REVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE. 1. Introduction We look at the problem of reversibility for operators of the form

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1 REVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE OMAR RIVASPLATA, JAN RYCHTÁŘ, AND BYRON SCHMULAND Abstract. Why is the rift coefficient b associate with a reversible iffusion on R given by a graient? Our explanation is inspire by Hana s recent results on reversibility an quasi-invariance of the invariant measure. 1. Introuction We look at the problem of reversibility for operators of the form Lf(x) = 1 f(x) + b i (x) i f(x), f Cc (R ). (1) i=1 For simplicity we assume that the function b i is smooth for every 1 i. We call L reversible if there exists a measure m on R so that (Lf)(x)g(x) m(x) = (Lg)(x)f(x) m(x), f, g Cc (R ). This terminology erives from the time reversal property of the corresponing iffusion process X = (X t ), with initial state chosen ranomly using the measure m. If the generator L of X is reversible, an L(X ) = m, then for any T >, the two finite horizon processes (X t ) t T an (X T t ) t T have ientical finiteimensional istributions. That is, X t has the same probabilistic properties whether the time parameter t runs forwars or backwars. A classical result of Kolmogorov [5] tells us that a iffusion process in R with infinitesimal generator L is reversible if an only if the vector fiel b(x) = (b 1 (x),..., b (x)) is conservative, i.e., given by a graient. In this note, we offer an alternative proof using the concept of quasi-invariance. Here is some notation an terminology that we use throughout the paper. A measure m is a non-zero Borel measure that is finite on compact sets. By transformation we will mean a measurable bijection with measurable inverse. The space C (R ) is the space of smooth functions on R, while Cc (R ) are the smooth functions with compact support. The brackets x, y will refer to the usual inner prouct on R. Mathematics Subject Classification. 6J6, 8C1, 47F5. Key wors an phrases. Diffusion process, infinitesimal generator, reversible measure, quasiinvariant measure, cocycle ientity. This work was one when the first name author was a grauate stuent at the University of Alberta. 1

2 OMAR RIVASPLATA, JAN RYCHTÁŘ, AND BYRON SCHMULAND. Quasi-invariant measures Let R be equippe with its Borel σ-algebra B(R ). Let S = {S v } v V enote a group of transformations on R, inexe by a vector space V. In other wors, for each v V, the mapping S v : R R is a transformation, an the following properties hol: S u+v (x) = S u (S v (x)), S (x) = x. Since these mappings are bimeasurable, we can efine the image measure by m S v (B) := m(s v (B)). This image measure is characterize by the fact that for all g Cc (R ) g(s v (x)) (m S v )(x) = g(x) m(x). () Definition 1. Let S = {S v } v V be a transformation group on (R, B(R )). A measure m is calle S-quasi-invariant if, for each v V, the measures m an m S v are equivalent. If m is S-quasi-invariant, then we can write the ensity of m S v with respect to m, for some measurable function Λ(v, x), as (m S v ) m (x) = eλ(v,x), m-a.s., an we say that m is S-quasi-invariant with cocycle Λ. This terminology is explaine by the proposition below, where (3) is calle the cocycle ientity. See [3], [4], an [6] for similar results in other contexts an [1] for more information on cocycles. Proposition 1. If m is S-quasi-invariant with cocycle Λ, then for any u, v V, we have: Λ(u + v, x) = Λ(u, S v (x)) + Λ(v, x), m-a.s. (3) Proof. For every v V the measures m an m S v are mutually absolutely continuous. Then on one han, from the efinition of Raon-Nikoym ensity we have m S u+v (x) = e Λ(u+v,x) m(x). On the other han, using the transformation group properties we have m S u+v (x) = m S u (S v (x)) = e Λ(u,Sv(x)) m(s v (x)) = e Λ(u,Sv(x)) e Λ(v,x) m(x) = e Λ(u,Sv(x))+Λ(v,x) m(x). Hence e Λ(u+v,x) = e Λ(u,Sv(x))+Λ(v,x) m-a.s., which gives the result. 3. Reversibility an Quasi-invariance In the remainer of the paper we take the group S of transformations to be S v (x) := x + v for v R. The next proposition provies a host of examples of quasi-invariant measures on R.

