Bakry-Emery Calculus For Diffusion With Additional Multiplicative Term
BBAKRY-EMERY CALCULUS FOR DIFFUSION WITH ADDITIONALMULTIPLICATIVE TERM
C. ROBERTO, B. ZEGARLINSKI
Abstract.
We extend the Γ calculus of Bakry and Emery to include a Carr´e du champ operator with multiplicative term, providing results which allow to analyse inhomoge-neous diffusions.
The aim of this paper is to extend Bakry-Emery approach [BE85, Bak94] to deal withsome quantities involving operators not only of order one, but also including order zero.One of the motivations for that is a possible application to analysis of hypercontractivityproperties for some classes of inhomogeneous Markov semi-groups, see e.g. [RZ].The setting is as follows: ( Q t ) t ≥ is the semi-group associated to a diffusion operator L = ∆ − ∇ U · ∇ , on R n , where the dot sign stands for the Euclidean scalar product.We assume that U : R n → R is twice differentiable and satisfies R e − U dx = 1 so that µ ( dx ) = e − U ( x ) dx is a probability measure on R n . By construction L is symmetric in L ( µ ). Following Bakry-Emery, we denote by Γ the carr´e du champs bilinear formΓ( f, g ) := 12 ( L ( f g ) − f Lg − gLf )and set Γ( f ) := Γ( f, f ). For the diffusion L considered here, we have Γ( f, g ) = ∇ f · ∇ g .The iterated operator Γ is defined asΓ ( f, g ) = 12 ( L Γ( f, g ) − Γ( Lf, g ) − Γ( f, Lg )) , and again, for simplicity, we set Γ ( f ) := Γ( f, f ). One can see that Hess( U ) ≥ ρ (asa matrix), ρ ∈ R , implies Γ ( f ) ≥ ρ Γ( f ) for all smooth enough f (see e.g. [ABC + ≥ ρ Γ (the so-called Γ -condition) is equivalent to the following commutation property between the semi-groupand the gradient operator (equivalently Γ):(1) Γ( Q t f ) ≤ e − ρt Q t (Γ( f )) , t ≥ f for which Γ( f ) is well defined, see e.g. [ABC +
00, Proposition 5.4.1].We will prove that a similar equivalence holds for an extended operator we introducenow. Let W : R n → R + be smooth enough so that in particular W belongs to the domainof L . Then, for f, g smooth enough, we setΓ W ( f, g ) := Γ( f, g ) + W f g which is therefore a positive bilinear form. The operator Γ W acts as a derivative andmultiplicatively. In fact, one can see Γ W as two-dimensional operator, call it D , that actsas Df := ( ∇ f, W f ). With this notation, Γ W ( f ) = | Df | = |∇ f | + W f is nothing but Date : February 23, 2021.
Key words and phrases. Γ calculus.Supported by the grants ANR-15-CE40-0020-03 - LSD - Large Stochastic Dynamics, ANR 11-LBX-0023-01 - Labex MME-DII and Fondation Simone et Cino del Luca in France, and the grant ... in theUK.. a r X i v : . [ m a t h . F A ] F e b C. ROBERTO, B. ZEGARLINSKI the Euclidean norm squared of the 2 dimensional vector Df . Similarly, we introduce theiterated operatorΓ W ( f, g ) := 12 (cid:16) L Γ W ( f, g ) − Γ W ( Lf, g ) − Γ W ( f, Lg ) (cid:17) = Γ ( f, g ) + 12 f gL ( W ) + W Γ( f, g ) + 2 W ∇ W ∇ ( f g )where the last equality follows from some algebra.It should be clear from the definition that we are not dealing with Γ calculus for the op-erator L W := L − W , even though by simple algebra we have (cid:16) L W ( f g ) − f L W g − gL W f (cid:17) =Γ W ( f, g ). The point is that we want to derive commutation formulas for the semi-group( Q t ) t ≥ associated to L , and not for the semi-group associated to L W . This also explainswhy Γ W is defined through the operator L and not L W .Our first main result reads as follows. Theorem 1.
