NONLINEAR MARKOV SEMIGROUPS AND INTERACTING LÉVY TYPE PROCESSES

Transcription

1 NONLINEAR MARKOV SEMIGROUPS AND INTERACTING LÉVY TYPE PROCESSES Vassili N. Kolokoltsov November 21, 2006 To appear in Journal of Statistical Physics. Abstract Semigroups of positivity preserving linear operators on measures of a measurable space X escribe the evolutions of probability istributions of Markov processes on X. Their ual semigroups of positivity preserving linear operators on the space of measurable boune functions B(X) on X escribe the evolutions of averages over the trajectories of these Markov processes. In this paper we introuce an stuy the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Eucliean space R (or more generally, a manifol) showing how these semigroups arise from the general kinetic equations of statistical mechanics an evolutionary biology that escribe the ynamic law of large numbers for Markov moels of interacting particles. Well poseness results for these equations are given together with applications to interacting particles: ynamic law of large numbers an central limit theorem, the latter being new alreay for the stanar coagulation-fragmentation moels. Key wors. Positivity preserving measure-value evolutions, conitionally positive operators, Markov moels of interacting particles, ynamic law of large numbers, normal fluctuations, rate of convergence, kinetic equations, interacting stable jump-iffusions, Lévy type processes, coagulation-fragmentation. Running Hea: Nonlinear Markov semigroups. 1 Introuction 1.1 Aims of the paper An important class of Markov semigroups is given by the so calle Feller semigroups, i.e. the semigroups of strongly continuous operators T t, t 0, on C (X) (the space Department of Statistics, University of Warwick, Coventry CV4 7AL UK. v.kolokoltsov@warwick.ac.uk 1

2 of continuous functions on a locally compact topological space X vanishing at infinity) such that 0 u 1 implies 0 T t u 1 for all t 0. In a wier sense, by Feller semigroups one unerstans the semigroups of positivity preserving linear operators in the space of boune continuous functions C(X), our basic reference for Feller semigroups being [28]. It is easy to see that if the evolution on C (X) given by the equation f = Af with a linear (possibly unboune) operator A in C (X) preserves positivity, then A is conitionally positive, i.e. (Ag)(x) 0 whenever a function g belongs to the omain of A, is non-negative an vanishes at x. If X is the Eucliean space R or a manifol, the Courrège theorem [12] states that a linear operator A in C (X) whose omain contains the space Cc 2 (X) of two times continuously ifferentiable functions with a compact support is conitionally positive if an only if it has the Levy-Khintchine form with variable coefficients, i.e. if Ag(x) =a(x)g(x) + (b(x), )g(x) + 1 (c(x), )g(x) 2 + (g(x + y) g(x) χ(y)(y, )g(x))ν(x; y), (1) where c(x) is a positive efinite -matrix, b(x) is a -vector, χ(y) is some boune non-negative function with a compact support that equals one in a neighborhoo of the origin, an ν(x, ) is a Lévy measure, i.e. a positive Borel measure such that min(1, y 2 )ν(x; y) <, ν(x)({0}) = 0. (2) For what follows the main role will belong to the ual formulation of Feller semigroups. Namely, a Feller semigroup T t on C (X) clearly gives rise to a ual positivity preserving semigroup Tt on the space M(X) of boune Borel measures on X through the uality ientity (T t f, µ) = (f, Tt µ), where the pairing (f, µ) is given by the integration, of course. If A is the generator of T t, then µ t = Tt µ can be characterize by the equation in the weak form t (g, µ t) = (Ag, µ t ) = (g, A µ t ), (3) where A is the ajoint to A, that hols for all g from the omain of A. Here we shall eal with nonlinear analogs of (3): t (g, µ t) = Ω(µ t )g, (4) where Ω is a nonlinear transformation from a ense omain of M(X) to the space of linear functionals on C(X). Of special interest are the equations of the form t (g, µ t) = (A(µ t )g, µ t ) = (g, A (µ t )µ t ), (5) where A(µ) is a nonlinear mapping from a ense omain of M(X) to (possibly unboune) linear operators in C (X), an even more specifically the equations with A in (5) epening polynomially on µ, i.e. the equations of the form t (g, µ t) = K k=1 (A k g)(x 1,..., x k )µ t (x 1 ) µ t (x k ), (6) 2