3 REVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE 3 Proposition. Let U : R R be a continuously ifferentiable function. The measure m(x) = e U(x) x is S-quasi-invariant with cocycle Λ(v, x) = 1 U(S tv (x)), v t. (4) Proof. Define Λ(v, x) = U(S v (x)) U(x), so that for any v R, (m S v )(x) = e U(Sv(x)) x = e U(Sv(x)) U(x) e U(x) x = e Λ(v,x) m(x), so m is S-quasi-invariant with cocycle Λ. Note that U(S v (x)) U(x) = U(S v (x)) U(S (x)) = 1 t U(S tv(x))t. We finish by using the chain rule to obtain t U(S tv(x)) = U(S tv (x)), v. Let us comment on how this result helps our intuition for the reversible case. Consier an operator L as in (1), an assume that the measure m is reversible for L. Formally, m shoul be given by m(x) = e U(x) x for some potential function U. Then by the result above m is S-quasi-invariant with cocycle (4). Furthermore, we know from Kolmogorov s criterion that in the reversible case we have b(x) = 1 U(x), where b(x) is the rift of the operator L. So we expect that Λ(v, x) = 1 b(s tv (x)), v t. (5) Our main theorem, Theorem 1, makes this intuition rigorous by showing that a measure m on R is a reversible measure for the operator L if an only if m is quasi-invariant uner the group {S v } v R of translations with cocycle (5). The following three technical lemmas will be use in proving Theorem 1. Lemma 1. Let Λ be given by (5). Fix an arbitrary v R. For any given g Cc (R ) an t R, efine Then g t C c g t (x) = g(s tv (x)) exp{λ(tv, x)}. (R ) for all t R, an t g t(x) = b(x), v g t (x) + v, g t (x). (6) Proof. Set F t (x) = Λ(tv, x) an rewrite g t (x) = g(s tv (x))e Ft(x). From (5) we see that x Λ(v, x) is smooth so that F t C (R ) an g t Cc (R ). The prouct rule gives us [ ] t g t(x) = t g(s tv(x)) e Ft(x) + g t (x) t F t(x) (7) By the chain rule g t (x) = [ g(s tv (x))] e Ft(x) + g t (x) F t (x). (8) t g(s tv(x)) = v, g(s tv (x)). (9)

4 4 OMAR RIVASPLATA, JAN RYCHTÁŘ, AND BYRON SCHMULAND Take the inner prouct of (8) with v, an subtract from (7) (using (9)) to get ( ) t g t(x) v, g t (x) = g t (x) t F t(x) v, F t (x). (1) So we nee to analyze the function F t. A change of variables gives F t (x) = t b(s rv (x)), v r. (11) Using the auxiliary function h(x) := b(x), v, we ifferentiate (11) to get F t (x) = t (h S rv )(x) r. Then using the same calculation as in (9) with h instea of g we get v, F t (x) = t v, (h S rv )(x) r = t r h(s rv(x)) r = [h(s tv (x)) h(s (x))] = [ b(s tv (x)), v b(x), v ] = t F t(x) b(x), v. Substituting this back into (1) gives the result. Lemma. Let L be an operator of the form (1), an let m be a measure on R. If m is reversible for L, then R Lf(x) m(x) =, f C c (R ). (1) Proof. Let f Cc (R ) be arbitrary. Since both f an Lf have compact support, we can fin an open ball B r () centere at the origin with raius big enough to contain the supports of both these functions. Take a function g Cc (R ) such that g = 1 on B r (). We have Lf(x) m(x) = Lf(x)g(x) m(x) = f(x)lg(x) m(x) =, R R R where in the last step we use the fact that Lg = on B r (). Lemma 3. Let L be an operator of the form (1), an let m be a measure on R. Then m is a reversible measure for L if an only if, for any f, g Cc (R ), (Lf)(x)g(x) m(x) = 1 f(x), g(x) m(x). (13) Proof. Suppose that m is reversible for L, an fix f, g Cc (R ), so that (Lf)(x)g(x) m(x) = f(x)(lg)(x) m(x). (14) Now, a irect computation shows that (Lf)(x)g(x) + f(x)(lg)(x) L(fg)(x) = f(x), g(x). (15) Also, fg Cc (R ), so L(fg)(x) m(x) = by (1). Then integrating both sies of (15) with respect to m we get (13).

5 REVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE 5 Conversely, assume that (13) hols. Since the right han sie of this equation is symmetric in f an g, this implies that the left han sie is also symmetric; i.e., (14) hols. Theorem 1. Let L be an operator of the form (1), an let m be a measure on R. Then m is a reversible measure for L if an only if m is quasi-invariant uner the group {S v } v R of all translations with cocycle Λ(v, x) = 1 b(s tv (x)), v t. (16) Proof. Let us take g C c (R ) an v R arbitrary. Integrating both sies of (6) with respect to m we get b(x), v g t (x) m(x) + 1 v, g t (x) m(x) = 1 t g t(x) m(x). Consier a ball B r () big enough to contain the supports of all the functions g t, for t 1 (see Lemma 1 for the efinition of g t ), an take ϕ Cc (R ) with ϕ(x) = 1 for x B r (). Now efine the function f(x) := x, v ϕ(x). We may regar this kin of function f as a truncate polynomial of first egree. Note that for x B r () we have f(x) = x, v, f(x) = v, an Lf(x) = b(x), v. For such f (Lf)(x)g t (x) m(x) + 1 f(x), g t (x) m(x) = 1 t g t (x) m(x). (17) We know that g t C c (R ) by Lemma 1. If m is a reversible measure for L, then the left-han sie of (17) vanishes by (13). Therefore g t (x) m(x) is constant in t an in particular g 1 (x) m(x) = g (x) m(x), or equivalently g(s v (x)) exp{λ(v, x)} m(x) = g(x) m(x). This gives us () with the appropriate ensity, an therefore m is quasi-invariant uner S with the esire cocycle. Conversely, assume that the measure m is quasi-invariant uner S with the given cocycle Λ. Fix an arbitrary g Cc (R ), an consier equation (17) for g an f(x) = x, v ϕ(x) for some v R. It is clear that the right-han sie of (17) vanishes (see Lemma 1 an equation ()). Thus (13) hols in this case. Next we note that, for fixe g Cc (R ), the set of functions f Cc (R ) that satisfy equation (13) is close uner multiplication. To see this, assume that

