( τ Yε ) + Z B δ ( v ) \{ v } (cid:0) − e − d ( v,g ) (cid:1) ν Yε ( dg ) . As P Yv (cid:0) Y τ Yε = ∆ (cid:1) = 0 and P Yv (cid:0) Y τ Yε ∈ ∁ B δ ( v ) (cid:1) E Yv ( τ Yε ) = Z ∁ B δ ( v ) ν Yε ( dg ) ≥ Z ∁ B δ ( v ) (cid:0) − e − d ( v,g ) (cid:1) ν Yε ( dg ) , (4.1)we get K Xε ≥ P Yv (cid:0) Y τ Yε = ∆ (cid:1) E Yv ( τ Yε ) + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) ν Yε ( dg )= K Yε . Thus, by following the proof of Feller’s theorem (see [6, Section 3]), we getlim ε ↓ (cid:16) f ( v ) P Yv (cid:0) Y τ Yε = ∆ (cid:1) E Yv ( τ Yε ) K Xε + Af ( v ) 1 K Xε − Z G\{ v } (cid:0) f ( g ) − f ( v ) (cid:1) ν Yε ( dg ) K Xε (cid:17) = 0 . However, it is P Yv ( Y τ Yε = ∆) = 0, and the exit distributions of Y decompose into ν Yε ( A ) = P Yv (cid:0) Y τ Yε ∈ A ∩ B δ ( v ) (cid:1) E Yv ( τ Yε ) + P Yv (cid:0) Y τ Yε ∈ A ∩ B δ ( v ) ∁ (cid:1) E Yv ( τ Yε ) , with P Yv (cid:0) Y τ Yε ∈ A ∩ B δ ( v ) (cid:1) E Yv ( τ Yε ) = P Yv (cid:0) Y τ Yε ∈ A | Y τ Yε ∈ B δ ( v ) (cid:1) P Yv (cid:0) Y τ Yε ∈ B δ ( v ) (cid:1) E Yv ( τ Yε )= P Xv (cid:0) X τ Xε ∈ A | X τ Xε ∈ B δ ( v ) (cid:1) P Xv (cid:0) Y τ Xε ∈ B δ ( v ) (cid:1) E Xv ( τ Xε )= P Xv (cid:0) X τ Xε ∈ A ∩ B δ ( v ) (cid:1) E Xv ( τ Xε )= ν Xε ( A ) , and P Yv (cid:0) Y τ Yε ∈ A ∩ B δ ( v ) ∁ (cid:1) E Yv ( τ Yε ) = P Yv (cid:0) Y τ Yε ∈ A | Y τ Yε ∈ B δ ( v ) ∁ (cid:1) P Yv (cid:0) Y τ Yε ∈ B δ ( v ) ∁ (cid:1) E Xv ( τ Xε )= κ v ( A ) P Xv ( X τ Xε = ∆) E Xv ( τ Xε ) . ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 11
Therefore, we havelim ε ↓ (cid:16) Af ( v ) 1 K Xε − Z G\{ v } (cid:0) f ( g ) − f ( v ) (cid:1) ν Xε ( dg ) K Xε − P Xv ( X τ Xε = ∆) E Xv ( τ Xε ) K Xε Z ∁ B δ ( v ) (cid:0) f ( g ) − f ( v ) (cid:1) κ v ( dg ) (cid:17) = 0 , and knowing that K Xεn , ν Xεn ( dg ) K Xεn , P Xv ( X τXεn =∆) E Xv ( τ Xεn ) K Xεn converge along the same sequence( ε n , n ∈ N ) given by Feller’s theorem [6, Theorem 1.2] for X , we conclude that c v, ∞ f ( v ) − X l ∈L ( v ) c v,l f ′ l ( v ) + c v Af ( v ) − Z G\{ v } (cid:0) f ( g ) − f ( v ) (cid:1) c v ( dg ) − c v, ∆1 Z ∁ B δ ( v ) (cid:0) f ( g ) − f ( v ) (cid:1) κ v ( dg ) = 0 . In case every point in the support of κ v has distance + ∞ from v , equation (4.1)shows that K Xε = K Yε holds true, and therefore the above set of boundary condi-tions at v for Y coincides with the Feller–Wentzell data of Y at v . (cid:3) The reader may notice that the resulting boundary data for Y given in Lemma 4.1might not satisfy the normalization condition of the Feller–Wentzell data, as givenin [6, Theorem 1.2], in case the support of κ v does not have infinite distance from v . Remark . We observe in Lemma 4.1 that the revival of a process upon its deathwith a revival distribution κ only transforms the “real” killing parameter c ∆1 into anadditional jump part c ∆1 κ , while leaving the artificial killing portion c ∞ intact. Themain explanation is that c ∞ does not represent the effect of “killing” in the senseof proper jumps to the cemetery point ∆. It is rather caused by an explosion ofthe process, triggered by ever-growing jumps when the process approaches a vertexpoint, and this effect is not transformed by the revival technique.In the Brownian context, we do not expect any effects which would contributeto c ∞ , and we indeed showed in [6, Theorem 1.4] that c ∞ vanishes for all Brownianmotions on star graphs. As these processes will form the building blocks of theBrownian motions on the general metric graph, the Feller–Wentzell data of allprocesses considered here will satisfy ∀ v ∈ V : c v, ∞ = 0 . Gluing the Graphs Together
We are going to discuss the main construction method, namely the pasting of thesubgraphs and their Brownian motions thereon. As already disclosed in section 2,this technique will compromise several steps:5.1.