3 where each A k is a (possibly unboune) operator from a ense subspace D of C (X) to the space C sym (X k ) of symmetric continuous boune functions of k variables from X. Our aim is two fols: 1) pure analytic problem (Sections 2-4) - to introuce an appropriate analog of the notion of conitional positivity for the generators Ω an A in (4), (5) that is characteristic for positivity preserving measure-value evolutions, to give an explicit general structure of conitionally positive generators (a nonlinear analog of the Courrège theorem) an to provie basic well poseness results for the corresponing evolution equations (5), (6); 2) application (Sections 5-7) - to evelop analytic tools for proving the law of large numbers an the central limit theorems together with precise rates of convergence for a wie class of interacting Markov processes with pseuo-ifferential generators, i.e. for interacting Lévy type processes. 1.2 Motivation: Markov moels of interacting particles A system of mean fiel interacting particles (or Markov processes) with an unerlying motion escribe by generator (1) is a system of N particles in X moving accoring to a generator of type (1), where the coefficients a j, b j, c j, γ j specifying the motion of every j -th particle epen not only on the position x j of this particle, but also on the point measure µ = h(δ x1 + + δ xn ) on X (mean fiel) specifie by the position of all other particles x 1,..., x N of the system (h being a positive scaling parameter, e.g. h = 1/N), i.e. a j (x j ) = a(x j, µ), b j (x j ) = b(x j, µ), c j (x j ) = c(x j, µ), ν j (x j ) = ν(x j, µ). It is well known (an can be easily seen) that passing to the mean fiel or McKean - Vlasov limit in this system (a scaling limit with the number of particles N tening to infinity in such a way that the scale measures h(δ x1 + + δ xn ) ten to some finite measures) leas formally to a measure-value ynamics on X escribe by equation (5) with A(µ)g(x) = a(x, µ)g(x) + (b(x, µ), )g(x) + 1 (c(x, µ), )g(x) 2 + (g(x + y) g(x) χ(y)(y, )g(x))ν(x, µ; y) (7) that specifies the eterministic ynamic law of large numbers for mean fiel interacting particles, see a nice informal iscussion in [16], where also other popular types of interaction geometry (e.g. -imensional lattice an the hierarchical interactions) are touche upon. Another special class of interactions, namely k-ary group interaction of arbitrary orer k (possibly non-binary, an possibly non preserving the total number of particles), was aresse in [7] (eveloping further some ieas from [6]), where it was shown that if the k-ary interactions of inistinguishable particles that preserve the number of particles are escribe by generators (conitionally positive operators) B k in C (X k ), k = 1,..., K, an the k-ary interactions that change the number of particles (from k to m) are given by symmetrical transitional kernels P k (x 1,..., x k ; y 1 y m ), the formal measure-value 3

4 limit of the large number of particles is given by the equation t (g, µ t) = + m K k=1 1 k! [(B k g + )(x 1,..., x k ) (g + (y 1,..., y m ) g + (x 1,..., x k ))P k (x 1,..., x k ; y 1 y m )]µ t (x 1 ) µ t (x k ), where g + (x 1,..., x l ) = g(x 1 ) + + g(x l ) (the weak form (8) being introuce in [39], see also Sections 5 an 7 for etails). Notice that though the original interacting particle systems leaing to (5), (7) (mean fiel interaction preserving the number of particles) an to (8) (incluing k-ary jumps with possible fragmentation or coagulation) are quite ifferent, equation (8) can be written in form (5). Moreover, it is easy to unerstan that equation (8) escribing the limit of systems with interactions not preserving the number of particles, can be also written in form (6). In fact, if all m k, then (ue to the symmetry of transition kernels) g + (y 1,..., y m )P k (x 1,..., x k ; y 1 y m ) m = k g+ (y 1,..., y k )P k (x 1,..., x k ; y 1 y m ) = m g + (y 1,..., y k ) P k m (x 1,..., x k ; y 1 y k ) with P m k (x 1,..., x k ; y 1 y k ) = 1 k an introucing the operators X m k P k (x 1,..., x k ; y 1 y m ) B k f(x 1,..., x k ) = B k f(x 1,..., x k ) + [m(f(y 1,..., y k ) kf(x 1,..., x k )] P k m (x 1,..., x k ; y 1 y k )) (9) m (8) one can rewrite (8) as t (g, µ t) = K k=1 1 k! (B k g + )(x 1,..., x k )µ t (x 1 ) µ t (x k ) (10) with conitionally positive operators B k in C (X k ). By similar manipulations (that we omit) one can eal with the case m < k. Equation (10) has the form (6) with A k g = 1 k! B kg +. (11) Characterizing positivity preserving evolutions in case of X being a Eucliean space (or a manifol) we shall firstly specify the class of polynomial equations (6) that can be presente in form (10) with conitionally positive operators B k, then give a structure 4