6 6 OMAR RIVASPLATA, JAN RYCHTÁŘ, AND BYRON SCHMULAND f 1, f Cc (R ) satisfy (13). Then using (15) with f 1 f replacing f we have (Lf 1 f )(x)g(x) m(x) = (Lf 1 )(x)f (x)g(x) m(x) + f 1 (x)(lf )(x)g(x) m(x) + f 1 (x), f (x) g(x) m(x) = 1 f 1 (x), (f g)(x) m(x) 1 f (x), (f 1 g)(x) m(x) + f 1 (x), f (x) g(x) m(x) = 1 f (x) f 1 (x), g(x) m(x) 1 f 1 (x) f (x), g(x) m(x) = 1 (f 1 f )(x), g(x) m(x), so the prouct f 1 f also satisfies (13). Thus (13) can be extene to all functions f(x) = x, v 1 x, v k ϕ(x) with v 1,..., v k R. Using the linearity in f of (13) it follows that this expression must be true for all truncate polynomials f of arbitrary egree. A suitable approximation proceure (see e.g. [, Appenix 7]) shows that (13) is vali for all f Cc (R ). Since g Cc (R ) was fixe arbitrarily, this establishes the reversibility of m. As consequence, we give now our explanation that an operator L as in (1) is reversible precisely when its rift b is of graient form. Corollary 1. The operator L as in (1) has a non-zero reversible measure m, if an only if b has a potential, i.e., there is a function F : R R such that b = F. Proof. If b = F, then set U(x) = F (x) an m(x) = e U(x) x. Proposition an Theorem 1 show that m is reversible for L. Now suppose that L has a non-zero reversible measure m. Define Λ as in (16) using the function b associate with L. By Proposition 1 an Theorem 1, Λ satisfies the cocycle ientity: Λ(u + v, x) = Λ(u, x + v) + Λ(v, x), m-a.s. (18) On the other han, Theorem 1 also says that m is quasi-invariant with respect to shifts, therefore it has full support on R. Since both sies of the equation (18) are continuous functions, the equation must be true for all x R. To show that b has a potential it is enough to prove that v b(x), u = u b(x), v, u, v R, x R. (19) For if (19) hols, then taking u = e i, v = e j we get j b i (x) = i b j (x), an this conition is sufficient for b to have a potential, since the omain R is simply

7 REVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE 7 connecte. Now let us establish (19). From the cocycle ientity we get or equivalently Λ(u, x + v) + Λ(v, x) = Λ(v, x + u) + Λ(u, x), Λ(u, x + v) Λ(u, x) = Λ(v, x + u) Λ(v, x). Then using the efinition of Λ we get 1 b(x + v + tu) b(x + tu), u t = 1 Replace u by δu an v by εv, for some δ, ε >. Then 1 b(x + εv + δtu) b(x + δtu), u ε t = 1 b(x + u + tv) b(x + tv), v t. b(x + δu + εtv) b(x + εtv), v δ an letting first δ an then ε we get v b(x), u = u b(x), v. t, References [1] J. Aaronson, An Introuction to Infinite Ergoic Theory. AMS Mathematical Surveys an Monographs, No. 5, Provience, Rhoe Islan, [] S. N. Ethier an T. Kurtz, Markov Processes: Characterization an Convergence. Wiley, New York, [3] K. Hana, Quasi-invariant measures an their characterization by conitional probabilities. Bull. Sci. Math. 15 (1), [4] K. Hana, Quasi-invariance an reversibility in the Fleming-Viot process. Probab. Theory Relat. Fiels 1 (), [5] A. Kolmogoroff, Zur Umkehrbarkeit er statistischen Naturgesetze. Math. Annalen 113 (1937), [6] B. Schmulan an W. Sun, A cocycle proof that reversible Fleming-Viot processes have uniform mutation. C.R. Math. Rep. Aca. Sci. Canaa 4 () Omar Rivasplata, Sección Matemáticas, Pontificia Universia Católica el Perú, Av. Universitaria cra 18, San Miguel, Lima 3, PERU aress: orivasplata@pucp.eu.pe Jan Rychtář, Department of Mathematical Sciences, University of North Carolina at Greensboro, Greensboro, NC 74, USA aress: rychtar@uncg.eu Byron Schmulan, Department of Mathematical an Statistical Sciences, University of Alberta, Emonton, T6G G1, CANADA aress: schmu@stat.ualberta.ca