Let ρ ∈ R . The following are equivalent ( i ) for all f smooth enough Γ W ( f ) ≥ ρ Γ W ( f )( ii ) for all f smooth enough and all t ≥ , Γ W ( Q t f ) ≤ e − ρt Q t (Γ W ( f ))( iii ) for all f smooth enough and all t ≥ , Q t ( f ) − ( Q t f ) + 2 Z t Q s (cid:16) W ( Q t − s f ) (cid:17) ds ≤ − e − ρt ρ Q t (Γ W ( f )) . Remark 2.
Above, when ρ = 0, the ratio − e − ρt ρ is understood as its limit (i.e. 2 t ).Notice that it is always non-negative.Observe that, applying ( ii ) to constant functions f ≡ C , C = 0, leads to W ≤ e − ρt Q t ( W ). Therefore, if R W dµ < ∞ and ρ >
0, taking the limit t → ∞ and byergodicity, we would conclude that W ≡
0. Therefore, for the inequality Γ W ( f ) ≥ ρ Γ W ( f )to hold for a non trivial W , either ρ ≤ R W dµ = ∞ . But we have no this restrictionremoving mean value µ ( f ) of the function f .Taking the mean with respect to µ in ( iii ) and passing to the limit t → ∞ , we get byinvariance and ergodicity that for ρ >
0, it holds Z f dµ − (cid:18)Z f dµ (cid:19) − Z W Z ∞ ( Q s f ) dsdµ ≤ ρ (cid:18)Z |∇ f | dµ + Z f W dµ (cid:19) . This is a sort of Poincar´e inequality in particular for a function f with mean value zero.Following Bakry-Emery (see e.g. [ABC +
00, proposition 5.5.4]), one can actually proovethat the latter holds under the weaker assumption that R Γ W ( f ) dµ ≥ ρ R Γ W ( f ) dµ . Proof.
The proof mimics the usual case ( W = 0).To prove that ( i ) implies ( ii ), fix t > s ∈ [0 , t ] → Ψ( s ) = Q s (cid:16) Γ W ( Q t − s f ) (cid:17) = Q s (Γ( Q t − s f ) + W ( Q t − s f ) ). Then, setting g := Q t − s f , itholds Ψ ( s ) = Q s (cid:16) L Γ( g ) + L ( W g ) − g, Lg ) − W gLg (cid:17) = 2 Q s (cid:16) Γ W ( g ) (cid:17) ≥ ρQ s (cid:16) Γ W ( g ) (cid:17) = 2 ρ Ψ( s )from which the result of Item ( ii ) follows. AKRY-EMERY CALCULUS FOR DIFFUSION WITH ADDITIONAL MULTIPLICATIVE TERM 3
Now we prove that ( ii ) implies ( iii ). LetΨ( s ) := Q s (( Q t − s f ) ) + 2 Z s Q u (cid:16) W ( Q t − u f ) (cid:17) du, s ∈ [0 , t ] . Then, setting again g := Q t − s f , it holdsΨ ( s ) = Q s (cid:16) L ( g ) − gLg + 2 W ( Q t − s f ) (cid:17) = 2 Q s (Γ W ( g )) . Therefore Ψ( t ) − Ψ(0) = Z t Ψ ( s ) ds = 2 Z t Q s (cid:16) Γ W ( Q t − s f ) (cid:17) ds ≤ Z t e − ρ ( t − s ) Q s ( Q t − s (Γ W ( f ))) ds = 1 − e − ρt ρ Q t (Γ W ( f )) . This corresponds to the expected result of Item ( iii ).Last we prove that ( iii ) implies ( i ). We may use the following expansions left to thereader: Q t f = f + tLf + t L ( Lf ) + o ( t )from which we deduce that Q t ( f ) − ( Q t f ) = 2 t Γ( f ) + t [ L (Γ( f )) + 2Γ( f, Lf )] + o ( t ) . On the other hand,2 Z t Q s (cid:16) W ( Q t − s f ) (cid:17) ds = 2 tW f + t [ L ( W f ) + 2 W f Lf ] + o ( t )and 1 − e − ρt ρ Q t (Γ W ( f )) = 2 t Γ W ( f ) + t [ − ρ Γ W ( f ) + 2 L (Γ W ( f ))] + o ( t ) . Plugging these expansions into ( iii ) leads precisely to ( i ). This ends the proof. (cid:3) In the next result, we give a condition for Inequality ( i ) of Theorem 1 to hold. Observefirst thatΓ W ( f ) = Γ ( f ) + 12 f L ( W ) + W Γ( f ) + 2 W ∇ W ∇ ( f )= Γ ( f ) + f [ W ∆ W + |∇ W | − W ∇ W · ∇ U ] + W |∇ f | + 4 f W ∇ W · ∇ f. Set ∂ ij for the second order derivative with respect to the variables x i and x j . SinceΓ ( f ) = P ni,j =1 ( ∂ ij f ) +( ∇ f ) T (Hess U )( ∇ f ), we observe that the condition Γ ( f ) ≥ ρ Γ( f )is satisfied as soon as Hess( U ) ≥ ρ . Theorem 3.