5 of arbitrary equations (6) preserving positivity, an at last show (uner mil technical assumptions) that a nonlinear operator A in positivity preserving equations (5) has to be of form (7). It will turn out, in particular, that in case K = 2 the positivity preserving equations (6) can be always written in form (10) with conitionally positive operators B k (which is not the case neither for a iscrete X, nor for the Eucliean X an K > 2). 1.3 Content of the paper an bibliographical comments At the en of this introuction we are going first to illustrate the ifference between positivity preserving evolutions (6) an kinetic equations (10) (in the terminology of the next section this means the ifference between conitionally positive an strongly conitionally positive operators) on a simple case of a iscrete state space X. At last we shall fix some basic notations to be use in the paper without further reminer. In Section 2 our main structural results on the positivity preserving generators A, A k in (5), (6) are obtaine (Theorems ) for the case of X being a Eucliean space (or similarly a manifol). Some examples are given. At the en of Section 2 we iscuss shortly the lifting of the nonlinear evolutions (6) to linear evolutions on the multi-particle state space. As an important previous contribution to the theory of positivity preserving equations of type (4) one shoul mention book [54], where these equations were analyze by introucing a natural infinite-imensional smooth manifol structure on the space of measures an its tangent spaces an where one can fin a result, which is essentially equivalent to our Theorem 2.4. As in the case of linear generators, one thing is to give a structure of formal generators, an another problem is to istinguish those of these formal generators that actually generate semigroups. Though lots of work in this irection is one both for linear an nonlinear cases, many questions remain open. Recent achievements in the linear theory are nicely presente in [28]. There exists an extensive literature on the well poseness for equation (5), (7) with ifferential generators A that appear in the analysis of interacting iffusions (secon orer operators A), see [49], [55], [56], [24], [59] an references therein, an of the eterministic processes (first orer operators A), see e.g. [10], [19]. Integral operators A, A k in (6), (7) stan for interacting pure jump processes an their stuy was motivate by the Boltzmann moel of collisions an the Smoluchovski moel of coagulation an fragmentation, see e.g. [44], [52], [46], [38], [39], [42] an references therein. Some particular combinations of ifferential an integral generators are mostly centere aroun spatially nontrivial versions of Boltzmann an Smoluchovski moels, see [25], [27], [50],[35] an references therein, see also [23] for moels with spontaneous births. In recent years one observes a new wave of interest to the Lévy processes (see e.g. books [3], [4]) an their spatially nontrivial extensions - Lévy type processes, the latter being basically just the Markov processes with pseuo-ifferential generators of Lévy- Khinchin type, see [28] for a comprehensive analytic stuy of these processes, [29] for a probabilistic analysis, where these processes are calle jump-iffusions, an also [53] for some connections between these approaches. Lévy an Lévy type processes enjoy quite ifferent properties (e.g. fat tails an iscontinuous trajectories) as usual Wiener process an relate iffusions, which make Lévy processes inispensable for applications in many situations where iffusion processes turn out to be un-aequate. On the other han, the analysis of these processes is surely more complicate than that of usual iffusions, 5