Decomposition of the Graph G into ee G − , ee G +1 . Let G = (cid:0) V , E , I , ∂, ρ (cid:1) be ametric graph. We partition G into two graphs by choosing two disjoint, non-emptysets V − and V +1 with V = V − ⊎ V +1 , and decompose the set of edges into E = E − ⊎ E +1 , with E j := { e ∈ E : ∂ ( e ) ∈ V j } , I = I − ⊎ I +1 ⊎ I s , with I j := { i ∈ I : ∂ − ( i ) ∈ V j , ∂ + ( i ) ∈ V j } , I s := I − s ⊎ I +1 s , with I js := { i ∈ I : ∂ − ( i ) ∈ V j , ∂ + ( i ) / ∈ V j } . As most of the following construction will be performed for both partial graphsin parallel, we will always assume that j ∈ {− , +1 } when nothing else is said.We define the metric graphs ee G − , ee G +1 by ee G j := (cid:0) V j , E j ∪ E js , I j , ∂ j , ρ j (cid:1) , equipped with additional external “shadow” edges E js := { e ji , i ∈ I s } , with ∀ i ∈ I s : e ji / ∈ E ∪ E − js ∪ I , where the combinatorial structure and edge lengths of the original graph are natu-rally transfered to ee G − , ee G +1 by setting ∂ j (cid:12)(cid:12) E j ∪ ( I j × I j ) := ∂ (cid:12)(cid:12) E j ∪ ( I j × I j ) , ∂ j ( e ji ) := ( ∂ − ( i ) , i ∈ I js ,∂ + ( i ) , i ∈ I − js ,ρ j (cid:12)(cid:12) E j ∪ I j := ρ (cid:12)(cid:12) E j ∪ I j , ρ j (cid:12)(cid:12) E js := + ∞ . For later use, we also define the “shadow length” of an external “shadow” edge by ρ s ( e ji ) := ρ ( i ) , e ji ∈ E − s ∪ E +1 s . The excrescent parts of the shadow edges, which will be removed in the followingdevelopment before gluing both subgraphs together, are named e G js := [ e ∈E js (cid:0) { e } × [ ρ s ( e ) , + ∞ ) (cid:1) . Introducing the Brownian Motion ee X j on ee G j . Let ee X − , ee X +1 be Brownianmotions on ee G − , ee G +1 respectively, which admit the hypotheses of right processes,feature infinite lifetimes, have the Feller–Wentzell data (cid:0) , , ( p v,l ) l ∈ ee L j ( v ) , p v , p v (cid:1) v ∈V j , are continuous inside every edge (cf. [7, Theorem 4.3]), and satisfy for all v ∈ V j ∀ ε < δ : P jv (cid:0) ee X j ee τ jε ∈ e G js (cid:1) = 0 , (5.1)with δ := min { ρ i , i ∈ I s } and ee τ jε := inf (cid:8) t ≥ d ( ee X jt , ee X j ) > ε (cid:9) .By gluing the graphs ee G − and ee G +1 (and thus the Brownian motions ee X − and ee X +1 thereon) together, we are going to show the following main result of this section: Theorem 5.1.
There exists a Brownian motion X on G with Feller–Wentzell data (cid:0) c v , ( c v,l ) l ∈L ( v ) , c v , c v (cid:1) v ∈V , such that for each v ∈ V , it holds that c v = 0 , c v = p v , c v = p v ◦ ( ψ j ) − , with ψ j being defined by equation (5.2) , and i ∈ I ( v ) : c v,i = ( p v,i , i ∈ I − ( v ) ∪ I +1 ( v ) ,p v,e ji , i ∈ I s ( v ) , with j ∈ {− , +1 } such that v ∈ V j , e ∈ E ( v ) : c v,e = p v,e . We construct this process X explicitly via alternating copies of transformedprocesses X − , X +1 of ee X − , ee X +1 . Before that, we need to kill the original processes ee X − and ee X +1 on the excrescent shadow edges and reorientate the remaining partsin order to comply with the direction of the original internal edges of G . ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 13
Defining e X j by Killing ee X j on e G js . Consider the first entry time into e G js ofthe prototype Brownian motion ee X j on ee G j , ee τ j := inf (cid:8) t ≥ ee X jt ∈ e G js (cid:9) . We define e X j to be the process obtained by killing ee X j at the terminal time ee τ j , e X jt := ( ee X jt , t < ee τ j , ∆ , t ≥ ee τ j , on the topological subspace e G j of ee G j given by e G j := ee G j \ e G js = V j ∪ [ l ∈E j ∪I j (cid:0) { l } × (0 , ρ l ) (cid:1) ∪ [ e ∈E js (cid:0) { e } × (cid:0) , ρ s ( e ) (cid:1)(cid:1) . Lemma 5.2. e X j is a right process on e G j with lifetime ee τ j .Proof. ee X j is a right process with infinite lifetime. By employing [4, Corollary 12.24],it suffices to observe that ee τ j is the debut of the closed, thus nearly optional set e G js ,and the regular set of the killing time ee τ j reads F := (cid:8) g ∈ ee G j : P jg ( ee τ j = 0) = 1 (cid:9) = e G js , as ee X j is a right continuous, normal process and e G js is closed. (cid:3) We would like to point out that the just introduced processes e X j are not Brown-ian motions on a metric graph in the sense of [6, Definitions 2.1, A.1, A.2] anymore,as e G j is not a metric graph. Thus, we will not be able to apply any results on Brow-nian motions for e X j in the upcoming development.5.4. Letting X j be the Mapping of e X j to the Subspace G j ⊆ G . We needto fit the subspaces e G j of ee G j to the corresponding subspaces of G . To this end, weintroduce the topological subspaces G − , G +1 of G by G j := V j ∪ [ l ∈E j ∪I j ∪I s (cid:0) { l } × (0 , ρ l ) (cid:1) , and consider the mapping ψ j : e G j → G j defined by(5.2) ∀ i ∈ I s , x ∈ (0 , ρ i ) : ψ j (cid:0) ( e ji , x ) (cid:1) := ( ( i, x ) , i ∈ I js , ( i, ρ i − x ) , i ∈ I − js ,ψ j = id otherwise . Clearly, ψ j is a bijective mapping, with its inverse ( ψ j ) − =: ϕ j : G j → e G j beinggiven by ∀ i ∈ I s , x ∈ (cid:0) , ρ i (cid:1) : ϕ j (cid:0) ( i, x ) (cid:1) := ( ( e ji , x ) , i ∈ I js , ( e ji , ρ i − x ) , i ∈ I − js ,ϕ j = id otherwise . Furthermore, ψ j is a continuous mapping, as it is continuous inside every edge andits preimages of balls with sufficiently small radius around vertices v ∈ V j coincidewith the corresponding balls of e G j . e X j is a right process on e G j , ψ j is a bijective and measurable map from e G j onto G j , and t ψ j ( e X jt ) is right continuous (as ψ j is continuous and t e X jt isright continuous). Thus, the following result is a direct consequence of [4, Corol-lary (13.7)]: Lemma 5.3.
