Invariant, super and quasi-martingale functions of a Markov process - PowerPoint PPT Presentation

Invariant, super and quasi-martingale functions of a Markov process Lucian Beznea Simion Stoilow Institute of Mathematics of the Romanian Academy and University of Bucharest Based on joint works with Iulian Cîmpean and Michael Röckner Probability and Analysis May 15, 2017, Bedlewo, Poland

• E : a Lusin topological space endowed with the Borel σ -algebra B X = (Ω , F , F t , X t , P x , ζ ) be a right Markov process with state space • E , transition function : • ( P t ) t � 0 : the transition function of X , P t u ( x ) = E x ( u ( X t ); t < ζ ) , t � 0 , x ∈ E . Proposition The following assertions are equivalent for a non-negative real-valued B -measurable function u and β � 0 . (i) ( e − β t u ( X t )) t � 0 is a right continuous F t -supermartingale w.r.t. P x for all x ∈ E. (ii) The function u is β -excessive. First aim : To show that this connection can be extended between the space of differences of excessive functions on the one hand, and quasimartingales on the other hand, with concrete applications to semi-Dirichlet forms.

Supermedian and excessive functions • For β � 0, a B -measurable function f : E → [ 0 , ∞ ] is called β -supermedian if P β t f � f , t � 0; ( P β t ) t � 0 denotes the β -level of the semigroup of kernels ( P t ) t � 0 , P β t := e − β P t . • If f is β -supermedian and lim t → 0 P t f = f pointwise on E , then it is called β -excessive . • A B -measurable function f is β -excessive if and only if α U α + β f � f , α > 0, and lim α →∞ α U α f = f pointwise on E , where U = ( U α ) α> 0 is the resolvent family of the process X , � ∞ 0 e − α t P t dt . U α := • U β := the β -level of the resolvent U , U β := ( U β + α ) α> 0 ; • E ( U β ) := the convex cone of all β -excessive functions. If β = 0 we drop the index β from notations.

⇒ ( ii). If ( e − β t u ( X t )) t � 0 is a right-continuous Proof. (i) = supermartingale then by taking expectations we get that e − β t E x u ( X t ) � E x u ( X 0 ) , hence u is β -supermedian. - If u is β -supermedian then to prove that it is β -excessive reduces to prove that u is finely continuous, which in turns follows by the well known characterization for the fine continuity: u is finely continuous if and only if u ( X ) has right continuous trajectories P x -a.s. for all x ∈ E. (ii) = ⇒ (i). Since u is β -supermedian and by the Markov property we have for all 0 � s � t E x [ e − β ( t + s ) u ( X t + s ) |F s ] = e − β ( t + s ) E X s u ( X t ) = e − β ( t + s ) P t u ( X s ) � e − β s u ( X s ) , hence ( e − β t u ( X t )) t � 0 is an F t -supermartingale. The right-continuity of the trajectories follows by the fine continuity of u via the previously mentioned characterization.

I. Differences of excessive functions and quasimartingales of Markov processes Theorem. The following assertions are equivalent for a non-negative real-valued B -measurable function u. (i) u ( X ) is an F t -semimartingale w.r.t. all P x , x ∈ E. (ii) u is locally the difference of two finite 1 -excessive functions. [E. Çinlar, J. Jacod, P . Protter, M.J. Sharpe, Z. W. verw. Gebiete 1980]

Quasimartingales Let (Ω , F , F t , P ) be a filtered probability space satisfying the usual hypotheses. An F t -adapted, right-continuous integrable process ( Z t ) t � 0 is called P - quasimartingale if n � Var P ( Z ) := sup E { | E [ Z t i − Z t i − 1 |F t i − 1 ] | + | Z t n |} < ∞ , τ i = 1 where the supremum is taken over all partitions τ : 0 = t 0 � t 1 � . . . � t n < ∞ .

M. Rao’s characterization of the quasimartingales A real-valued process on a filtered probability space (Ω , F , F t , P ) satisfying the usual hypotheses is a quasimartingale if and only if it is the difference of two positive right-continuous F t -supermartingales. [P .E. Protter, Stochastic Integration and Diff. Equations. Springer 2005]

Remark. If u ( X ) is a quasimartingale, then the following two conditions for u are necessary: (i) sup P t | u | < ∞ t > 0 and (ii) u is finely continuous. The first assertion is clear since for each x ∈ E E x | u ( X t ) | � Var P x ( u ( X )) < ∞ . sup P t | u | ( x ) = sup t t The second one follows from the Blumenthal-Getoor’s characterization of the fine continuity.

