EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES

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1 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES VLADIMIR I. BOGACHEV, GIUSEPPE DA PRATO, AND MICHAEL RÖCKNER Abstract. Let A = (a ij ) be a Borel mapping on [, 1 with values in the space of nonnegative operators on an let b = (b i ) be a Borel mapping on [, 1 with values in. Let Lu(t, x) = t u(t, x) + a ij (t, x) xi xj u(t, x) + b i (t, x) xi u(t, x), u C ((, 1) ). Uner broa assumptions on A an b, we construct a family µ = (µ t ) t [,1) of probability measures µ t on which solves the Cauchy problem L µ = with initial conition µ = ν, where ν is a probability measure on, in the following weak sense: 1 Lu(t, x) µ t (x) t =, u C ((, 1) ), an lim t ζ(x) µ t (x) = ζ(x) ν(x), ζ C ( ). Such an equation is satisfie by transition probabilities of a iffusion process associate with A an b provie such a process exists. However, we o not assume the existence of a process an allow quite singular coefficients, in particular, b may be locally unboune or A may be egenerate. An infinite imensional analogue is iscusse as well. Main methos are L p -analysis with respect to suitably chosen measures an reuction to the elliptic case (stuie previously) by piecewise constant approximations in time. AMS Subject Classification: 35K1, 35K12, 6J35, 6J6, 47D7 1. Introuction Accoring to Kolmogorov s classical result, the transition probabilities of the iffusion process in governe by the stochastic ifferential equation ξ t = σ(t, ξ t )W t + b(t, ξ t )t with time-epenent coefficients satisfy the parabolic equation L p =, where A = (a ij ), A = σ 2 /2, an L is the formal ajoint to the operator u(t, x) Lu(t, x) := + a ij (t, x) xi xj u(t, x) + b i (t, x) xi u(t, x). t In the case of uniformly boune coefficients an classical solutions, the above mentione parabolic equation for probability measures was investigate in the important paper by Il in an Hasminskii [8, where the main objective was to stuy the behaviour of solutions an their stabilization as t. In recent years, there has been a growing interest in such equations for measures (efine in the weak sense as explaine below) in the situation, where it is not assume in avance that there exists a corresponing iffusion an if a ij an b i are not regular. In [5, families of measures satisfying equations of the above type have been stuie in connection with flows of probability measures. An approach to constructing generalize iffusions via solutions to weak parabolic equations has been recently evelope by W. Stannat [15. However, in this paper, we o not aress probabilistic issues (which 1

2 2 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER will be the subject of another work) an consier only analytic problems, although these problems still keep a certain probabilistic flavour (in particular, we are intereste in solutions that are probability measures). If the elliptic part H of L is time inepenent an there exists a probability measure µ on such that H µ = in the sense that Hϕ µ = ϕ C ( ), then, uner broa assumptions (involving the existence of a suitable Lyapunov function), there exists a strongly continuous Markov semigroup (T t ) t on L 1 (µ) (actually, on all L p (µ), p [1, )) whose generator coincies with H on C ( ) an which serves as the transition semigroup of a iffusion process (see [14). As shown in [1, Section 4, the ajoint semigroup (Tt ) t can be extene even to boune measures on an, for any probability measure ν on, the family (Tt ν) t consists of probability measures with ensities p(t, x) an the measure µ = Tt ν t on [, 1) satisfies the weak parabolic equation L µ =. In this work, we investigate the existence problem in the time-epenent case. The problem is as follows. Let A(t, x) = (a ij (t, x)) 1 i,j be a Borel mapping on [, 1 with values in the space of nonnegative operators on an let b(t, x) = (b i (t, x)) be a Borel mapping on [, 1 with values in. Let u(t, x) Lu(t, x) := + a ij (t, x) xi xj u(t, x) + b i (t, x) xi u(t, x) (1.1) t for u C ((, 1) ). The elliptic part of L is enote by Hu(t, x) := a ij (t, x) xi xj u(t, x) + b i (t, x) xi u(t, x). (1.2) We shall say that a family of Raon measures µ = (µ t ) t [,1) on parabolic equation satisfies the weak L µ = (1.3) if the functions a ij an b i are integrable on every compact set in (, 1) with respect to the measure µ t t an, for every u C ((, 1) ), one has 1 Lu(t, x)µ t (x) t =. (1.4) We shall say that µ satisfies the initial conition µ := ν at t = if ν is a measure on an lim ζ(x) µ t (x) = ζ(x) ν(x) (1.5) t for all ζ C ( ). In this case we write µ = (µ t ) t [,1). Our principal goal is to construct solutions to such equations so that µ t is a probability measure for every t [, 1) (in finite an infinite imensions) uner the assumption that there exists a suitable Lyapunov function. The corresponing results generalize those obtaine in [2 in the elliptic case (however, the latter are use here). It turns out that the existence of solutions can be prove uner quite general assumptions, in particular, without any regularity of a ij an b i. In contrast to the case of classical weak solutions, such solutions may be extremely irregular. For example, if = 1, b =, a(t, x) = α(x) is an arbitrary positive measurable function such that 1/α is integrable, then the family of measures µ t α 1 x solves the corresponing problem. Our approach to constructing solutions is base on establishing certain a priori estimates of two types: global, which involve suitable Lyapunov functions an yiel uniform tightness, an local, which yiel uniform bouneness

3 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 3 of L p -norms on compact sets an enable us to verify (1.3) for measures constructe by means of tightness. In the case when A an b are inepenent of t an satisfy the hypotheses of Theorem 3.1 below, our solution coincies with the aforementione family (Tt ν) t [,1). In general, uner the hypotheses of Theorem 3.1, all possible solutions possess certain local ifferentiability properties (in particular, the ensity p t of µ t belongs to the Sobolev class H p,1 loc ( )), which enables one to write equation (1.3) in the classical weak form 1 [ u t p t a ij xi p t xj u p t xi a ij xj u + p t b i xi u x t =. However, typically the functions p t are neither boune nor square integrable over with respect to Lebesgue measure, i.e., o not belong to the functional classes consiere, e.g., in [7, [11, [12, [13. If the coefficients a ij are not ifferentiable (as in the corollaries to the main theorem), then the functions p t may not be ifferentiable as well (in the case of egenerate A they even nee not exist). As shown in [1, for nonegerate A, the ifferentiability properties of p t are essentially the same as those of a ij an simple examples show that they may be not better. The main novelty of the metho in this paper is first to use extensively L p -analysis with respect to suitably chosen measures (ifferent from Lebesgue measure) which are eeply relate to the coefficients of the given operator. Secon, a piecewise constant approximation enables us to use our previously obtaine results in the elliptic case. 2. Auxiliary results Let us recall some stanar notation for various Sobolev classes on or on open sets U. The class H p,1 (U) consists of all functions f L p (U) with generalize partial erivatives xi f L p (U). This space is equippe with its natural Sobolev norm f p,1. Let H p,1 (U) be the closure in H p,1 (U) of C (U) (the class of infinitely ifferentiable functions with compact support in U). For a function u on (, 1), we set t u(t, x) := u(t, x)/ t. Let B be an open ball in. Set S := ([, 1) B) ({} B). The space p,1 ([, 1, B) consists of all measurable functions w on [, 1 B such that, for each t [, 1), one has w(t, ) H p,1 (B) an ( 1 ) 1/p w p,1 ([,1,B) := w(t, ) p p,1 t <. p,1 The space ([, 1, B) is the subspace in p,1 ([, 1, B) forme by the functions with w(t, ) H p,1 (B) for all t [, 1). For convenience of notation all functions efine on [, 1) will be consiere also as functions on [, 1 Lemma 2.1. If µ = (µ t ) t [,1) satisfies (1.3) an (1.5), then for almost all t [, 1) one has t ζ(x) µ t (x) Lζ(s, x) µ s (x) s = ζ(x) ν(x) for every ζ C ( ). (2.1) If, for each ζ C ( ), the function ζ(x) µ t (x) is continuous on [, 1), then (2.1) hols for all t [, 1) an is equivalent to (1.3) an (1.5). Proof. Let u(t, x) = ϕ(t)ζ(x), where ϕ C (, 1) an ζ C ( ), an let f(t) := ζ(x) µ t (x), h(t) := Lζ(t, x) µ t (x)..