Assume that Γ ≥ ρ Γ for some ρ ∈ R and that γ := inf x ∈ R n : W ( x ) =0 ∆ WW − |∇ W | W − ∇ U · ∇ WW ! > −∞ . Then, we have Γ W ( f ) ≥ min ( ρ, γ )Γ W ( f ) for all f smooth enough. In the above, by convention we set inf ∅ = + ∞ . C. ROBERTO, B. ZEGARLINSKI
Example 4.
Consider U ( x ) = c + (1+ | x | ) p/ p and W ( x ) = (1+ | x | ) q/ q , x ∈ R n , p, q ≥ c so that R e − U ( x ) dx = 1. Here, as usual, | x | = ( P x i ) / is the Euclidean norm. The(spurious) form of U and W is here to guarantee smoothness (indeed it would have beeneasier to work with W ( x ) = | x | q that is, however, not smooth on the whole Euclideanspace).We observe that ∇ U ( x ) = x (1 + | x | ) ( p − / , ∇ W ( x ) = x (1 + | x | ) ( q − / and ∆ W ( x ) = n (1 + | x | ) ( q − / + ( q − | x | (1 + | x | ) ( q − / . Therefore∆ WW − |∇ W | W − ∇ U · ∇ WW = qn | x | − q | x | | x | (cid:18) q + 1)1 + | x | + (1 + | x | ) ( p − / (cid:19) is bounded below if and only if p ≤
2, in which case, Theorem 3 applies and leads to anon-trivial statement.
Proof of Theorem 3.
Form the expression of Γ W above, we infer thatΓ W ( f ) ≥ ρ |∇ f | + f [ W ∆ W + |∇ W | − W ∇ U · ∇ W ] + W |∇ f | + 4 f W ∇ W · ∇ f. Now 4 f W ∇ W · ∇ f ≥ − f |∇ W | − W |∇ f | so thatΓ W ( f ) ≥ ρ |∇ f | + W f ∆ WW − |∇ W | W − ∇ U · ∇ WW ! . The expected result follows. (cid:3)
As an immediate corollary, we get the following useful result.
Corollary 5.
Assume that Γ ≥ ρ Γ for some ρ ∈ R and that γ := inf x ∈ R n : W ( x ) =0 ∆ WW − |∇ W | W − ∇ U · ∇ WW ! > −∞ . Then, for all f smooth enough, it holds (2) Γ W ( Q t f ) ≤ e − ρ,γ ) t Q t (Γ W ( f )) , t ≥ . Next, we show that the quantity min( ρ, γ ), that appears in Theorem 3 and Corollary5, is optimal, in the sense that, for some examples of U and W , it cannot be improved.Observe first that, if W ≡
0, then γ = ∞ and therefore Γ W ≥ min ( ρ, γ )Γ W is equivalentto Γ ≥ ρ Γ which is known to be optimal (for example for the Gaussian potential U ( x ) = | x | / ρ = 1).In fact, consider the Gaussian potential U ( x ) = | x | − n log(2 π ) in R n , and W ( x ) = p | x | , x ∈ R n . One has∆ WW − |∇ W | W − ∇ U · ∇ WW = n −
21 + | x | + 3(1 + | x | ) − γ = − /
12 if n = 1 and γ = − n ≥
2. Since in thatspecific case ρ = 1, min( ρ, γ ) = γ and therefore, Corollary 5 asserts that, in dimension 2or higher, Γ W ( Q t f ) ≤ e t Q t (Γ W ( f )), for t ≥