6 ue in particularly to technically more involve analysis of the corresponing stochastic ifferential equations (see e.g. [3] an [5] for the latter). Roughly speaking, the results of our Section 2 state that positivity preserving measure value evolutions uner mil technical assumptions have generators of non-linear spatially nontrivial Lévy-Khinchin type thus representing nonlinear analogs of Lévy type processes an semigroups. In Sections 3 we start our analysis of general positivity preserving evolutions by establishing (using a fixe point argument that is well known in nonlinear analysis, see e.g. [48] for its application to nonlinear Schroinger equation an [55] for stochastic Ito s equations escribing the evolution (5) with ifferential generators) a general well poseness result (Theorem 3.1) allowing to reuce the stuy of nonlinear problems (5)-(7) to some regularity properties of the corresponing linear problems. This result can be applie to a variety of moels with pseuo-ifferential generators (beyon stanar iffusions), where the corresponing regularity results for linear generators are available, for instance in case of ecomposable generators (see [37]), regular enough egenerate iffusions like interacting curvilinear Ornstein-Uhlenbeck processes or stochastic geoesic flows on the cotangent bunles to Riemannian manifols (see [1], [33]), or stable jump iffusions. As the latter are of special interest among Lévy type processes, ue to their applications in a wie variety of moels from plasma physics to finances (see e.g. [3], [8], [4], [58]), we shall concentrate further on stable type processes ening Section 3 with a simple illustration of Theorem 3.1 in the case of non-egenerate secon orer part of a pseuo-ifferential generator. In Section 4 we introuce our basic moel of interacting stable jump-iffusions, prove a well poseness result (reucing exposition for simplicity to the case where the stable part is not affecte by interaction) an analyze the core of the generator of the corresponing (linear) Feller process in C(M + (X)). In Sections 5 an 6 the general analytic tools for proving the law of large numbers an central limit theorems respectively are evelope for interacting Markov processes with pseuo-ifferential generators. These tools are illustrate on our basic example - interacting stable jump-iffusions, but they seem to be of interest even for usual interacting iffusions. Unlike a probabilistic approach (compactness in Skoroho space of trajectories) normally use for obtaining these type of results (see e.g. [24], [27], [52], [38] an references therein for the law of large numbers, [13], [17], [57], [55], [56], [22] an references therein for the central limit of interacting iffusions an [18], [20], [23], [50] for the central limit to some moes with jumps), our metho, being base exclusively on the theory of contraction semigroups, yiels new insights in the analytic aspects of the processes uner consieration (Feller property, structure of the cores of the generators, precise rate of convergence, etc). Moreover, instea of escribing the limiting Gaussian process of fluctuations in the very large space S (X), as is usual in the literature, our metho allows to o it in a natural smaller space(c 1 (X)) that is just a bit larger, than the initial space M(X) (which itself is usually unappropriate for escribing the fluctuations). Another specific technical feature is the systematic use of the erivatives of the solutions of the kinetic equations with respect to the initial ata (in case of the Botzmann equation these erivatives being stuie in etail in [40]). Section 7 is evote to the interaction changing the number of particles, whose law of large number is escribe by equation (8). As we note above this equation can be also written in form (10) with conitionally positive operators B k in C(X k ), which allows to consier them as particular cases of (10). However, for the application to interacting 6

7 particles an alternative reuction to (10) becomes more natural, namely by observing that (8) has alreay form (10) but with B k presenting not just the operators in C sym (X k ) (as we always suppose in (10)), but the operators from C sym (X ) to C sym (X k ) given by B k f(x 1,..., x k ) = Bf(x 1,..., x k ) + (f(y 1,..., y m ) f(x 1,..., x k ))P k (x 1,..., x k ; y 1 y m ) (12) m=1 that are conitionally positive in the sense that if a non-negative f C sym (X ) vanishes at x = (x 1,..., x k ), then B k f(x) 0. The theory of Sections 5 an 6 is presente in a way that more or less straightforwarly extens to the Markov moels of interactions not preserving the number of particles thus incluing the processes of, say, coagulation an fragmentation. In Section 7 we illustrate this evelopment by obtaining new results on the rate of convergence to the law of large numbers an the central limit in the stanar Smoluchovski moel of coagulation-fragmentation (thus giving a solution to Problem 10 from [2] for the case of boune intensities). In Appenix two simple auxiliary results are presente, the first of which is evote to the possibility of approximation of continuous functions on measures by analytic functions in such a way that this approximation respects the erivation (that oes not follow from the stanar Stone-Weierstrass theorem on approximation). It is worth noting that the obtaine limits of fluctuation processes supply a large class of natural examples for infinite imensional Mehler semigroups, whose analysis is well uner way in the present literature, see e.g. [45] an references therein for general theory, [51] for some properties of Gaussian Mehler semigroups an [15] for the connection with branching processes with immigration. 1.4 Remarks on iscrete case As a warming up we shall iscuss shortly the case of X being the set of natural numbers an the r.h.s of (6) being a homogeneous quaratic polynomial. In this case equation (6) takes the form of the system of quaratic equations ẋ j = (A j x, x), j = 1, 2,..., N, (13) where N is a natural number or N =, unknown x = (x 1, x 2,...) is N-vector an all A j are given square N N-matrices. If X is the set of natural numbers an only binary interactions preserving the number of particles are allowe, measure-value kinetic equation (8) is reuce to the iscrete system ẋ j = 1 2 N k=1 K x k x l l=1 n m Pkl nm (δn j + δm j δ j k δj l ), j = 1, 2,..., N, (14) where Pkl nm is an arbitrary collection of positive numbers that is symmetric with respect to the exchange of n an m (efining the rates of transformation of any pair of particles j of type k an l to the pair of particles of type m an n), an where δn j is the Kronecker symbol (equals 1 if j = n an 0 otherwise). Proposition 1.1 Suppose N is finite an N j=1 Aj = 0 (this assumption is mae in orer to avoi aitional problems with the existence of the solutions). 7