4 4 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER Then we erive from (1.3), where we use the ientity ζ/ t =, that f = h in the sense of istributions on (, 1). Taking into account (1.5) we arrive at (2.1), where, however, the full measure set in t may epen on ζ. Hence such a set can be chosen common for any given countable set of functions ζ j. This enables one to fin a common full measure set in (, 1) for all ζ C ( ), since there exists a countable family of functions ζ j C ( ) such that every function ζ C ( ) can be approximate uniformly with the first an secon erivatives by a sequence of functions in {ζ j } with support in a common ball. Certainly, if ζ(x) µt (x) is continuous in t, then we obtain the above equality for all t. The converse is also clear. It is worth noting that the reason why we require that all measures µ t (an not just almost all) in the next lemma be probabilities is that this is the case when one eals with transition probabilities. From the analytical point of view, this is not essential, of course. Lemma 2.2. Let µ = (µ t ) t [,1) be a family of probability measures on satisfying (1.3) an (1.5), where ν is a probability measure on. Suppose that there exists a nonnegative function Ψ C 2 ( ) such that Ψ L 1 (ν), Ψ(x) = +, an lim LΨ(t, x) C µ t t-a.e. (2.2) with some constant C. Then Ψ(x) µ t (x) Ct + Ψ(x) ν(x) (2.3) for a.e. t [, 1). If the functions t ζ(x) µ t (x), ζ C ( ), are continuous on [, 1), then (2.3) is true for all t [, 1). Proof. It is clear that (2.1) remains true also for ζ Cb ( ) such that ζ(x) = q = const outsie some ball. Inee, the function ζ := ζ(x) q is of compact support, Lζ = Lζ an q µ t (x) = q ν(x) = q. Furthermore, ue to the local integrability of the functions a ij an b i with respect to µ t t, (2.1) is clearly still true for ζ Cb 2( ) (in place of C ) such that ζ(x) = q = const outsie some ball. Now let us fix k an take a function θ k C 2 ( ) such that θ k (r) = r if r k, θ k (r) = k + 1 if r k + 2, θ k (r) 1, an θ k (r). By our assumption on Ψ, θ k Ψ is constant outsie a sufficiently large ball. Hence, as explaine above, (2.1) is true with ζ(x) = ζ k (x) := θ k (Ψ(x)). We observe that Lζ k = θ k (Ψ)LΨ + θ k (Ψ) A Ψ, Ψ θ k (Ψ)LΨ. Therefore, for a.e. t [, 1) t θ k Ψ µ t = θ k Ψ ν + Lζ k µ s s t θ k Ψ ν + (θ k Ψ)LΨ µ s s Ψ ν + Ct. By Fatou s lemma we arrive at the estimate Ψ µ t Ψ ν + Ct.

5 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 5 Corollary 2.3. Let (µ t ) t [,1) be as in Lemma 2.2 an let Ψ(x) := x 2. Suppose that LΨ(t, x) C an trace A(t, x) C + C Ψ(x) µ t t-a.e. for some nonnegative constants C, C an C, an that ν has all moments. Then, for every m, there exists C m, which epens only on m, C, an x 2m ν(x), such that for a.e. t [, 1). If the functions t [, 1), then (2.4) is true for all t [, 1). x 2m µ t (t) C m (2.4) ζ(x) µ t (x), ζ C ( ), are continuous on Proof. If m = 1 we can take C 1 := C + x 2 ν(x) accoring to Lemma 2.2. Suppose that our claim is true for some m. Let us consier the function Ψ m+1 (x) := x 2m+2. Then we obtain where LΨ m+1 (t, x) = 4m(m + 1)Ψ m 1 (x) A(t, x)x, x + 2(m + 1)Ψ m (x)lψ(t, x) κ 1 Ψ m + κ 2 Ψ m+1, κ 1 := 2C(m + 1) + 4C m(m + 1), κ 2 := 4C m(m + 1). Now the same reasoning as in the proof of the above lemma yiels the estimate t θ k Ψ m+1 µ t Ψ m+1 ν + θ k (Ψ m+1) ( ) κ 1 Ψ m + κ 2 Ψ m+1 µs s t Ψ m+1 ν + κ 1 C m + κ 2 θ k(ψ m+1 )Ψ m+1 µ s s, where the last step follows by the hypothesis of inuction. Let f k (t) := θ k Ψ m+1 µ t. We observe that θ k k+2 (t)t θ k k(t). Inee, if t < k, then θ k (t) = t an θ k (t) = 1, if t > k + 2, then θ k (t) =, an if k t k + 2, then θ k(t) k an θ k (t) 1. Hence θ k (Ψ m+1)ψ m+1 k+2 θ k k(ψ m+1 ) an the above estimate yiels that f k (t) C m + κ 2 k + 2 k t f k (s) s, where C m is some constant. By Gronwall s inequality we obtain ( f k (t) C m k + 2 ) exp κ 2 k t. Letting k, we arrive at the esire conclusion by Fatou s lemma. Corollary 2.4. Let µ = (µ t ) t [,1) be a family of probability measures on satisfying (1.3) an (1.5), where ν is a probability measure on. Let Ψ C 2 ( ) be a nonnegative function such that lim t t-a.e., where C is a constant. Then one can fin a nonnegative function Ψ C 2 ( ) such that lim Ψ (x) = +, LΨ C µ t t-a.e., an Ψ L 1 (ν).