The process X j := ψ j ( e X j ) , resulting from the state space mappingof e X j by ψ j , is a right process on ψ j ( e G j ) = G j with lifetime ζ j = e ζ j = ee τ j . Constructing X as Alternating Copies Process of X − , X +1 . We applythe technique of [8] to define the process X obtained by forming alternating copiesof X − and X +1 via the transfer kernels K − and K +1 , given by(5.3) K − := X i ∈ I − s ε ∂ + ( i ) { i } (cid:0) π ( X − ζ − − ) (cid:1) + X i ∈ I +1 s ε ∂ − ( i ) { i } (cid:0) π ( X − ζ − − ) (cid:1) ,K +1 := X i ∈ I − s ε ∂ − ( i ) { i } (cid:0) π ( X +1 ζ +1 − ) (cid:1) + X i ∈ I +1 s ε ∂ + ( i ) { i } (cid:0) π ( X +1 ζ +1 − ) (cid:1) . That is, the transfer kernels implement the following rules for j ∈ {− , +1 } :(i) X is revived as X +1 at v = ∂ − j ( i ), if X − dies on i ∈ I js ;(ii) X is revived as X − at v = ∂ j ( i ), if X +1 dies on i ∈ I js .For later use, we give the following combined formula of the above definitions forthe transfer kernels K j , j ∈ {− , +1 } : K j = k j ( i ) := ( ε ∂ + ( i ) , i ∈ I js ,ε ∂ − ( i ) , i ∈ I − js , for i := π ( X jζ j − ) . (5.4) Lemma 5.4. K j is a transfer kernel from X j to E − j .Proof. With probability 1, the process ee X j cannot realize ee τ j through a direct jumpfrom any vertex v ∈ V j : Otherwise, this would imply P jv (cid:0) ee X j ee τ jε ∈ e G js (cid:1) >
0, as ee τ j ≥ ee τ jε holds for ε < δ , contradicting our fundamental assumption (5.1). Furthermore, ee X j is continuous on every edge, so ee X j ee τ j − exists and is equal to ee X j ee τ j . Thus, e X jζ j − = lim t ⇈ ζ j e X jt = lim t ⇈ ee τ j ee X jt = ee X j ee τ j exists in (cid:8)(cid:0) e, ρ s ( e ) (cid:1) , e ∈ E js (cid:9) , and π (cid:0) X jζ j − (cid:1) = π (cid:0) ψ j ( e X jζ j − ) (cid:1) = π (cid:0) ψ j ( ee X j ee τ j ) (cid:1) (5.5)exists in I s . Therefore, π (cid:0) X jζ j − (cid:1) ∈ F j [ ζ j − ] , so K j is indeed a probability kernel K from (Ω j , F j [ ζ j − ] ) to ( E − j , E − j ), that is, a transfer kernel. (cid:3) Let • τ − − be the first entry time of X − into G − \G +1 , ζ − the lifetime of X − , • τ +1+1 be the first entry time of X +1 into G +1 \G − , ζ +1 the lifetime of X +1 .Then, according to [8, Theorem 1.6], X is a right process on G = G − ∪ G +1 in casethe following conditions hold true for all g ∈ G − ∩G +1 , f ∈ b B ( G ), h − ∈ b B ( G − ), h +1 ∈ b B ( G +1 ):(i) E − g (cid:16) Z τ − − e − αt f ( X − t ) dt (cid:17) = E +1 g (cid:16) Z τ +1+1 e − αt f ( X +1 t ) dt (cid:17) ; ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 15 (ii) E − g (cid:0) e − ατ − − h − ( X − τ − − ); τ − − < ζ − (cid:1) = E +1 g (cid:0) e − αζ +1 K +1 h − ; ζ +1 < τ +1+1 (cid:1) , E +1 g (cid:0) e − ατ +1+1 h +1 ( X +1 τ +1+1 ); τ +1+1 < ζ +1 (cid:1) = E − g (cid:0) e − αζ − K − h +1 ; ζ − < τ − − (cid:1) .We are preparing the proof of these equalities: By construction, we have G − ∩ G +1 = [ i ∈I s (cid:0) { i } × (0 , ρ i ) (cid:1) , G j \G − j = V j ∪ [ l ∈E j ∪I j (cid:0) { l } × (0 , ρ l ) (cid:1) . By using the definition of X j and observing that ϕ j ( G j \G − j ) = G j \G − j , we get τ jj = inf (cid:8) t ≥ X jt ∈ G j \G − j (cid:9) = inf (cid:8) t ≥ ψ j ( e X jt ) ∈ G j \G − j (cid:9) = inf (cid:8) t ≥ e X jt ∈ G j \G − j (cid:9) . The process e X j was constructed by killing ee X j at ee τ j . Thus, by introducing the firstexit times of ee X j from the shadow edges ee τ jj := inf (cid:8) t ≥ ee X jt ∈ G j \G − j (cid:9) = inf (cid:8) t ≥ ee X jt / ∈ S i ∈I s (cid:0) { e ji } × (0 , ∞ ) (cid:1)(cid:9) , we obtain the relation τ jj ∧ ζ j = ee τ jj ∧ ee τ j . (5.6)Turning to the actual proof of (i) and (ii), let g ∈ G − ∩G +1 , that is, g = ( i, x ) forsome i ∈ I s , x ∈ (0 , ρ i ). Choose j ∈ {− , +1 } such that i ∈ I js . By tracing X j backto ee X j and employing that the latter is a Brownian motion on ee G j , [6, Lemma 2.4and Corollary 2.10] yield E − jg (cid:16) Z τ − j − j e − αt f (cid:0) X − jt (cid:1) dt (cid:17) = E − j ( ψ − j ) − ( g ) (cid:16) Z τ − j − j e − αt f (cid:0) ψ − j (cid:0) ee X − jt , ee τ − j (cid:1)(cid:1) dt (cid:17) = E − j ( e − ji ,ρ ( i ) − x ) (cid:16) Z ee τ − j − j ∧ ee τ − j e − αt f (cid:0) ψ − j ( ee X − jt ) (cid:1) dt (cid:17) (5.7) = E Bρ ( i ) − x (cid:16) Z τ ∧ τ ρ ( i ) e − αt f (cid:0) ψ − j ( e − ji , B t ) (cid:1) dt (cid:17) = E Bρ ( i ) − x (cid:16) Z τ ∧ τ ρ ( i ) e − αt f (cid:0) i, ρ ( i ) − B t (cid:1) dt (cid:17) , and analogously(5.8) E jg (cid:16) Z τ jj e − αt f (cid:0) X jt (cid:1) dt (cid:17) = E j ( e ji ,x ) (cid:16) Z ee τ jj ∧ ee τ j e − αt f (cid:0) ψ j ( ee X jt ) (cid:1) dt (cid:17) = E Bx (cid:16) Z τ ∧ τ ρ ( i ) e − αt f (cid:0) i, B t (cid:1) dt (cid:17) . By the spatial homogeneity and reflection invariance of the one-dimensional