G¨ unter Last Institut f¨ ur Stochastik Karlsruher Institut f¨ ur Technologie Invariant transports of random measures and the extra head problem G¨ unter Last (Karlsruhe) joint work with Peter M¨ orters (Bath) and Hermann Thorisson (Reykjavik) presented at the conference Stochastic Networks and Stochastic Geometry Paris, January 12-14, 2015
1. Four problems on random shifts 1. Extra head problem Consider a two-sided sequence of independent and fair coin tosses. Find a coin that landed heads so that the other coin cosses are still independent and fair. 2. Marriage of Lebesgue and Poisson Let η be a stationary Poisson process in R d . Find a point T of η such that d θ T η − δ 0 = η. G¨ unter Last Extra head problem
3. Poisson matching Let η and ξ be two independent stationary Poisson processes with equal intensity. Find a point T of ξ such that θ T ( η + δ 0 , ξ ) d = ( η, ξ + δ 0 ) 4. Unbiased shifts of Brownian motion Let B = ( B t ) t ∈ R be a two-sided standard Brownian motion. Find a random time T such that the space-time shifted process ( B T + t − B T ) t ∈ R is a Brownian motion, independent of B T . G¨ unter Last Extra head problem
2. Invariant transports of random measures Setting (Ω , F , P ) is a σ -finite measure space. For the first three problems P can be taken as probability measure. Definition A random measure on R d is a random element in the space of all locally finite measures on R d equipped with the Kolmogorov product σ -field. G¨ unter Last Extra head problem
Setting We consider mappings θ s : Ω → Ω , s ∈ R d , satisfying θ 0 = id Ω and the flow property s , t ∈ R d . θ s ◦ θ t = θ s + t , The mapping ( ω, s ) �→ θ s ω is supposed to be measurable. We assume that P is stationary, that is s ∈ R d . P ◦ θ s = P , Definition A random measure ξ is invariant if ω ∈ Ω , s ∈ R d , B ∈ B d . ξ ( θ s ω, B − s ) = ξ ( ω, B ) , G¨ unter Last Extra head problem
Definition Let ξ be an invariant random measure on R d . The measure �� Q ξ ( A ) := 1 { θ s ω ∈ A , s ∈ B } ξ ( ω, ds ) P ( d ω ) , A ∈ F , is called the Palm measure of ξ (with respect to P ), where B ∈ B d satisfies 0 < λ d ( B ) < ∞ . Theorem (Refined Campbell theorem) Let ξ be an invariant random measure on R d . Then � � f ( θ s , s ) ξ ( ds ) = E Q ξ f ( θ 0 , s ) ds E P for all measurable f : Ω × R d → [ 0 , ∞ ) . G¨ unter Last Extra head problem
Definition An allocation rule is a measurable mapping τ : Ω × R d → R d that is equivariant in the sense that s , t ∈ R d , P -a.e. ω ∈ Ω . τ ( θ t ω, s − t ) = τ ( ω, s ) − t , Theorem (L. and Thorisson ’09) Let ξ and η be two invariant random measures with positive and finite intensities. Let τ be an allocation rule and define T := τ ( · , 0 ) . Then Q ξ ( θ T ∈ · ) = Q η iff τ is balancing ξ and η , that is � 1 { τ ( s ) ∈ ·} ξ ( ds ) = η P -a.e. G¨ unter Last Extra head problem
Remark The previous result extends to weighted transport kernels and to LCSC-groups G ; see L. and Thorisson ’09 and L. ’10a. It can even be extended to random measures on a space, on which G operates; see L. ’10b and Kallenberg ’11. G¨ unter Last Extra head problem
Example Assume that ξ = λ d is Lebesgue measure and that η is a simple point process. An allocation rule τ is balancing ξ and η , iff P -a.e. λ d ( C τ ( t )) = 1 , t ∈ η, where the cell C τ ( t ) is given by C τ ( t ) := { s ∈ R d : τ ( s ) = t } . Theorem (Holroyd and Peres ’05) Assume that η is a stationary unit-rate Poisson process and let τ be an allocation rule. Then τ is balancing Lebesgue measure and η iff P ( θ τ ( 0 ) η ∈ · ) = P ( η + δ 0 ∈ · ) . G¨ unter Last Extra head problem
Example Assume that ξ and η are simple point processes. An allocation rule τ is balancing ξ and η , iff τ is a perfect matching ( P -a.e.) of the points of ξ with the points of η . Theorem (Holroyd, Pemantle, Peres, Schramm ’09) Assume that ξ and η are independent stationary unit-rate Poisson processes (defined on their canonical probability space) and let τ be an allocation rule. Then τ is balancing ξ and η iff θ T ( ξ + δ 0 , η ) d = ( ξ, η + δ 0 ) , where T := τ (( ξ + δ 0 , η ) , 0 ) . G¨ unter Last Extra head problem