6 6 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER Moreover, if is a uniformly tight family of probability measures on an for every ν there exists a solution µ ν = (µ ν t ) t [,1) of the problem (1.3), (1.5) with initial conition ν, then one can fin a function Ψ as above such that { } { } sup Ψ µ ν t, ν, t [, 1) sup C + Ψ ν, ν <. Proof. Inee, one can fin a function θ C 2 ( ) nonnegative on + such that Ψ := θ Ψ L 1 (ν), lim θ(r) = +, r + θ 1, an θ. Then LΨ θ ΨLΨ C. Moreover, for any uniformly tight family of probability measures, such a function θ can be foun common for all ν with sup Ψ ν <. ν Remark 2.5. It is obvious from the proof that the conition LΨ C in Lemma 2.2 can be relaxe as follows: there exists a measurable set E such that LΨ(t, x) C µ t t-a.e. 1 on (, 1) ( \E) an C := LΨ(t, x) µ t (x) t <. Then in the left-han sie of E (2.3) one shoul a C. It is also worth mentioning that if in the situation of Lemma 2.2 the functions a ij are boune on boune subsets of [, 1 an (2.2) hols only for x outsie some boune set, then one can fin another nonnegative function Ψ C 2 ( ) such that Ψ coincies with Ψ outsie some boune set an LΨ (t, x) C µ t t-a.e., where C is a positive constant. Inee, let θ C ( 1 ) be such that θ(s) = if s 1, θ(s) = s if s 1, θ 1. There exists k such that LΨ C µ t t-a.e. if x k. Let M := sup θ (s) sup A(t, x) Ψ(x), Ψ(x) an take Ψ (x) := θ(ψ(x) k). Then s t [,1, x k+2 LΨ (t, x) = θ (Ψ(x) k 1)LΨ(t, x) + θ (Ψ(x) k 1) A(t, x) Ψ(x), Ψ(x) C + M, since θ (Ψ(x) k 1) = if x k, LΨ(x) C if x k, an θ (Ψ(x) k 1) = if x k + 2. Let us introuce the following conitions on A, b, p [1, + ), an an open ball B (C1) there exist two constants M 1 = M 1 (B) an M 2 (B) such that for all i, j one has et A(t, x) M 1 an sup a ij (t, ) H p,1 (B) M 2. (t,x) [,1 B t [,1 (C2) there exists M 3 = M 3 (B) such that for all i one has sup b i (t, ) L p (B) M 3. t [,1 It follows from (C1) an the Sobolev embeing theorem that if p >, then every functions a ij has a jointly measurable version such that all functions x a ij (t, x), t (, 1), are Höler continuous of orer 1 /p an boune on B uniformly in t (their Höler an sup-norms on B are estimate by a constant epening on p,, B an M 2 ). Below we use the same notation a ij for these particular versions. We nee a technical result on weak solutions of the equation t w a ij xi xj w = xi h i (2.5) with zero bounary conition on S := ([, 1) B) ({} B), where h i L p ((, 1) B). A function w on [, 1) is sai to be a solution of (2.5) with zero bounary conition if p,1 (i) w is continuous an belongs to the space ([, 1, B), :

7 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 7 (ii) for every function ϕ C ( 1 ) with support in [ 1, 1) B, one has 1 [ 1 w t ϕ + a ij xi w xj ϕ + ϕ xi a ij xj w x t = h i xi ϕ x t. (2.6) Note that (2.6) implies also that, for every ξ C (B), one has w(, x)ξ(x) x =, which expresses the zero bounary value at t =, whereas the zero bounary value on [, 1 B is taken care of by the conition w(t, ) H p,1 (B). Inee, let [ f(t) := w(t, x)ξ(x) x, g(t) := a ij xi w xj ξ + ξ xi a ij xj w + h i xi ξ x. Then, for every ψ C ( 1 ) with support in [ 1, 1), we have 1 ψ (t)f(t) t = 1 ψ(t)g(t) t. Hence f = g in the sense of istributions. Therefore, f is absolutely continuous an there is a constant C such that f(t) = C t g(s) s, which yiels C =. Let p > + 2. Suppose that there exist two constants M 1 an M 2 (which epen only on A an B) such that et A(t, x) M 1 an sup a ij (t, ) H p,1 (B) M 2. Then, it follows from t the proof of [1, Lemma 3.5 that there exist R > an M > (which epen only on B,, p, M 1, an M 2 ) such that, for every open ball B R of raius R in B an any solution w of equation (2.5) with zero bounary conition on ([, 1) B R ) ({} B R ) one has w p,1 ([,1,B R ) M (h i ) L p ((,1) B R ). (2.7) In a similar manner, for b i L p ([, 1) B), we say that a continuous function w on [, 1) satisfies the equation t w a ij xi xj w b i xi w =, w(, x) = ζ(x), (2.8) on [, 1), where b i L p ([, 1) ) an ζ C ( ), if, for every ball B, w belongs to the space p,1 ([, 1, B), an, for every function ϕ C ( 1 ) with support in [ 1, 1), one has 1 [ w t ϕ + a ij xi w xj ϕ + ϕ xi a ij xj w ϕb i xi w x t = ζ(x)ϕ(, x) x (2.9) One can verify that (2.9) inee implies that, for every ξ C ( sup t [,1) ), one has w(, x) = ζ(x). Lemma 2.6. Suppose w is a continuous function on [, 1) satisfying (2.8), where ζ C ( ), a ij (t, ) H p,1 loc ( ), b i (t, ) L p loc ( ), t [, 1), p > + 2, an for every centere ball B R of raius R in, there exists a constant C(R) such that [ a ij (t, ) H p,1 (B R ) + b i (t, ) L p (B R ) + inf et A 1 (t, x) C(R). x B R Suppose that sup w 1. Then, for all R (, 1), x, y B R/2 an all t, s [, R, one has w(t, x) w(s, y) M(R, C(R), T,, p, ζ) [ t s γ + x y γ, where M(R, C(R),, p, ζ) an γ = γ(, p) epen on the inicate objects only.

8 8 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER Proof. Let us fix a function θ C ( ) such that θ 1, θ BR/2 = 1, supp θ B R. The function v(t, x) := θ(x)w(t, x) θ(x)ζ(x) satisfies the equation where for H as in (1.2) one has h := whθ + 2a ij xi w xj θ H(θζ) t v(t, x) a ij (t, x) xi xj v(t, x) b i xi v(t, x) = h, = whθ H(θζ) 2w xi ( a ij xj θ ) + 2 xi ( wa ij xj θ ), with zero bounary conition on ([, 1 B R ) ({} B R ). We shall prove that there exist functions f i L p ([, 1 B R ) such that an xi f i = h (f i ) L p ([,1 B R ) K(R, C(R),, p, ζ, θ), where K(R, C(R),, p, ζ, θ) epens only on the inicate objects. Inee, for every fixe t, let g(t, ) be the solution of the Dirichlet problem g = whθ H(θζ) 2w xi a ij xj θ 2wa ij xi xj θ, x B R, with zero bounary conition on B R. Then we can set f i (t, x) := xi g(t, x) + 2w(t, x)a ij (t, x) xj θ(x) an employ the fact that the L p -norm of the graient of the solution of the above Dirichlet problem is estimate via the L p -norm of the right-han sie, which is majorize by [ a ij (t, ) L p (B R ) sup xi xj θ(x) + xi xj (θζ)(x) + 2 xi a ij (t, ) L p (B R ) sup xi θ(x) x B R x B R [ + b i (t, ) L p (B R ) sup xi θ(x) + xi (θζ)(x). x B R Therefore, accoring to (2.7) with h i := b i + f i, we majorize w p,1 ([,1,B R/2 ) by a number, epening only on R, C(R),, p, ζ an θ (but θ is fixe for each R). This yiels (as in [1, Theorem 3.8 by an embeing theorem from [1), the uniform bouneness of the Höler norm of a certain orer γ (epening only on an p) of the function w on [, R B R. We shall say that a function V on is compact if lim V (x) = +. We recall that x + if A an b o not epen on t satisfy (C1) an (C2) with p > for every ball an if there is a compact function V such that lim HV (x) = (such a function is calle a Lyapunov function), then there exists a unique probability measure µ on satisfying the equation H µ = (see [2, [3). In aition, µ = ϱ x, where ϱ H p,1 loc ( ) is strictly positive an locally Höler continuous. Finally, as shown by W. Stannat [14, there exists a unique strongly continuous semigroup (T µ t ) t on L 1 (µ) whose generator extens H. This semigroup is Markovian an µ is its invariant measure an also a unique solution of H µ = (see [3). Moreover, if p > + 2, there is a semigroup of probability kernels K t (x, y) = ϱ t (x, y) y such that for every f L 1 (µ) the function K µ t f(x) := f(y)ϱ t (x, y) y is a version of T µ t f an (t, x) K µ t f(x) is locally Höler continuous on (, ). In particular, the semigroup (K µ t ) t of kernels is strong Feller. The measure µ is a unique