Brow-nian motion B , we have E Bρ ( i ) − x (cid:16) Z τ ∧ τ ρ ( i ) e − αt f (cid:0) i, ρ ( i ) − B t (cid:1) dt (cid:17) = E Bx (cid:16) Z τ ∧ τ ρ ( i ) e − αt f (cid:0) i, B t (cid:1) dt (cid:17) , which proves the equality of (5.7) and (5.8), and thus concludes (i). Coming to (ii), we will prove both assertions simultaneously, as they only differin the initial process. Let j ∈ {− , +1 } . We start by reducing the first expectationto ee X j , and obtain with the help of equation (5.6) the identity E − jg (cid:0) e − ατ − j − j h − j (cid:0) X − jτ − j − j (cid:1) ; τ − j − j < ζ − j (cid:1) = E − j ( ψ − j ) − ( g ) (cid:16) e − α ee τ − j − j h − j (cid:0) ψ − j ( ee X − j ee τ − j − j ) (cid:1) ; ee τ − j − j < ee τ − j (cid:17) , where ( ψ − j ) − ( g ) = ( e − ji , ρ i − x ) or ( ψ − j ) − ( g ) = ( e − ji , x ) depending on whether i ∈ I js or i ∈ I − js . For all that follows, we define for any g ∈ ee G j the first hittingtime ee H jg of the set { g } by the process ee X j . By the continuity of ee X j inside the edges,we see that P j ( e ji ,y ) -a.s. for any i ∈ I s , the relation ee τ jj = ee H jv on (cid:8)ee τ jj < ee τ j (cid:9) = (cid:8) ee H jv < ee H j ( e ji ,ρ s ( e ji )) (cid:9) holds true with v := ∂ ( e ji ), so we have ψ − j ( ee X − j ee τ − j − j ) = ∂ ( e − ji ) = ( ∂ + ( i ) , i ∈ I js ,∂ − ( i ) , i ∈ I − js . Therefore, we get E − jg (cid:0) e − ατ − j − j h − j (cid:0) X − jτ − j − j (cid:1) ; τ − j − j < ζ − j (cid:1) = E − j ( ψ − j ) − ( g ) (cid:16) e − α ee τ − j − j h − j (cid:0) ψ − j ( ee X − j ee τ − j − j ) (cid:1) ; ee τ − j − j < ee τ − j (cid:17) = E − j ( e − ji ,ρ ( i ) − x ) (cid:16) e − α ee H jv h − j (cid:0) ∂ + ( i ) (cid:1) ; ee H − jv < ee H − j ( e − ji ,ρ s ( e − ji )) (cid:17) , i ∈ I js , E − j ( e − ji ,x ) (cid:16) e − α ee H jv h − j (cid:0) ∂ − ( i ) (cid:1) ; ee H − jv < ee H − j ( e − ji ,ρ s ( e − ji )) (cid:17) , i ∈ I − js . But ee X − j is a Brownian motion on ee G − j , so [6, Corollary 2.10, Remark 2.9] togetherwith ρ s ( e − ji ) = ρ ( i ) yield(5.9) E − jg (cid:0) e − ατ − j − j h − j (cid:0) X − jτ − j − j (cid:1) ; τ − j − j < ζ − j (cid:1) = ( h − j (cid:0) ∂ + ( i ) (cid:1) E Bρ ( i ) − x (cid:0) e − ατ ; τ < τ ρ ( i ) (cid:1) , i ∈ I js ,h − j (cid:0) ∂ − ( i ) (cid:1) E Bx (cid:0) e − ατ ; τ < τ ρ ( i ) (cid:1) , i ∈ I − js . Next, we employ the same techniques as above in order to compute the right-hand sides of (ii). Equations (5.5) and (5.6) give E jg (cid:0) e − αζ j K j h − j ; ζ j < τ jj (cid:1) = E j ( ψ j ) − ( g ) (cid:16) e − α ee τ j k j (cid:0) π (cid:0) ψ j ( ee X j ee τ j ) (cid:1)(cid:1) h − j ; ee τ j < ee τ jj (cid:17) . We observe that ee τ j = ee H j ( e ji ,ρ s ( e ji )) on (cid:8)ee τ j < ee τ jj (cid:9) = (cid:8) ee H j ( e ji ,ρ s ( e ji )) < ee H jv (cid:9) , as ee τ jj ≤ ee τ j in case ee τ j = ee H j ( e jk ,ρ s ( e jk )) for some other k = i . Thus, we have π (cid:0) ψ j ( ee X j ee τ j ) (cid:1) = π (cid:0) ψ j (cid:0) ( e ji , ρ s ( e ji )) (cid:1)(cid:1) P ( e ji ,x ) -a.s. on { ee τ j < ee τ jj } , ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 17 and because ψ j maps e ji to i , the definition of the transfer kernel K j , which wassummarized in equation (5.4), gives K j = k j (cid:0) π (cid:0) ψ j (cid:0) ( e ji , ρ s ( e ji )) (cid:1)(cid:1) = ( ε ∂ + ( i ) , i ∈ I js ,ε ∂ − ( i ) , i ∈ I − js . This results in(5.10) E jg (cid:0) e − αζ j K j h − j ; ζ j < τ jj (cid:1) = E j ( ψ j ) − ( g ) (cid:16) e − α ee τ j k j (cid:0) π (cid:0) ψ j ( ee X j ee τ j ) (cid:1)(cid:1) h − j ; ee τ j < ee τ jj (cid:17) = E j ( e ji ,x ) (cid:16) e − α ee H j ( eji ,ρs ( eji )) h − j (cid:0) ∂ + ( i ) (cid:1) ; ee H j ( e ji ,ρ s ( e ji )) < ee H jv (cid:17) , i ∈ I js , E j ( e ji ,ρ ( i ) − x ) (cid:16) e − α ee H j ( eji ,ρs ( eji )) h − j (cid:0) ∂ − ( i ) (cid:1) ; ee H j ( e ji ,ρ s ( e ji )) < ee H jv (cid:17) , i ∈ I − js = ( h − j (cid:0) ∂ + ( i ) (cid:1) E Bx (cid:0) e − ατ ρ ( i ) ; τ ρ ( i ) < τ (cid:1) , i ∈ I js ,h − j (cid:0) ∂ − ( i ) (cid:1) E Bρ ( i ) − x (cid:0) e − ατ ρ ( i ) ; τ ρ ( i ) < τ (cid:1) , i ∈ I − js . Now, the first passage time formulas for the one-dimensional Brownian motion B (cf. [2, Section 1.7]) give E Bρ ( i ) − x (cid:0) e − ατ ; τ < τ ρ ( i ) (cid:1) = sinh (cid:0) √ α x (cid:1) sinh (cid:0) √ α ρ ( i ) (cid:1) = E Bx (cid:0) e − ατ ρ ( i ) ; τ ρ ( i ) < τ (cid:1) , E Bx (cid:0) e − ατ ; τ < τ ρ ( i ) (cid:1) = sinh (cid:0) √ α ( ρ ( i ) − x ) (cid:1) sinh (cid:0) √ α ρ ( i ) (cid:1) = E Bρ ( i ) − x (cid:0) e − ατ ρ ( i ) ; τ ρ ( i ) < τ (cid:1) . A comparison of the equations (5.9) and (5.10) then proves the equalities in (ii).We have shown that the conditions of [8, Theorem 1.6] are fulfilled and thus haveproved:
Lemma 5.5.