3. Local time of Brownian motion Setting B = ( B t ) t ∈ R is a two-sided standard Brownian motion starting in 0 ( B 0 = 0) defined on its canonical probability space (Ω , F , P 0 ) . Definition An unbiased shift (of B ) is a random time T (negative values are allowed) such that: B ( T ) := ( B T + t − B T ) t ∈ R is a Brownian motion, B ( T ) is independent of B T . G¨ unter Last Extra head problem
Example If T ≥ 0 is a stopping time, then ( B T + t − B T ) t ≥ 0 is a one-sided Brownian motion independent of B T . However, the example T := inf { t ≥ 0 : B t = a } shows that ( B T + t − B T ) t ∈ R need not be a two-sided Brownian motion. Example Consider a deterministic T ≡ t 0 . Then B ( T ) = ( B t 0 + t − B t 0 ) t ∈ R is a two-sided Brownian. However, since B ( T ) − t 0 = − B t 0 , this two-sided motion is not independent of B T = B t 0 . G¨ unter Last Extra head problem
Remark An unbiased shift with B T = 0 is characterized by d ( B T + t ) t ∈ R = B . According to Mandelbrot (The Fractal Geometry of Nature) ”...the process of Brownian zeros is stationary in a weakened form.“ He is using the (non-rigorous) concept of conditional stationarity. However, the stopping time T := inf { t ≥ 1 : B t = 0 } has the property B T = 0. But clearly B ( T ) is not a Brownian motion. The missing link will be provided by balancing local times at different levels. G¨ unter Last Extra head problem
Definition Let ℓ 0 be the local time (random measure) at zero. Its right-continuous (generalised) inverse is defined as � sup { t ≥ 0 : ℓ 0 [ 0 , t ] = r } , r ≥ 0 , T r := sup { t < 0 : ℓ 0 [ t , 0 ] = − r } , r < 0 . Theorem Let r ∈ R . Then T r is an unbiased shift. Idea of the proof: The intervals [ T n , T n + 1 ] , n ∈ Z , split B into iid-cycles. The distribution of these cycles is time-reversible. G¨ unter Last Extra head problem
Definition The local time measure ℓ x at x ∈ R can be defined by 1 � ℓ x ( C ) := lim 1 { s ∈ C , x ≤ B s ≤ x + h } ds . h h → 0 Hence � �� f ( x , s ) ℓ x ( ds ) dx f ( B s , s ) ds = P 0 -a.s. for all measurable f : R 2 → [ 0 , ∞ ) . G¨ unter Last Extra head problem
Definition For t ∈ R the shift θ t : Ω → Ω is given by ( θ t ω ) s := ω t + s , s ∈ R . For x ∈ R let P x := P 0 ( B + x ∈ · ) , x ∈ R , where B is the identity on Ω . Remark It is a possible to choose a perfect version of local times, that is a (measurable) kernel satisfying for all x ∈ R and P x -a.e. that ℓ x is diffuse and ℓ x ( θ t ω, C − t ) = ℓ x ( ω, C ) , C ∈ B , t ∈ R , ℓ x ( B , · ) = ℓ 0 ( B − x , · ) . G¨ unter Last Extra head problem
Definition Let � P := P x dx be the distribution of a Brownian motion with a ”uniformly distributed“ starting value. Remark Stationary increments of B imply that P is stationary, that is P = P ◦ θ s , s ∈ R . G¨ unter Last Extra head problem
Theorem (Geman and and Horowitz ’73) The Palm (probability) measure of the local time ℓ x is P x . Definition Let ν be a probability measure on R . Define � � ℓ ν := ℓ x ν ( dx ) . P ν := P x ν ( dx ) , Corollary P ν is the Palm probability measure of ℓ ν . Remark In the language of stochastic analysis ℓ ν is a continuous additive functional with Revuz measure ν . G¨ unter Last Extra head problem
4. Existence of unbiased shifts Definition (Skorokhod embedding problem) Let ν be a probability measure on R . A random time T embeds ν if B T has distribution ν . Theorem Let T be a random time and ν be a probability measure on R . Then T is an unbiased shift embedding ν if and only if the allocation rule τ defined by τ T ( s ) := T ◦ θ s + s is balancing ℓ 0 and ℓ ν . G¨ unter Last Extra head problem
Example Let r > 0. Then τ ( s ) := inf { t > s : ℓ 0 ([ s , t ]) = r } , s ∈ R . Then τ is an allocation rule balancing ℓ 0 with itself. Hence T r = τ ( · , 0 ) is an unbiased shift (embedding δ 0 ). G¨ unter Last Extra head problem
Theorem Let ν be a probability measure on R with ν { 0 } = 0 . Then the stopping time T := inf { t > 0 : ℓ 0 [ 0 , t ] = ℓ ν [ 0 , t ] } embeds ν and is an unbiased shift. Remark The above stopping time above was introduced in Bertoin and Le Jan (1992) as a solution of the Skorokhod embedding problem. Theorem (L., M¨ orters and Thorisson ’14) Let ν be a probability measure on R . Then there is a non- negative stopping time that is an unbiased shift embedding ν . G¨ unter Last Extra head problem
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