9 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 9 invariant probability for the semigroup (K t ) t. The proofs of these statements are given in [1. Lemma 2.7. Let H = a ij xi xj + b i xi, where a ij an b i are inepenent of t an satisfy conitions (C1) an (C2) with p > + 2 for every ball. Assume that there exists a compact function V C 2 ( ) such that lim HV (x) =. Let µ be a probability measure satisfying the equation H µ = an let (T µ t ) t, (K µ t ) t be the corresponing semigroups introuce above. Then there exists a sequence of locally boune measurable mappings b l : such that, letting H l := a ij xi xj + b i l x i, one has (i) b l b in measure µ an b l (x) b(x), (ii) lim lv (x) = uniformly in l, (iii) there exist unique probability measures µ l satisfying Hl µ l = such that µ l = ϱ l x, the functions ϱ l are locally uniformly Höler continuous an converge locally uniformly to ( the ensity ϱ of µ, an, for the corresponing Markovian strongly continuous semigroups T (t, l) )t given by strong Feller kernels ( K(t, l) ) µ, one has T (t, l)f T t t f in L 1 (µ) for all boune measurable functions f. In aition, if f C ( ), then K(t, l)f(x) K µ t f(x) for all x. Proof. Let us fix l an consier the close set Z := { } x: V (x) =. The function α(x) := a ij (x) xi xj V (x) is continuous an HV (x) = α(x) if x Z. Let us consier the sets S m := {m x < m + 1}, m =, 1,.... One can pick N l,m so large that the set {x S m : b(x) > N l,m } has Lebesgue measure less than 2 l m, There exists δ m > such that if x, x S m an x x < δ m, then α(x) α(x ) } < 1. We shall take δ m = δ m (l) so small that the set D m := {x S m : < ist (x, Z) δ m has Lebesgue measure less than 2 l m. Now, for every x D m we set b l (x) :=. Then we have H l V (x) sup HV (y) + 1 y S m for all x D m. If x S m Z an b(x) N l,m, we set b l (x) := b(x) an if b(x) > N l,m, we set b l (x) :=. Next we consier x S m \(D m Z). Let M l,m := N l,m + ( inf { V (y) : y S m \(D m Z) }) 1 ( m + sup y S m α(y) Suppose that b(x), V (x) 1. If b(x) M l,m, then we set b l (x) := b(x). If b(x) > M l,m, then we set b l (x) := M l,m V (x)/ V (x). We observe that in the latter case, { } H l V (x) = α(x) M l,m V (x) sup α(y) M l,m inf V (y) : y S m \(D m Z) y S m m. Finally, if b(x), V (x) > 1, then b l (x) is efine to be b(x) or epening on whether b(x) N l,m or b(x) > N l,m. It follows from the above efinitions that b l is locally boune, b l (x) = b(x) outsie a set of Lebesgue measure less than 2 l an that at every point x S m one has H l V (x) sup HV (y) + 1 or H l V (x) m. Therefore, we have (i) an (ii). y S m As explaine above, there exist unique probability measures µ l such that Hl µ l = an that there exist unique Markovian strongly continuous semigroups ( T (t, l) ) on t L1 (µ l ) whose ( generators ) exten the operators H l an which are given by strongly Feller kernels K(t, l) such that µ t l is invariant for ( K(t, l) ). In aition, accoring to [2, the measures µ l are uniformly tight an possess strictly positive locally uniformly Höler continuous t ensities ϱ l. Hence every subsequence in {µ l } has a weak cluster point µ. Obviously, µ has ).

10 1 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER a ensity with respect to Lebesgue measure, hence it is reaily verifie that H µ =. Since µ is a unique probability measure satisfying this equation, we obtain µ = µ. Consequently, {µ l } converges weakly to µ. Together with the locally uniform Höler continuity of the functions ϱ l this yiels that these functions converge locally uniformly to the ensity ϱ of µ. So, in particular, lim µ l µ =, where enotes the variation norm. l Let ϕ C ( ) an let u l (t, x, ϕ) := K(t, l)ϕ(x). Since b l (x) b(x) an b l b in measure, one can reaily euce from the results in [1 (see [1, Corollary 3.11 an Proposition 4.4) that there is a subsequence {l k } such that the functions u lk converge locally uniformly in [, + ) to a boune continuous function u(t, x, ϕ). Let us observe that, given any countable collection {ϕ n }, one can choose such a subsequence common for all ϕ n. In orer to simply the notation we shall enote this subsequence again by {l} (we shall see below that the assertion is inee true for the whole sequence). We can pick the countable family {ϕ n } such that, for every boune measurable function f, there exists a sequence {ϕ nk } in {ϕ n } convergent to f (Lebesgue) a.e. such that ϕ nk (x) sup f + 1. Let f be a boune Borel measurable function. We claim that T t f := lim l T (t, l)f (2.1) exists in L 1 (µ) an efines a Markovian strongly continuous semigroup of contractions on every L p (µ), p [1, ). To prove the claim we may assume that f 1. We note that for all t, since the measures µ an µ l are equivalent, there is no ambiguity in consiering T (t, l)f as an element of L 1 (µ). Let us fix t > an ε >. There exists l such that µ µ l < ε for all l l. One can fin ϕ n such that ϕ n (x) 2 an f ϕ n L 1 (µ) < ε. Next we choose l 1 > l such that T (t, l)ϕ n T (t, k)ϕ n L 1 (µ) < ε for all l, k l 1, which is possible, since T (t, l)ϕ n u(t,, ϕ n ) pointwise, hence also in L 1 (µ) by the ominate convergence theorem. We observe that for every boune measurable function g, one has g µ g µ l + µ µ l sup g(x), g µ l g µ + µ µ l sup g(x). (2.11) Therefore, taking into account that T (t, l) is contractive on L 1 (µ l ) an T (t, l)f 1, T (t, l)ϕ n 2 for all l, we obtain for all l, k l 1 that T (t, l)f T (t, k)f L 1 (µ) T (t, l)f T (t, l)ϕ n L 1 (µ) + T (t, l)ϕ n T (t, k)ϕ n L 1 (µ) + T (t, k)ϕ n T (t, k)f L 1 (µ) T (t, l)f T (t, l)ϕ n L 1 (µ l ) + 3ε + ε + T (t, k)ϕ n T (t, k)f L 1 (µ k ) + 3ε f ϕ n L 1 (µ l ) + f ϕ n L 1 (µ k ) + 7ε 2 f ϕ n L 1 (µ) + 13ε 15ε. So, T t f := lim T (t, l)f exists in L 1 (µ). Clearly, T t is Markovian. Furthermore, using (2.11) l again we obtain T t f µ = f µ, since µ l is invariant for T (t, l). In particular, T t extens to a contraction on every L p (µ), p [1, + ). A similar reasoning shows that (T t ) t is a semigroup on L 1 (µ). Inee, for all t, s, T t+s f is the limit of T (t + s, l)f = T (t, l)t (s, l)f in L 1 (µ) an T t T s f is the limit of