The process X which is obtained by forming alternating copies of X − and X +1 via the transfer kernels K − and K +1 , as defined by equation (5.3) , is aright process on G − ∪ G +1 = G . Proving that X is a Brownian Motion on G . As just seen, X is a rightprocess and therefore a strong Markov process on G . In regard to [6, Theorem 2.5],it suffices to analyze the stopped resolvent and the exit behavior from any edge inorder to show that X is indeed a Brownian motion on G : Lemma 5.6. X is a Brownian motion on G .Proof. For mutual edges i ∈ I s , we choose j ∈ {− , +1 } such that i ∈ I js . Then wehave X t = X jt for all t < τ jj and X R ∈ ∂ ( i ), P ( i,x ) -a.s., for the first revival time R = inf (cid:8) t ≥ X t ∈ G − j \G j (cid:9) . Therefore, H X = τ jj ∧ R holds true, and withequation (5.8) we get E ( i,x ) (cid:16) Z H X e − αt f ( X t ) dt (cid:17) = E j ( i,x ) (cid:16) Z τ jj ∧ ζ j e − αt f (cid:0) X jt (cid:1) dt (cid:17) = E Bx (cid:16) Z τ ∧ τ ρ ( i ) e − αt f (cid:0) i, B t (cid:1) dt (cid:17) = E Bx (cid:16) Z H B e − αt f ( i, B t ) dt (cid:17) . For non-mutual edges l / ∈ I s , on the other hand, choose j ∈ {− , +1 } suchthat ( l, x ) ∈ ee G j . Then X jt = ee X jt holds for all t < τ jj , P ( l,x ) -a.s., and as ee X j isitself a Brownian motion on ee G j , the above identity follows immediately.Coming to the exit distribution from an edge, the identity P ( l,x ) ◦ (cid:0) H X , X H X (cid:1) − = P Bx ◦ (cid:0) H B , ( l, B H B ) (cid:1) − follows for edges l / ∈ I s from the corresponding property of ee X − or ee X +1 by [6,Theorem 2.5]. In case i ∈ I s , we choose j ∈ {− , +1 } with i ∈ I js . By employingequations (5.9), (5.10) and H X = τ jj ∧ R P ( i,x ) -a.s., we get for all α > h ∈ b B ( G ) E ( i,x ) (cid:0) e − αH X h ( X H X ) (cid:1) = E j ( i,x ) (cid:0) e − ατ jj h ( X jτ jj ); τ jj < ζ j (cid:1) + E j ( i,x ) (cid:0) e − αζ j K j g ; ζ j < τ jj (cid:1) = E Bx (cid:0) e − ατ h (cid:0) ∂ − ( i ) (cid:1) ; τ < τ ρ ( i ) (cid:1) + E Bx (cid:0) e − ατ ρ ( i ) h (cid:0) ∂ + ( i ) (cid:1) ; τ ρ ( i ) < τ (cid:1) = E Bx (cid:0) e − αH B h ( i, B H B ) (cid:1) , which results in P ( i,x ) ◦ (cid:0) H X , X H X (cid:1) − = P Bx ◦ (cid:0) H B , ( i, B H B ) (cid:1) − . (cid:3) Computing the Feller–Wentzell Data of X . The Feller–Wentzell dataof X , as given in [6, Theorem 1.2], is derived from its exit distributions from anyarbitrarily small neighborhood of each vertex. X is constructed via alternatingcopies of X − and X +1 , so we first need to analyze their respective exit behavior.To this end, we consider the exit times of X j τ jε := inf (cid:8) t ≥ d (cid:0) X jt , X j (cid:1) > ε (cid:9) together with the exit distributions X jτ jε for all small ε >
0. As we only have infor-mation on ee X j , we need to trace back the required data to these original processes.Fix v ∈ V and choose j ∈ {− , +1 } such that v ∈ V j , and let ee τ jε := inf (cid:8) t ≥ d (cid:0) ee X jt , ee X j (cid:1) > ε (cid:9) . Using the definition of X j and the isometric property of ψ j , we get for all ε > τ jε = inf (cid:8) t ≥ d (cid:0) ψ j ( e X jt ) , ψ j ( e X j ) (cid:1) > ε (cid:9) = inf (cid:8) t ≥ d (cid:0) e X jt , e X j (cid:1) > ε (cid:9) =: e τ jε . By its definition, e X jt = ee X jt holds for all t < ee τ j , and as ∁ B ε ( v ) ⊇ e G js , we obtain ∀ ε < δ : ee τ jε ≤ ee τ j P jv -a.s. . More precisely, we even get ∀ ε < δ : ee τ jε < ee τ j , if ee τ jε = + ∞ , P jv -a.s. , because P jv (cid:0)ee τ jε = ee τ j , ee τ jε < + ∞ (cid:1) = P jv (cid:0)ee τ jε = ee τ j , ee X ee τ j ∈ e G js , ee τ jε < + ∞ (cid:1) = P jv (cid:0)ee τ jε = ee τ j , ee X ee τ jε ∈ e G js , ee τ jε < + ∞ (cid:1) ≤ P jv (cid:0) ee X ee τ jε ∈ e G js (cid:1) = 0 . ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 19
Therefore, we see that for all ε < δ , ee τ jε = inf (cid:8) t ∈ [0 , ee τ j ) : d ( ee X jt , ee X j ) > ε (cid:9) ∧ ee τ j = inf (cid:8) t ∈ [0 , ee τ j ) : d ( e X jt , e X j ) > ε (cid:9) ∧ ee τ j = inf (cid:8) t ≥ d ( e X jt , e X j ) > ε (cid:9) , where we used that e X j is a subprocess of ee X j with lifetime ee τ j , that is d ( e X j ee τ j , e X j ) = d (∆ , e X j ) = + ∞ > ε. We have thus shown:
Lemma 5.7.
Let v ∈ V j . For all ε < δ , it holds P jv -a.s. that τ jε = e τ jε = ee τ jε , and ee τ jε < ee τ j , if ee τ jε < + ∞ . Corollary 5.8.