0. We stress that this goes in the oppositedirection of (1), which, in the Gaussian setting, can be recast as |∇ Q t f | ≤ e − t Q t ( |∇ f | )with optimal decay e − t . As we may prove now, e t is also optimal.For that purpose, consider the following family of functions f a ( x ) := e a · x , a = ( a , . . . , a n ) , x = ( x , . . . , x n ) ∈ R n where as usual a · x := P i a i x i is the scalar product in R n . Optimality can be obtainedequivalently (thanks to Theorem 1) either from the bound Γ W ( Q t f ) ≤ e t Q t (Γ W ( f )) (us-ing Melher’s representation formula for the Ornstein-Uhlenbeck semi-group) or in Γ W ( f ) ≥− Γ W ( f ). We will dig on the latter by computing Γ W ( f a ) and Γ W ( f a ) for all a, x ∈ R n . AKRY-EMERY CALCULUS FOR DIFFUSION WITH ADDITIONAL MULTIPLICATIVE TERM 5
On the one hand we haveΓ W ( f a ) = |∇ f a | + W f a = (cid:16) | a | + 1 + | x | (cid:17) f a . On the other hand,Γ W ( f a ) = n X i,j =1 ( ∂ ij f a ) + ( ∇ f a ) T (Hess U )( ∇ f a ) + f a [ W ∆ W + |∇ W | − W ∇ W · ∇ U ]+ W |∇ f a | + 4 f a W ∇ W · ∇ f a = f a (cid:16) | a | + 2 | a | + n + | x | ( − | a | ) + 4 x · a (cid:17) Therefore, lim | x |→∞ Γ W ( f a )Γ W ( f a ) = − | a | . Finally, in the limit | a | →
0, we conclude that the biggest constant κ satisfying Γ W ( f ) ≥ κ Γ W ( f ) for all f must satisfy κ ≤ − κ = − √ Γ instead of Γ. Unfortunately,there is not a clean commutation result, as in the usual Bakry-Emery theory, for Γ W .However, we may prove the following proposition, that is already useful for applications. Inparticular, such a result was used by the authors to deal with hypercontractivity propertiesfor some class of inhomogeneous Markov semi-groups [RZ]. Proposition 6.
Assume the following: ( i ) there exists ρ ∈ R such that for all f smooth enough it holds Γ ( f ) ≥ ρ Γ( f ) ; ( ii ) c := max (cid:18) k|∇ W |k ∞ , sup x : W ( x ) =0 (cid:16) LWW − ρ (cid:17) − (cid:19) < ∞ .Then, for all f non-negative, it holds q Γ( P t f ) + W P t f ≤ e ( c − ρ ) t P t (cid:18)q Γ( f ) + W f (cid:19) . Proof.
Following Bakry-Emery, see [ABC +
00, proof of Proposition 5.4.5], introduce Ψ( s ) = e − ρs P s (cid:16)p Γ( P t − s f ) + W P t − s f (cid:17) , s ∈ [0 , t ], t being fixed. Therefore, setting g := P t − s f ,one hasΨ ( s ) = − ρ Ψ( s ) + e − ρs P s L q Γ( g ) + P s ( L ( W g )) − e − ρs P s Γ( g, Lg ) p Γ( g ) ! − e − ρs P s ( W Lg ) . Now L q Γ( g ) = L Γ( g )2 p Γ( g ) − Γ(Γ( g ))4Γ( g ) / Hence, after some algebra, we getΨ ( s ) = e − ρs P s (cid:18) g )(Γ ( g ) − ρ Γ( g )) − Γ(Γ( g ))Γ( g ) / (cid:19) + e − ρs P s ( L ( W g ) − W Lg − ρW g ) . Assumption ( i ) ensures that the first term of the right hand side of the latter is non-negative (see [ABC +
00, Lemma 5.4.4]). On the other hand, L ( W g ) − W Lg − ρW g = g ( LW − ρW ) + 2 ∇ W · ∇ g ≥ − c ( |∇ g | + W g ) . It follows that Ψ ( s ) ≥ − c Ψ( s ). In turn, Ψ( t ) ≥ Ψ(0) e − ct from which the desired resultfollows. (cid:3) C. ROBERTO, B. ZEGARLINSKI
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