11 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 11 T (t, l)t s f. Let ε > an let us take l such that T (s, l)f T s f L 1 (µ) < ε an µ µ l < ε for all l l. Then T (t, l)t (s, l)f T (t, l)t s f L 1 (µ) T (t, l)t (s, l)f T (t, l)t s f L 1 (µ l ) + 2ε T (s, l)f T s f L 1 (µ l ) + 2ε T (s, l)f T s f L 1 (µ) + 4ε < 5ε. Since t T t ϕ n is continuous from [, + ) to L p (µ) for all n, {ϕ n } is ense in L p (µ) an each T t is a contraction on L p (µ), we obtain the strong continuity of (T t ) t. The claim is prove. To complete the proof of this lemma it remains to show that T µ t f = T t f t >, f L 1 (µ). (2.12) Inee, (2.12) implies that T t is inepenent of the subsequence chosen in its efinition, so T (t, l)f T µ t f in L 1 (µ) for all boune measurable functions f. Furthermore, for all ϕ C ( ) an t >, by continuity K µ t ϕ(x) = u(t, x, ϕ) for all x, so the last part of assertion (iii) follows from (2.12). Let ( Ĥ, D(Ĥ)) be the generator of (T t ) t on L 1 (µ). By virtue of the uniqueness result mentione before this lemma, it suffices to show that for any ϕ C ( ), one has ϕ D(Ĥ) an Ĥϕ = Hϕ, which in turn follows from the ientity t T t ϕ g µ = ϕg µ + T s (Hϕ) g µ s, t >, g C ( ). (2.13) To prove (2.13) we nee some preparations. Let l. Consier the operator ( H l, C ( ) ) on L 2 (µ l ) efine by H lϕ := a ij xi xj ϕ + (2 xi a ij + 2a ij xi ϱ l /ϱ l b i l) xj ϕ. Then (H l ) µ l = an by [14 the closure of this operator generates a Markovian strongly continuous semigroup of contractions T (t, l) on L 1 (µ l ) such that for all boune measurable f an all g one has T (t, l)f g µ l = ft (t, l)g µ l. By [3, Remark 2.14 the functions xi ϱ l /ϱ l have uniformly boune local L p -norms. Therefore, by the same arguments as above, T t f := lim T (t, l)f exists in L 1 (µ) for all boune l measurable functions f (for some subsequence of {l k } again enote by {l}). By (2.11) we obtain for all boune measurable f an g T t f g µ = ft t g µ. (2.14) Clearly, (2.14) extens to all g L 1 (µ). Now we can prove (2.13). Let ϕ, g C ( T t ϕ g µ = lim T (t, l)ϕ g µ = lim T (t, l)ϕ g µ l l l ( t ) = lim ϕg µ l + T (t, l) T (s, l)h l ϕ g µ l s = l t ϕg µ + lim T (t, l) l T (s, l)hϕ g µ l s, ). Then

12 12 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER since lim H l ϕ H l ϕ L 2 (µ l ) ue to the uniform bouneness of L 2 (U)-norms of ϱ l on every l ball U. It remains to note that on the support of ϕ the ensities ϱ l converge uniformly to ϱ (which is strictly positive continuous), which yiels lim l T (s, l)hϕ g µ l = lim Hϕ T (s, l) g µ l = Hϕ T s g µ = T s Hϕ g µ. l In aition, these integrals are uniformly boune in s. Hence we arrive at (2.13). Remark 2.8. We observe that the existence of a Lyapunov function V was only use in orer to obtain the following three properties: (a) the very existence of the measures µ l an µ satisfying Hl µ l =, H µ = an generating the semigroups ( T (t, l) ) µ an (T t t ) t, for which they are invariant; (b) the convergence of µ l to µ in the variation norm; (c) the uniqueness of a probability measure satisfying H µ = an the uniqueness of the associate Markovian semigroup. Therefore, the same assertion is true if we require (a), (b) an (c) in place of the existence of V, assuming that b l b in measure µ an b l (x) b(x) + 1 (an keeping our usual local assumptions on a ij an b i ). Lemma 2.9. Let p > +2. Let us assume that there exist points = τ < τ 1 < < τ n < 1 such that A(t, x) = A(τ k, x) an b(t, x) = b(τ k, x) if t [τ k, τ k+1 ). Suppose also that, for every fixe ball U, one has inf et A(τ k, x) >, a ij (τ k, ) H p,1 (U), b i (τ k, ) L p (U) x U for each k. Finally, assume that there exist nonnegative compact functions V l C 2 ( ) such that one has [ lim a ij (τ k, x) xi xj V k (x) + b i (τ k, x) xi V k (x) =. (2.15) Then, for every probability measure ν, there exists a family µ = (µ t ) t [,1) of probability measures on satisfying (1.3) an (1.5) such that t ζ µ t is continuous on [, 1) for all ζ C ( ). Proof. Suppose first that the mappings b(τ k, ) are locally boune. As explaine in the proof of Lemma 2.7, for every k there exist a probability measure µ k on an a strongly continuous Markovian semigroup ( T (t, k) ) on t L1 (µ k ) such that its generator extens the operator H k := a ij (τ k, x) xi xj + b i (τ k, x) xi an µ k is invariant for T (t, k). Let K(t, k) enote the corresponing Markovian kernels. Let us set µ t := K(t τ k, k) K(τ k, k 1) K(τ 1, ) ν if t [τ k, τ k+1 ). We shall verify that (µ t ) t [,1) satisfies (1.3) an (1.5). Inee, (1.5) is true accoring to [1, Proposition 4.4. In orer to verify (1.3), it suffices to consier u of the form ϕ(t)u(x) with ϕ C (, 1) an u C ( ). For each k we have τk+1 [ ϕ (t)u(x) + ϕ(t)h k u(x) K(t τ k, k) µ τk (x) t τ k = ϕ(τ k+1 ) u(x) µ τk+1 (x) ϕ(τ k ) u(x) µ τk (x). Here we use that t T (t τ k, k)u = T (t τ k, k)h k u, where t is taken in L1 (µ τk ). Inee, this hols by the following reasoning: we know that this is true if t is taken in L 1 (µ k ),