8 (i) System (13) efines a positivity preserving semigroup (i.e. if all coorinates of the initial vector x 0 are non-negative, then the solution x(t) is globally uniquely efine an all coorinates of this solution are non-negative for al times), if an only if for each j the matrix Ãj obtaine from A j by eleting its j-th column an j-th row is such that (Ãj v, v) 0 for any (N 1)-vector v with non-negative coorinates. (ii) System (13) can be written in form (14) (with some positive numbers Pkl nm ) if an only if the entries A j kl are non-negative whenever k j, l j, i.e. if an only if the matrix Ãj has only non-negative entries for all j. We omit a simple proof of this fact. It was presente to give a feeling of the ifference between the class of positivity preserving evolutions an the class of kinetic equations (10). This ifference will be stuie systematically in the next section. 1.5 Notations for spaces an semigroups Throughout the paper the letter X enotes a locally compact metric space, an C(X) (respectively C (X)) enotes the Banach space of boune continuous functions on X (respectively its close subspace consisting of functions vanishing at infinity) equippe with the norm f C(X) = sup x X f(x). Denoting by X 0 a one-point space an by X j the powers X X (j-times), we shall enote by X their isjoint union X = j=0 Xj, which is again a locally compact space. The elements of X will be esignate by bol letters, e.g. x, y. Instea of X it is often more convenient to work with its symmetrization SX which is the quotient space obtaine by ientifying any two pairs x, y that iffer only by a permutation of its elements. We shall enote by C sym (X ) = C(SX ) (resp. B sym (X )) the Banach spaces of symmetric continuous (respectively boune measurable) functions on X an by C sym (X k ) (resp. B sym (X k )) the corresponing spaces of functions on the finite powers X k. M(X) (respectively M sym (X )) enotes the Banach space of finite Borel measures on X (respectively symmetric measures on X ) with the full variation as the norm. By weak topology in the space of measures we shall always unerstan the -weak topology. The upper subscript + for all these spaces (e.g. M + (X)) will enote the corresponing cones of non-negative elements. The lower subscript fin for the spaces of functions on X, e.g. C sym fin (X ), will enote a subspace with only finite number of non-vanishing components (polynomial functionals). We shall work mostly with X = R or X being a close manifol. In this case by C k (X) we shall enote the Banach space of boune k times continuously ifferentiable functions with boune erivatives equippe with the norm k f C k (X) = f (l) C(X). l=0 Of course C(X) = C 0 (X) an Cc k (X) (respectively C (X)) k enotes the subspace of C k (X) consisting of functions with a compact support (respectively vanishing at infinity together with all their erivatives up to an incluing the orer k). By L 1 (X) we enote the usual Banach space of integrable functions, an W k (X) is the Sobolev space of measurable functions having finite norm k f W k (X) = f (t) L1 (X), t=0 8