For all v ∈ V j , ε < δ , the exit distribution of X j is given by X jτ jε = ( ψ j ( ee X j ee τ jε ) , ee τ jε < + ∞ , ∆ , ee τ jε = + ∞ . We are ready to compute the Feller–Wentzell data of X . By Lemmas 5.7 and 5.3,we have for all ε < δ τ jε = ee τ jε < ee τ j = ζ j on { ζ j < + ∞} , so τ jε < ζ j a.s. holds. On the other hand, X t = X jt holds for all t < R = ζ j (moreformally, X jt ( ω i ) = X t (cid:0) ( ω , ω , . . . ) (cid:1) with i = 1 if j = −
1, and i = 2 if j = +1) bythe construction of X , yielding P v ◦ (cid:0) τ ε , X τ ε (cid:1) − = P jv ◦ (cid:0) τ jε , X jτ jε (cid:1) − . Thus, if v is not a trap, then ee τ jε < + ∞ holds P jv -a.s. for all sufficiently small ε > τ ε < + ∞ holds P v -a.s. as well. By using thenotations of [6, Theorem 1.2] and backtracking X to ee X j , we compute for ε < δ , forall A ∈ B (cid:0) G\{ v } (cid:1) : ν vε ( A ) = P v (cid:0) X τ ε ∈ A (cid:1) E v ( τ ε ) = P jv (cid:0) X jτ jε ∈ A (cid:1) E jv ( τ jε ) = P jv (cid:0) ψ j ( ee X j ee τ jε ) ∈ A (cid:1) E jv ( ee τ jε ) = ee ν j,vε (cid:0) ( ψ j ) − ( A ) (cid:1) , where we naturally extend, here and in all that follows, the mapping ψ j : e G j → G j to ψ j : e G j → G . This gives K vε = 1 + P v (cid:0) X τ ε = ∆ (cid:1) E v ( τ ε ) + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) ν vε ( dg )= 1 + P jv (cid:0) ee X j ee τ jε = ∆ (cid:1) E jv ( ee τ jε ) + Z e G j \{ v } (cid:0) − e − d ( v,ψ j ( g )) (cid:1) ee ν j,vε ( dg )= 1 + P jv (cid:0) ee X j ee τ jε = ∆ (cid:1) E jv ( ee τ jε ) + Z ee G j \{ v } (cid:0) − e − d ( v,g ) (cid:1) ee ν j,vε ( dg )= ee K j,vε , because ψ j is an isometry with ψ j ( v ) = v , and as ee ν vε (cid:0) ee G j \ e G j (cid:1) = 0 holds due to theassumption (5.1). Renormalization yields, again because ψ is an isometry,(5.11) ∀ A ∈ B (cid:0) G\{ v } (cid:1) : µ vε ( A ) = Z A (cid:0) − e − d ( v,g ) (cid:1) ν vε ( dg ) K vε = Z ψ − ( A ) (cid:0) − e − d ( v,ψ ( g )) (cid:1) ee ν j,vε ( dg ) ee K vε = ee µ j,vε (cid:0) ( ψ j ) − ( A ) (cid:1) . Next, introduce the topological subspaces e G j \{ v } of ee G j \{ v } and G j \{ v } of G\{ v } ,and consider the continuous extension of ψ j : e G j → G j to ψ j : e G j \{ v } → G\{ v } .Continuity of ψ j dictates that the new points e G j \{ v }\ e G j are mapped to(5.12) i ∈ I js ( v ) : ψ j (cid:0) ( e ji , (cid:1) = lim x (cid:21) ψ j (cid:0) ( e ji , x ) (cid:1) = ( i, ,ψ j (cid:0) ( e ji , ρ i − ) (cid:1) = lim x ⇈ ρ ( i ) ψ j (cid:0) ( e ji , x ) (cid:1) = ( i, ρ i ) ,i ∈ I − js ( v ) : ψ j (cid:0) ( e ji , (cid:1) = lim x (cid:21) ψ j (cid:0) ( e ji , x ) (cid:1) = ( i, ρ i ) ,ψ j (cid:0) ( e ji , ρ i − ) (cid:1) = lim x ⇈ ρ ( i ) ψ j (cid:0) ( e ji , x ) (cid:1) = ( i, , and analogously(5.13) i ∈ I j ( v ) : ψ j (cid:0) ( i, (cid:1) = ( i, , if v = ∂ − ( i ), ψ j (cid:0) ( i, ρ i − ) (cid:1) = ( i, ρ i − ) , if v = ∂ + ( i ), e ∈ E j ( v ) : ψ j (cid:0) ( e, (cid:1) = ( e, ,e ∈ E j : ψ j (cid:0) ( e, + ∞ ) (cid:1) = ( e, + ∞ ) . Proceeding in the course of the proof of [6, Theorem 1.2] for ee X j , we extend themeasures ee µ j,vε to measures ee µ j,vε on ee G j \{ v } by ee µ j,vε ( A ) := ee µ vε (cid:0) A ∩ (cid:0) ee G j \{ v } (cid:1)(cid:1) , A ∈ B (cid:0) ee G j \{ v } (cid:1) , and choose a sequence of positive numbers ( ε n , n ∈ N ) converging to zero, suchthat (cid:0)ee µ j,vε n , n ∈ N (cid:1) converges weakly to a measure ee µ j,v . When also extending themeasures µ vε to measures µ vε on G\{ v } , we obtain with equation (5.11) ∀ A ∈ B (cid:0) G\{ v } (cid:1) : µ vε ( A ) = µ vε (cid:0) A ∩ (cid:0) G\{ v } (cid:1)(cid:1) = ee µ j,vε (cid:0) ( ψ j ) − (cid:0) A ∩ (cid:0) G\{ v } (cid:1)(cid:1) = ee µ j,vε (cid:0) ( ψ j ) − ( A ) ∩ (cid:0) e G j \{ v } (cid:1)(cid:1) = ee µ j,vε (cid:0) ( ψ j ) − ( A ) ∩ (cid:0) ee G j \{ v } (cid:1)(cid:1) = ee µ j,vε ◦ ( ψ j ) − ( A ) . By the continuous mapping theorem, ( µ vε n , n ∈ N ) converges weakly to the measure µ v = ee µ j,v ◦ ( ψ j ) − on G\{ v } . We summarize all of our results up to this point: ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 21
Lemma 5.9.
Let v ∈ V j , and K vε , µ vε , µ v and ee K j,vε , ee µ j,v , ee µ j,v be defined as in [6,Theorem 1.2] for the Brownian motions X , ee X j respectively. Then,(i) K vε = ee K j,vε for all ε < δ ,(ii) µ vε = ee µ j,vε ◦ ( ψ j ) − for all ε < δ ,(iii) ( µ vε n , n ∈ N ) converges weakly along the same sequence ( ε n , n ∈ N ) of posi-tive numbers for which ( ee µ j,vε n , n ∈ N ) converges weakly to ee µ j,v , and the limitof ( µ vε n , n ∈ N ) is µ v = ee µ j,v ◦ ( ψ j ) − . We are now ready to compute the Feller–Wentzell data of the glued process X ,thus completing the proof of Theorem 5.1: Proof of Theorem 5.1.