13 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 13 hence it can be taken in µ k -measure (which is equivalent to Lebesgue measure). Since µ τk is absolutely continuous (which is prove in [1) an since H k u is a boune function (this is the only place where we use the local bouneness of b(τ k, )), it follows by the ominate convergence theorem that can be taken in L 1 (µ t τk ). Taking into account the equality ϕ() = ϕ(1) =, we arrive at (1.3). Let us consier the general case. Accoring to Lemma 2.7, for each k =,..., n, there exists a sequence of locally boune mappings b l (τ k, ) such that b l (τ k, x) b(τ k, x) + 1 an H k,l V (x) = uniformly in l, where Let lim H k,l ϕ(x) := a ij (τ k, x) xi xj ϕ(x) + b i l(τ k, x) xi ϕ(x). H l (t, x) := a ij (τ k, x) xi xj ϕ(x) + b i l (τ k, x) xi ϕ(x) if t [τ k, τ k+1 ), k =,..., n. As shown above, for each b l there exists a solution (µ l t ) t [,1) of the problem (1.3) an (1.5) corresponing to H l. The semigroups corresponing to the rifts b l (τ k, x) (use above to construct solutions for H l ) will be enote by T (t, k, l) an K(t, k, l), respectively. As shown in [1, the measures µ l t have ensities ϱ(l, t) such that, for every interval [c, (, 1) an every close ball B, the functions Φ l : (t, x) ϱ(l, t)(x) are Höler continuous on [c, B uniformly in l. Therefore, passing to a subsequence, we may assume that the sequence {Φ l } converges locally uniformly on (, 1) to a continuous function Φ. It is reaily seen that the family of measures µ t := Φ(t, x) x satisfies (1.3). We observe that the sequence of measures µ l t is uniformly tight for each t [, 1). Inee, by Lemma 2.2, the family of measures µ l t, t [, τ 1, l, is uniformly tight. It follows by Corollary 2.4 that the family of measures µ l t, t [τ 1, τ 2, l, is uniformly tight as well. Repeately applying Corollary 2.4 we obtain the uniform tightness of the measures µ l t, t [, 1), l. Together with the convergence of ensities this yiels that the measures µ t are probabilities. It remains to verify (1.5). Let ζ C ( ). It suffices to show that the functions t g l (t) := ζ µ l t are uniformly continuous on [, 1) an are equal to ζ ν at t =. We have for t [, τ 1 an u(t,, l) := K(t,, l)ζ that g l (t) = u(t, x, l) ν(x). (2.16) So, g l () = ζ ν for all l. We claim that u(t,, l), t [, τ 1, satisfies (2.9). Suppose the claim is prove. Then by Lemma 2.6 an our assumptions on b l, the functions s u(t, x, l) are Höler continuous on [, τ 1 uniformly in l, x B, t [, τ 1, whence it follows that the functions g l are equicontinuous. Inee, given ε >, one can fin a ball B such that ν(b) > 1 ε/4. There exists δ = δ(ε) > such that u(t, x, l) u(t, x, l) < ε/4 for all x B an all l if t t < δ. Hence g l (t) g l (t ) ε 2 + 2ν( \B) < ε for all l if t t < δ. In particular, we obtain that lim l g l (t) = ζ ν uniformly in l, which yiels (1.5). It remains to prove the claim. To this en we first note that by exactly the same arguments as use above we see that t u(t,, l) = H,lu(t,, l) = T (t, l)h,l ζ

14 14 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER with taken in t L1 loc (x) (instea of L1 (µ )). Furthermore, u(t,, l) ζ an u(t,, l) H p,1 loc ( ) by [1, Corollary Let ϕ C (, τ 1). Then we have for all u(l) := u(,, l) integrating by parts 1 [ u(l) t ϕ + a ij xi u(l) xj ϕ + ϕ xi a ij xj u(l) ϕb i l x i u(l) x t = = 1 1 (u(l)ϕ) x t t ζ(x)ϕ(, x) x. This completes the proof. H,l u(l) ϕ x t + 3. Main results 1 u(l) ϕ x t t Theorem 3.1. Let p > + 2 an let A an b satisfy (C1) an (C2) for every ball. Assume that for a.e. t (, 1), there exist a nonnegative compact function V t C 2 ( ) such that one has [ lim a ij (t, x) xi xj V t (x) + b i (t, x) xi V t (x) =. (3.1) Finally, assume that there exists a nonnegative compact function Ψ C 2 ( C such that LΨ C a.e. in (, 1) ) an a constant. (3.2) Then, for every probability measure ν, there exists a family µ = (µ t ) t [,1) of probability measures on satisfying (1.3) an (1.5) such that t ζ µ t is continuous on [, 1) for every ζ C ( ). Proof. We may assume that (3.1) hols for all t (, 1), since for those t which o not satisfy this conition, we may reefine A an b by setting A(t, x) = A(t, x), b(t, x) = b(t, x), where t is any fixe point for which that conition is satisfie. Clearly, this oes not affect other hypotheses an (1.3) is not sensitive to such reefinitions of the coefficients. Our strategy of proof is as follows. We have alreay prove the result in the case when A(t, x) an b(t, x) are piece-wise constant in t. Now we consier a sequence of solutions µ k = (µ k t ) t [,1) corresponing to A k (t, x) an b k (t, x) which coincie with A(t k,j, x) an b(t k,j, x) if t [t k,j, t k,j+1 ), where = t k, < t k,1 <... < t k,nk = 1 is a suitably chosen partition of (, 1 into N k intervals. We shall verify that the measures µ k t, t [, 1), k, are uniformly tight on. Finally, the esire solution µ will be constructe as a weak cluster point of {µ k }. In orer to show that µ satisfies our equation, certain uniform a priori estimates for {µ k } will be establishe. First of all, let us choose a sequence of ecreasing partitions of [, 1) as follows (in the case when A an b are continuous in t, one can take t k,j = j2 k, N k = 2 k, which oes not work in general as we shall see). We recall a well-known result of Lebesgue accoring to which, for any integrable function f on [, 1, one can fin a ecreasing sequence of partitions [, t k,1 ), [t k,1, t k,2 ),...,[t k,nk, 1) (i.e., every partition contains all partition points of the previous one) such that the Riemann sums N k f(t k,j 1 )(t k,j t k,j 1 ), where t k, =, converge to the integral of f. We shall call such a sequence of partitions a Riemannian sequence of partitions. Moreover, given a countable family of integrable functions f n, one can choose common points t k,j for all f n. The simplest way to prouce such partitions is j=1