For a real-valued function u , a finite partition τ of R + , τ : 0 = t 0 � t 1 � . . . � t n < ∞ , and α > 0 we set n � V α P α t i − 1 | u − P α t i − t i − 1 u | + P α τ ( u ) := t n | u | , i = 1 V α ( u ) := sup V α τ ( u ) . τ where the supremum is taken over all finite partitions of R + . Admissible sequence of partitions: an increasing sequence ( τ n ) n � 1 of finite partitions of R + such that � τ k is dense in R + and if r ∈ � τ k k � 1 k � 1 then r + τ n ⊂ � τ k for all n � 1. k � 1

Theorem Let u be a real-valued B -measurable function and β � 0 such that P t | u | < ∞ for all t. Then the following assertions are equivalent. (i) ( e − β t u ( X t )) t � 0 is a P x -quasimartingale for all x ∈ E. V β (ii) u is finely continuous and sup τ n ( u ) < ∞ on E for one (hence all) n admissible sequence of partitions ( τ n ) n . (iii) u is a difference of two real-valued β -excessive functions. [L. Beznea, I. Cîmpean, Trans. Amer. Math. Soc. 2017]

Comments about the proof • Key idea: By the Markov property one can show that Var P x (( e − α t u ( X t ) t � 0 ) = V α ( u )( x ) for all x ∈ E , meaning that assertion (i) holds if and only if V α ( u ) < ∞ . V α ( u ) is a supremum of measurable functions taken over an • uncountable set of partitions, hence it may no longer be measurable. However, the set [ V α ( u ) < ∞ ] is of interest to us, not necessarily V α ( u ) . It turns out that [ V α ( u ) < ∞ ] is measurable and, moreover, it is • V α completely determined by sup τ n ( u ) for any admissible sequence of n partitions ( τ n ) n � 1 . This aspect is crucial in order to give criteria to check the quasimartingale nature of u ( X ) .

Criteria for quasimartingale functions on L p -spaces Assume that µ is a σ -finite sub-invariant measure for ( P t ) t � 0 ; i.e., µ ◦ P t � µ for all t > 0. Proposition The following assertions are equivalent for a B -measurable function � L p ( µ ) and β � 0 . u ∈ 1 � p � ∞ u of u such that ( e − β t � (i) There exists a µ -version � u ( X t )) t � 0 is a P x -quasimartingale for x ∈ E µ -a.e. (ii) For an admissible sequence of partitions of ( τ n ) n � 1 of R + , V β sup τ n ( u ) < ∞ µ -a.e. n (iii) There exist u 1 , u 2 ∈ E ( U β ) finite µ -a.e. such that u = u 1 − u 2 µ -a.e. Remark. If u is finely continuous and one of the above equivalent assertions is satisfied then all of the statements hold quasi everywhere, not only µ -a.e., since an µ -negligible finely open set is µ -polar. If in addition µ is a reference measure then the assertions hold everywhere on E .

The generator on L p -spaces Since µ is sub-invariant, ( P t ) t � 0 and U extend to strongly continuous semigroup resp. resolvent family of contractions on L p ( µ ) , 1 � p < ∞ . The corresponding generator ( L p , D ( L p ) ⊂ L p ( µ )) is defined as • D ( L p ) = { U α f : f ∈ L p ( m ) } , for all f ∈ L p ( µ ) , 1 � p < ∞ , L p ( U α f ) := α U α f − f with the remark that this definition is independent of α > 0. The analogous notations for the dual structure are � • P t and ( � L p , D ( � L p )) , and note that the adjoint of L p is � L p ∗ ; 1 p + 1 p ∗ = 1. We focus our attention on a class of β -quasimartingale functions which arises as a natural extension of D ( L p ) .

• Any function u ∈ D ( L p ) , 1 � p < ∞ , has a representation u = U β f = U β ( f + ) − U β ( f − ) with U β ( f ± ) ∈ E ( U β ) ∩ L p ( µ ) , hence u has a β -quasimartingale version for all β > 0; moreover, � � � t � � � P t u − u � p = 0 P s L p uds p � t � L p u � p . � � The converse is also true, namely if 1 < p < ∞ , u ∈ L p ( µ ) , and • � P t u − u � p � const · t , t � 0, then u ∈ D ( L p ) . But this is no longer the case if p = 1 (because of the lack of reflexivity of L 1 ), i.e. � P t u − u � 1 � const · t does not imply u ∈ D ( L 1 ) . However, it turns out that this last condition on L 1 ( m ) is yet enough to ensure that u is a β -quasimartingale function. Proposition Let 1 � p < ∞ and suppose A ⊂ { u ∈ L p ∗ + ( m ) : � u � p ∗ � 1 } , � P s A ⊂ A for all s � 0 , and E = � supp ( f ) m-a.e. If u ∈ L p ( m ) satisfies � f ∈A sup E | P t u − u | fdm � const · t for all t � 0 , f ∈A u of u such that ( e − β t � then there exists an m-version � u ( X t )) t � 0 is a P x -quasimartingale for all x ∈ E m-a.e. and every β > 0 .