We have already proved in Lemma 5.6 that X is a Brownianmotion on G . It remains to compute the Feller–Wentzell data of X by employingLemma 5.9. To this end, let v ∈ V and choose j ∈ {− , +1 } such that v ∈ V j .The killing parameters are given by c v, ∆1 = lim n →∞ P v ( X τ εn = ∆) E v ( τ ε n ) K ve n = lim n →∞ P jv ( ee X j ee τ jεn = ∆) E jv ( ee τ jε n ) ee K j,ve n = p v, ∆1 ,c v, ∞ = X e ∈E µ v (cid:0) { ( e, + ∞ ) } (cid:1) = X e ∈E j ∪E js ee µ j,v (cid:0) { ( e, + ∞ ) } (cid:1) = p v, ∞ , and thus vanish, as p v = p v, ∆1 + p v, ∞ = 0 holds by assumption.The reflection parameters are defined as c v,l = µ v (cid:0) { ( l, } (cid:1) , l ∈ E ( v ) ,µ v (cid:0) { ( l, } (cid:1) , l ∈ I ( v ) , v = ∂ − ( l ) ,µ v (cid:0) { ( l, ρ l − ) } (cid:1) , l ∈ I ( v ) , v = ∂ + ( l ) . For e ∈ E ( v ), the relation ( ψ j ) − (cid:0) ( e, (cid:1) = ( e, c v,e = p v,e .For i ∈ I ( v ), we need to distinguish some cases, using equations (5.12) and (5.13):For i ∈ I ( v ) with v = ∂ − ( i ), that is if i ∈ I j ( v ) ∪ I js ( v ), we have c v,i = µ v (cid:0) { ( i, } (cid:1) = (ee µ j,v (cid:0) { ( i, } (cid:1) = p v,i , i ∈ I j ( v ) , ee µ j,v (cid:0) { ( e ji , } (cid:1) = p v,e ji , i ∈ I js ( v ) , while for i ∈ I ( v ) with v = ∂ + ( i ), that is if i ∈ I − j ( v ) ∪ I − js ( v ), we have c v,i = µ v (cid:0) { ( i, ρ i − ) } (cid:1) = (ee µ j,v (cid:0) { ( i, ρ i − ) } (cid:1) = p v,i , i ∈ I − j ( v ) , ee µ j,v (cid:0) { ( e ji , } (cid:1) = p v,e ji , i ∈ I − js ( v ) . The diffusion parameter is given by c v = lim n →∞ K ve n = lim n →∞ ee K j,ve n = p v . For all A ∈ B ( G\{ v } ), the jump distribution is computed by c v ( A ) = Z A − e − d ( v,g ) µ v ( dg )= Z ( ψ j ) − ( A ) − e − d ( v,ψ j ( g )) ee µ j,v ( dg )= p v ◦ ( ψ j ) − ( A ) , as ψ j is an extension from ψ j : e G j → G and an isometry. (cid:3) Completing the Construction
We are ready to carry out the construction that was laid out in section 2.
Theorem 6.1.
Let G = ( V , E , I , ∂, ρ ) be a metric graph, and for every v ∈ V letconstants p v,l ≥ for each l ∈ L ( v ) , p v ≥ and a measure p v on G\{ v } be given,satisfying X l ∈L ( v ) p v,l + p v + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) p v ( dg ) = 1 , and p v (cid:0) G\{ v } (cid:1) = + ∞ , if X l ∈L ( v ) p v,l + p v = 0 , as well as p v (cid:0) ∁ B δ ( v ) (cid:1) = 0 for some δ ∈ (0 , min l ∈L ρ l ) . Then there exists a Brownianmotion X on G which has infinite lifetime, is continuous inside all edges, satisfies X τ ε ∈ B δ ( v ) P v -a.s. for all ε < δ , v ∈ V , and admits the Feller–Wentzell data (cid:0) , ( p v,l ) l ∈L ( v ) , p v , p v (cid:1) v ∈V . Proof.
We proceed via an induction over the count n := |V| of vertices. If n = 1,then G is a star graph, so the construction given in [7] together with [7, Theo-rem 4.33], [6, Lemma 4.1] and [7, Theorem 4.3] yield the result.Assume now that such Brownian motions exist for all metric graphs with lessthan n vertices. Let G be a metric graph with n vertices V = { v , . . . , v n } andboundary data as given in the theorem. We decompose the graph into ee G − and ee G +1 ,as done in section 5, for V − = { v , . . . , v n − } and V +1 = { v n } . Then the conditionsof the theorem are satisfied for these graphs ee G − , ee G +1 with n − p v,l ≥ , l ∈ ee L j ( v )), p v ≥ p v ◦ ψ j (as ψ j is an isometry, this data satisfies the normalization requirements).Therefore, there exist Brownian motions ee X j on ee G j with infinite lifetime which arecontinuous inside all edges, satisfy ee X j ee τ jε ∈ B δ ( v ) P jv -a.s. for all v ∈ V j and admitthe Feller–Wentzell data (cid:0) , ( p v,l ) l ∈ ee L j ( v ) , p v , p v ◦ ψ j (cid:1) v ∈V j with p v,e ji := p v,i for i ∈ I s ( v ), v ∈ V j . We then follow the construction of section 5in order to glue ee X − and ee X +1 together, and Theorem 5.1 concludes the proof. (cid:3) In order to implement the killing parameter and the non-local jumps, we firstneed to adjoin the “fake cemeteries” (cid:3) v for all v ∈ V : ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 23
Theorem 6.2.
Let G = ( V , E , I , ∂, ρ ) be a metric graph, and for every v ∈ V letconstants p v ≥ , p v,l ≥ for each l ∈ L ( v ) , p v ≥ and a measure p v on G\{ v } begiven with p v + X l ∈L ( v ) p v,l + p v + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) p v ( dg ) = 1 , and p v (cid:0) G\{ v } (cid:1) = + ∞ , if X l ∈L ( v ) p v,l + p v = 0 , as well as p v (cid:0) ∁ B δ ( v ) (cid:1) = 0 for some δ ∈ (cid:0) , min l ∈L ρ l (cid:1) . Then there exists a Brow-nian motion X on G ∪ { (cid:3) v , v ∈ V} with { (cid:3) v , v ∈ V} being an isolated, absorbingset for X , such that X has infinite lifetime, is continuous inside all edges, satisfies X τ ε ∈ B δ ( v ) ∪ { (cid:3) v } P v -a.s. for all ε < δ , v ∈ V , and has the Feller–Wentzell data (cid:0) , ( p v,l ) l ∈L ( v ) , p v , p v + p v ε (cid:3) v (cid:1) v ∈V . Proof.
This proof proceeds analogously to the proof of Theorem 6.1, except thatwe need to adjoin the isolated points (cid:3) v , v ∈ V , to the partial processes and revivethese processes there before gluing the partial graphs together.If |V| = 1, then G is a star graph, and the construction of [7] (again with [7,Theorem 4.33], [6, Lemma 4.1], and [7, Theorem 4.3]) gives a Brownian motionon G with the needed properties and Feller–Wentzell data (cid:0) p v , ( p v,l ) l ∈L ( v ) , p v , p v (cid:1) . By concatenating it with the constant process on { (cid:3) v } with the technique of [8],we revive this Brownian motion on a new, isolated, absorbing point (cid:3) v . Thena computation along the lines of Lemma 4.1 yields that the revived process is aBrownian motion on G ∪ { (cid:3) v } with its Feller–Wentzell data at v being given by (cid:0) , ( p v,l ) l ∈L ( v ) , p v , p v + p v ε (cid:3) v (cid:1) . Now let V = { v , . . . , v n } , and assume that the assertion of the theorem holds forany graph with less than n vertices. We decompose the graph G into ee G − and ee G +1 ,as done in section 5, for V − = { v , . . . , v n − } and V +1 = { v n } . By assumption,there exist Brownian motions ee X j on ee G j ∪ (cid:8) (cid:3) v , v ∈ V j (cid:9) with the needed pathproperties and Feller–Wentzell data (cid:0) , ( p v,l ) l ∈ ee L j ( v ) , p v , p v ◦ ψ j + p v ε (cid:3) v (cid:1) v ∈V j with p v,e ji := p v,i for i ∈ I s ( v ), v ∈ V j . We then again follow the construction ofsection 5 to glue ee X − and ee X +1 together, and Theorem 5.1 yields the result. (cid:3) In order to complete the proof of the existence theorem for Brownian motionson metric graphs with non-local boundary conditions, it remains to implement the“global” jumps:
Proof of Theorem 1.1.