15 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 15 to employ the fact establishe by Jessen [9 that, for any function f that has perio 1 an is integrable on [, 1, the Riemann sums R k (f, s) := 2 k 2k f(s + j2 k ) converge to the integral of f over [, 1 for almost all s. We shall apply Lebesgue s result to the following countable collection of functions. Let {ζ n } C (, 1) an {ϕ m } C ( ) be two sequences of functions with the following property: for every function u that is continuous an has compact support in (, 1), there exists a sequence h j of finite linear combinations of the functions ζ n ϕ m with rational coefficients such that the functions h j vanish outsie some compact set containing the support of u an converge to u uniformly. Let us consier functions ζ n (t)α i,j,m (t) an ζ n (t)β i,j,m (t), where α i,j,m (t) = ϕ m (x)a ij (t, x) x, β i,m (t) = ϕ m (x)b i (t, x) x. Let us choose for the obtaine countable set of functions ζ n α i,j,m, ζ n β i,j,m a common Riemannian sequence of ecreasing partitions forme by the points t k,j, j =,..., N k. Let us set A k = (a ik k ), b k = (b i k ), where a ij k (t, x) = aij (t k,l 1, x), b i k (t, x) = bi (t k,l 1, x) if t [t k,l 1, t k,l ). We observe that the following is true. Let u C ((, 1) ), let J be a close interval in (, 1), let K be a close ball in such that supp u J K, an let {ϱ n } be a sequence of continuous functions that converges to a function ϱ uniformly on J K. Then, for all i, j one has 1 1 lim a ij k (t, x)u(t, x)ϱ k(t, x) x t = a ij (t, x)u(t, x)ϱ(t, x) x t, (3.3) k j=1 1 lim k b i k (t, x)u(t, x)ϱ k(t, x) x t = 1 b i (t, x)u(t, x)ϱ(t, x) x t. (3.4) Let us verify (3.3). Consiering the functions uϱ n an uϱ we arrive at the case where u = 1 an the functions ϱ n vanish outsie J K. It follows from their uniform convergence that it suffices to show that 1 lim k a ij k (t, x)ϱ(t, x) x t = 1 a ij (t, x)ϱ(t, x) x t. In turn, ue to our choice of {ζ n } an {ϕ m }, it is enough to consier ϱ(t, x) = ζ n (t)ϕ m (x), i.e., to show that lim k N k l=1 Accoring to our construction, we have N k lim k l=1 tk,l 1 α i,j,m (t k,l 1 ) ζ n (t) t = ζ n (t)α i,j,m (t) t. t k,l 1 ζ n (t k,l )α i,j,m (t k,l 1 )(t k,l t k,l 1 ) = 1 ζ n (t)α i,j,m (t) t.

16 16 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER Now it suffices to note that, letting δ k = max l N k t k,l t k,l 1, one has N k tk,l α i,j,m (t k,l 1 ) ζ n (t) t t k,l 1 l=1 sup s (,1) N k l=1 α i,j,m (s) sup t,s: t s δ k ζ n (t) ζ n (s), ζ n (t k,l 1 )α i,j,m (t k,l )(t k,l t k,l 1 ) which tens to as k by the uniform continuity of ζ n. The same reasoning proves (3.4). We now turn to the main statement. Accoring to Lemma 2.9, for the k-th partition, we have a solution µ k = (µ k t ) t [,1) of our problem corresponing to the operator H k given by a ij k (t, x) = a(t k,j, x), b i k (t, x) = b(t k,j, x) if t [t k,j, t k,j+1 ). We have µ k t = ϱ k (t, x) x, where the functions ϱ k (t, x) are jointly continuous on (, 1). Accoring to Corollary 2.4, there is a nonnegative function Ψ C 2 ( ) such that Ψ(x) + as x +, LΨ C an Ψ L 1 (ν). By Lemma 2.2, the family of measures µ k t, t [, 1), k, is uniformly tight. By the same reasoning as in Lemma 2.9, passing to a subsequence, we may assume that the functions ϱ k (t, x) converge uniformly on compact subsets in (, 1) to a function ϱ(t, x). Therefore, we obtain probability measures µ t := ϱ(t, x) x, which satisfy (1.3). The last step is to show that µ := (µ t ) t [,1) satisfies (1.5). It suffices to show that, for every ζ C ( ), the functions g k (t) = ζ(x) µ k t (x) are equicontinuous on [, 1/2. However, this is clear from the analogous step in the proof of Lemma 2.9, since every g k is the limit of the functions g k,l corresponing to the approximations constructe in the lemma cite (see (2.16)), an those functions are equicontinuous also with respect to k, which follows from the proof of the equicontinuity of the functions in (2.16). Remark 3.2. (i) It is clear from the proof of the above theorem that in the case when the functions b i are boune on boune subsets of (, 1), the nonegeneracy conition on A can be slightly relaxe as follows: it suffices to have inf et A(t, x) > (t,x) [τ 1,τ 2 K for every [τ 1, τ 2 (, 1) an every compact set K. (ii) It follows from Remark 2.5 an the above proof that conition (3.2) can be relaxe as follows: there exists a compact set K such that LΨ(t, x) C a.e. in (, 1) ( \K). (iii) It is also clear that the solution constructe above has the following property: for a.e. t, the measure µ t has a ensity from the Sobolev class H p,1 loc ( ). As shown in [1, this is true for any solution of (1.3) uner our local assumptions on A an b. Hence, uner these assumptions, equation (1.3) can be written in the classical weak form after integrating by parts in the term with xi xj u. Below we consier more general equations whose solutions o not have such a property. Corollary 3.3. Suppose that et A(t, x) is uniformly boune on [, 1) compact set K an every [τ 1, τ 2 (, 1), one has inf (t,x) [τ 1,τ 2 K et A(t, x) >, sup (t,x) [,1) K b(t, x) <. an, for every

17 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 17 Assume also that lim b(t, x), x = for a.e. t (, 1) an sup b(t, x), x = M <. (3.5) (t,x) [,1) Then, for every probability measure ν on, there exists a family (µ t ) t [,1) of probability measures on satisfying (1.3) an (1.5) such that t ζ µ t is continuous on [, 1) for every ζ C ( ). If, in aition, the functions b i an a ij are continuous in x for a.e. fixe t, then the same is true without the assumption that et A is strictly positive. Proof. As in the theorem, we may assume that the first relation in (3.5) hols for each t. We shall fin a sequence of smooth mappings A j (t, x) such that sup j an, for every compact set K U sup A j (t, x) S + 1 < (3.6) (t,x) [,1) inf j an every [τ 1, τ 2 (, 1), one has inf et A j(t, x) >. (t,x) [τ 1,τ 2 K Then we shall verify that the sequence of the corresponing solutions (µ j t) t [,1) has a subsequence which is weakly convergent for each t an that its weak limit is the esire solution. The approximations A j can be given by A j θ(jx) + j 1 I, where θ is a smooth probability ensity on with support in the unit ball U. Then (3.6) is clear an, for each fixe j, the functions xm a ik j are uniformly boune on boune subsets in (, 1) an et A j j. In aition, given [τ 1, τ 2 (, 1) an k >, for every unit vector e an all (t, x) [τ 1, τ 2 ku, one has A j (t, x)e, e = j 1 + A(t, x y/j)e, e θ(y) y inf A(t, z)e, e (t,z) [τ 1,τ 2 (k+1)u inf (t,z) [τ 1,τ 2 (k+1)u ( et A(t, z)) 1/ >. (3.7) Let us enote by L j the parabolic operator obtaine from L by replacing A with A j. Let us set V t (x) = x, x. Then, since sup j,t,x A j (t, x) S + 1, we obtain for every fixe t that lim L jv t =. Let us fix a probability measure ν on. Letting Ψ (x) = x, x, one has LΨ 2S + 2M. Therefore, by Corollary 2.4, we can replace Ψ with a nonnegative compact function Ψ C 2 ( ) such that LΨ 2S + 2M an Ψ L 1 (ν). In aition, Ψ constructe in that corollary has the form Ψ = θ(ψ ) with θ 1 an θ. Hence L j Ψ(t, x) = 2θ ( x 2 )trace A j (t, x) + 4θ ( x 2 ) A j (t, x)x, x + 2θ ( x 2 ) b(t, x), x 2S + 2M. Accoring to the above theorem, for each j, there exists a family of probability measures (µ j t) t [,1) on satisfying (1.3) an (1.5) for the operator L j. In aition, µ j t = f j (t, x)x for t >. It follows by Lemma 2.2 that, for every t [, 1), the sequence of measures µ j t is uniformly tight. Let ϕ C ( ) be fixe. Since the functions a ik j, x i xk ϕ, an ϕ(x), b(t, x) are uniformly boune, we conclue that the functions [ v j (t) := a ik j xi xk ϕ + b i xi ϕ µ j t

18 18 V.I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER are uniformly boune in t [, 1) an j. Therefore, the functions t ϕ µ j t, whose generalize erivatives are the functions v j, are uniformly Lipschitzian. Hence, given a sequence of functions ϕ l C ( ), we can choose a subsequence j n such that all the sequences ϕ l µ jn t, l, will be convergent uniformly on [, 1). Together with the uniform tightness of {µ j t} for each fixe t, this yiels a subsequence such that, for every t [, 1), the measures µ jn t converge weakly to some probability measures µ t. Clearly, for every ϕ C ( ), the function ϕ µ t is Lipschitzian. Hence (1.5) is satisfie. Let us show that for almost every t the measure µ t is absolutely continuous an that (1.3) is satisfie. To this en, it suffices to show that, given a close ball B in an a close interval I (, 1), the functions f j (t, x) have uniformly boune norms in L r (I B), where r = ( + 1) = 1 + 1/. Inee, then, by Fatou s theorem, for a.e. fixe t, one has lim inf f jn (t, ) L r (B) <, whence µ t = f(t, x)x with f(t, ) L r (B) for such t. In aition, the sequence f jn contains a subsequence which converges weakly in L r (B), hence it converges to f (by the weak convergence of measures), which easily yiels (1.3). Now, the last step is to show inee the uniform bouneness of f j in L r (I B). Let us take a ball B with the same center as B an the raius by 1 bigger than that of B. Let us also take a close interval I (, 1) strictly containing I. Set Ω = I B, Ω = I B. We observe that for every smooth function ϕ with support in the interior of Ω, one has 1 [ t ϕ + a ik j x i xk ϕ µ j t C sup x ϕ, where C = sup Ω b(t, x). It is easily seen from the proof of Theorem 3.1 in [1 (see also [1, Corollary 3.2) an estimate (3.7) that there is a constant κ = κ(c,, Ω, Ω, inf et A) such Ω that f j L r (Ω) κ. Finally, note that in the case when the coefficients are continuous in x for a.e. t the reasoning is similar an even simpler, since there is no nee to have local L p -estimates. A more general result is vali if A an b are continuous in x. Corollary 3.4. Suppose that the functions x a ij (t, x) an x b i (t, x) are continuous for each t [, 1) an are boune on boune sets in [, 1). In aition, suppose that, for every fixe ball U, the functions x a ij (t, x), t [, 1), are equicontinuous on U. Assume further that for each t [, 1), there exist nonnegative compact functions V t C 2 ( ) such that [ a ij (t, x) xi xj V t (x) + b i (t, x) xi V t (x) =. lim Finally, assume that there exists a nonnegative compact function Ψ C 2 ( ) an a constant C such that LΨ C. Then, for every probability measure ν, there exists a family µ = (µ t ) t [,1) of probability measures on satisfying (1.3) an (1.5) such that t ζ µ t is continuous on [, 1) for every ζ C ( ). Moreover, if et A is separate from zero on compact subsets in (, 1) continuity of b in x is not neee., then the Proof. The reasoning is similar to the previous corollary. First of all, accoring to Remark 2.5 one can choose a nonnegative compact function Ψ C 2 ( ) such that LΨ C an Ψ L 1 (ν). Next, by using that the functions x a ij (t, x), t (, 1), are equicontinuous on

19 EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 19 every fixe ball U, i.e., have a common (for all t (, 1)) moulus of continuity, one can fin such approximations A j = (a ij n ) of A that the functions aij [ n (t, x) are smooth in x, inf (t,x) (,1 U et A j > for every ball U in, lim a ik j x i xk V t (x)+b i (t, x) xi V t (x) = [ for each t (, 1), an a ik j x i xk Ψ(x) + b i (t, x) xi Ψ(x) C + 1 for all j. Then the rest of the proof is the same as in the previous corollary. Remark 3.5. (i) All the above results remain vali, of course, for any time interval [, T ) in place of [, 1) provie that the corresponing hypotheses are vali for that interval. It is also worth noting that in place of we coul consier a general complete Riemannian manifol M. All local statements, of course, are vali automatically, an the techniques base on Lyapunov functions are justifie in a similar manner (see [2 an [4, where analogous ieas are applie to elliptic equations). In particular, construction of suitable Lyapunov on manifols is iscusse in [4. (ii) The problem of uniqueness of our solutions is open. We recall that even in the timeinepenent case with A = I an smooth b, there might be no uniqueness if there is no Lyapunov function (see [3). Another interesting open problem concerns the behaviour of the ensities p t (x) of the measures µ t as t. For example, if the initial value µ = ν has a continuous ensity an b is locally boune, then one can show that p t (x) is jointly continuous up to t =. We o not know whether this is true for general b. In a forthcoming paper we shall obtain some infinite imensional extensions of the above results. Let us give a sample result. Let X be a separable Hilbert space with inner prouct x, y. Let {e n } be an orthonormal basis in X. Set x i = x, e i. Let FC ((, 1) X, {e n }) enote the space of all functions of the form u(t, x) = u (t, x 1,..., x n ) with u C ((, 1) n ). The class of all functions u(x) = u (x 1,..., x n ) with u C ( n ) is enote by FC (X, {e n}). Suppose that we have a mapping b: [, 1 X X an a mapping A on [, 1 X with values in the space of nonnegative symmetric trace class operators on X such that b, e i an Ae i, e j are Borel measurable. We shall say that a family of Borel probability measures µ t, t [, 1), solves the problem L µ =, µ = ν, (3.8) where ν is a given Borel probability measure on X if, for every u FC ((, 1) X, {e n}) an every v FC (X, {e n}), one has 1 [ t u(t, x) + A(x)e i, e j ei ej u(t, x) X i,j + b(t, x), x u(t, x) µ t (x) t =, (3.9) lim v(x) µ t (x) = v(x) ν(x), (3.1) t X X where we assume also the existence of all integrals. A typical existence result which can be prove by our methos is the following. Theorem 3.6. Assume that, for some constants C, C 1,n, C 2,n, m n, one has for all t an n lim b(t, x), x =, b(t, x), x C, b(t, x), e n C 1,n + C 2,n x mn, trace A(t, x) C.