Let δ > δ < min l ∈L ρ l , and define for every v ∈ V q v := p v + p v (cid:0) ∁ B v ( δ ) (cid:1) , q v := p v (cid:12)(cid:12) B v ( δ ) . The introduction of the normalizing factor c v := (cid:16) q v + X l ∈L ( v ) p v,l + p v + Z G\{ v } (cid:0) − e − d ( v,g ) (cid:1) q v ( dg ) (cid:17) − enables us to employ Theorem 6.2 in order to construct a Brownian motion X on G ∪{ (cid:3) v , v ∈ V} which has infinite lifetime, is continuous inside all edges, satisfies X τ ε ∈ B δ ( v ) ∪ { (cid:3) v } P v -a.s. for all ε < δ , v ∈ V , and has the Feller–Wentzell data (cid:0) c v (cid:0) , ( p v,l ) l ∈L ( v ) , p v , q v + q v ε (cid:3) v (cid:1)(cid:1) v ∈V . As X has infinite lifetime, we can use an application of the concatenationtechniques (see [8] and [4, Proposition 14.20]) to adjoin a new, isolated, absorb-ing point (cid:3) to X , resulting in a Brownian motion X on the metric graph G ∪ { (cid:3) v , v ∈ V} ∪ { (cid:3) } with the same Feller–Wentzell data as X for all v ∈ V , andadditional Feller–Wentzell data (0 , , ,
0) at the new vertex (cid:3) .Let X be the right process on G ∪ { (cid:3) } which results from killing X on theabsorbing set { (cid:3) v , v ∈ V} (see Appendix A). As X is strongly Markovian and X t = X t for all t ≤ H V , X is a Brownian motion on G ∪ { (cid:3) } , and Lemma 3.2asserts that the Feller–Wentzell data of X reads (cid:0) c v (cid:0) q v , ( p v,l ) l ∈L ( v ) , p v , q v (cid:1)(cid:1) v ∈V . Now construct X as the revived process obtained from X by the identicalcopies method with revival distributions κ v := ( q v ) − (cid:0) p v ε (cid:3) + p v (cid:12)(cid:12) ∁ B δ ( v ) (cid:1) , v ∈ V . Then by Lemma 4.1, X is a Brownian motion on G ∪{ (cid:3) } , and its generator satisfies D ( A ) ⊆ n f ∈ C ( G ∪ { (cid:3) } ) : ∀ v ∈ V : − X l ∈L ( v ) c v p v,l f ′ l ( v ) + c v p v f ′′ ( v ) − Z ( G\{ v } ) ∪{ (cid:3) } (cid:0) f ( g ) − f ( v ) (cid:1) c v (cid:0) p v (cid:12)(cid:12) B δ ( v ) + p v ε (cid:3) + p v (cid:12)(cid:12) B δ ( v ) ∁ (cid:1) ( dg ) = 0 o = n f ∈ C ( G ∪ { (cid:3) } ) : ∀ v ∈ V : − X l ∈L ( v ) p v,l f ′ l ( v ) + p v f ′′ ( v ) − Z ( G\{ v } ) ∪{ (cid:3) } (cid:0) f ( g ) − f ( v ) (cid:1) (cid:0) p v + p v ε (cid:3) (cid:1) ( dg ) = 0 o . Finally, employ once more the transformation of Appendix A in order to kill X on the isolated, absorbing set { (cid:3) } and obtain the Brownian motion X on G .Lemma 3.1 asserts that the domain of its generator satisfies D ( A ) ⊆ n f ∈ C ( G ) : ∀ v ∈ V : p v f ( v ) − X l ∈L ( v ) p v,l f ′ l ( v ) + p v f ′′ ( v ) − Z G\{ v } (cid:0) f ( g ) − f ( v ) (cid:1) p v ( dg ) = 0 o . (cid:3) ROWNIAN MOTIONS ON METRIC GRAPHS II: CONSTRUCTION 25
Appendix A. Killing on an Absorbing Set
We present an easy technique to kill a right process on an absorbing set (see [4,Definition 12.27]), which will be used in the main construction of this article. Forthe role of the cemetery point ∆ and the lifetime conventions in the context of rightprocesses, the reader may consult [4, Section 11].Let e E = E ∆ ⊎ F be the topological union of two disjoint Radon spaces E ∆ and F ,and consider a right process X on e E , with F being an absorbing set for X . We killthe process X on this absorbing set F by mapping F to ∆ with ψ : e E → E ∆ , x ψ ( x ) := ( x, x ∈ E ∆ , ∆ , x ∈ F. By checking the consistency conditions for state space transformations of rightprocesses (see [4, Section 13] and [8, Section 3.1]), we show:
Theorem A.1. ψ ( X ) is a right process on E ∆ .Proof. The transformation ψ is clearly surjective and measurable, as ∀ B ∈ E u ∆ : ψ − ( B ) = ( B, ∆ / ∈ B,B ∪ F, ∆ ∈ B. Let H F be the first entry time of X into F . We have X t ∈ F for all t ≥ H F a.s.,as the strong Markov property at H F yields P (cid:0) X H F + t ∈ F for all t ≥ (cid:1) = E (cid:0) P X HF ( X t ∈ F for all t ≥ (cid:1) = 1 . Furthermore, it is evident that X t / ∈ F for all t < H F , so the transformed process t ψ ( X t ) = ( X t , t < H F , ∆ , t ≥ H F is a.s. right continuous.For all f ∈ b E u ∆ , x ∈ e E , we have for the semigroup ( T t , t ≥
0) of X : T t ( f ◦ ψ )( x ) = E x (cid:0) f ◦ ψ ( X t ) (cid:1) = E x (cid:0) f ( X t ) ; t < H F (cid:1) + f (∆) P x (cid:0) t ≥ H F (cid:1) = g ◦ ψ ( x ) , with g ∈ b E u ∆ being defined by g ( x ) := ( E x (cid:0) f ( X t ) ; t < H F (cid:1) + f (∆) P x (cid:0) t ≥ H F (cid:1) , x ∈ E,f (∆) , x = ∆ , as H F = 0 holds P x -a.s. for all x ∈ F . (cid:3) Acknowledgements
The main parts of this paper were developed during the author’s Ph.D. thesis [5]supervised by Prof. J¨urgen Potthoff, whose constant support the author gratefullyacknowledges.
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