9 where the erivatives are efine in the weak sense. At last, for a Banach space B an a positive number t 0, we shall enote by C([0, t 0 ], B) the Banach space of continuous functions from [0, t 0 ] to B equippe with the norm f t C([0,t0 ],B) = sup f t B. t [0,t 0 ] By a propagator (respectively a backwar propagator) in a family of sets M s, s 0, we shall mean a family U(t, s), t s 0 of transformations M s M t (respectively M t M s ) satisfying the co-cycle ientity U(t, s)u(s, r) = U(t, r) (respectively U(s, r)u(t, s) = U(t, r)) for t s r an such that U(t, t) is the ientity operator for any t. In particular, a propagator of probability in a family of the Borel subsets X t of X (shortly, in X) is a propagator of linear contractions V (t, s) : M + (X s ) M + (X t ). The ajoint operators U(t, s) = V (t, s) act backwars from B(X t ) to B(X s ) (an form a backwar propagator), the uality equation being (U(t, s)f t, µ s ) = (f t, V (t, s)µ s ) for functions f t B(X t ) an measures µ s M + (X s ). We say that such a propagator is Feller if the family U(t, s) is a strongly continuous family of operators C (X t ) C (X s ) (if U(t, s) epen only on the ifference t s, these operators form a Feller semigroup). A Markov process specifie by U(t, s), i.e. by the transition probabilities P (t, s, x, y) from X s to X t efine by E t,s x f t = f t (y)p (t, s, x, y) = (U(t, s)f t )(x), will be then calle a nonhomogeneous Feller process (where we use the stanar notations E x for the expectation of the process starting at x). 2 The structure of generators Definition. Suppose D is a linear hull of a ense subspace in C (X) an the set of constant functions. A mapping Ω from a ense subspace of M(X) containing finite linear combinations of Dirac measures to (possibly unboune) linear functionals in C(X) with omains containing D is calle x = (x 1,..., x n )-conitionally positive for a given finite collection of elements x of X, if Ω( n i=1 ω iδ xi )g is non-negative for arbitrary positive numbers ω 1,..., ω n whenever g D is non-negative an vanishes at points x 1,..., x n. Ω is calle conitionally positive if it is x-conitionally positive for all x X. In particular, applying this efinition to polynomial operators Ω (see the r.h.s. of equation (6)) we say that a linear map A = (A 1,..., A K ) : D (C(X), C sym (X2),..., C sym (X K )) (15) is conitionally positive if for any collection of ifferent points x 1,..., x m of X, for any non-negative function g D such that g(x j ) = 0 for all j = 1,..., m, an for any collection of positive numbers ω j one has K m k=1 i 1 =1 m ω i1 ω ik A k g(x i1,... x ik ) 0. (16) i k =1 9

10 Moreover, we shall say that A is strongly conitionally positive if A k g(x 1,..., x k ) is x = (x 1,..., x k )-conitionally positive for any k an x. Finally, a linear operator A k : D C sym (X k ) is sai to be strongly conitionally positive or conitionally positive if this is the case for the family A = (0,..., 0, A k ) of type (15). Remark. If Ω is linear, i.e. Ω(µ)g = (Ag, µ) with some linear operator A in C(X), the above efinition of conitional positivity is reuce to the stanar efinition of conitional positivity of the linear operator A (see introuction). Our efinitions are motivate by the following simple observations. Proposition 2.1 (i) If the solutions to (4) are efine at least locally an are positive for initial measures being positive finite linear combinations of Dirac elta-measures so that (4) hols for all g D, then Ω is conitionally positive. (ii) Suppose linear operators B k in C (X k ), k = 1,..., K, are conitionally positive in the usual sense (an thus specify a kinetic equation (10)). Then the operators g B k g + from C (X)(ense subspace of ) to C (X k ) are strongly conitionally positive (the notation g + is introuce in (8)). Proof. (i) Let a non-negative g D be such that g(x 1 ) = = g(x m ) = 0 for some points x 1,..., x m. Choosing µ 0 = ω 1 δ x1 + + ω m δ xm we see that (g, µ 0 ) = 0, an consequently the conition of positivity preservation implies that (g, µ t t) t=0 0, which means that Ω( n i=1 ω iδ xi )g is non-negative. Statement (ii) is obvious. Remark. Thus on the formal level, the ifference between the general positivity preserving evolutions of form (6) an the kinetic equations of k-ary interacting particles (10) is the ifference between conitionally positive an strongly conitionally positive generators. For the case of iscrete X this observation is illustrate by Proposition 1.1. Our first structural result characterizes the conitional positivity at fixe points. Theorem 2.1 Suppose X = R with a natural an the space D from the efinition above contains Cc 2 (X). If an operator is x = (x 1,..., x n )-conitionally positive, then ( n n [ Ω ω i δ xi )g = a j (x)g(x j ) + (b j (x), )g(x j ) (cj (x), )g(x j ) i=1 j=1 ] + (g(x j + y) g(x j ) χ(y)(y, )g(x j ))ν j (x; y) for g C 2 c (X), where each c j is a positive efinite matrix, each ν j is a Lévy measure, χ is an inicator like in (1), an with all a j, b j, c j, ν j epening on n i=1 ω iδ xi. Proof. Let us start with a comment on the Courrège theorem. A look on the proof of his theorem (see [12], [9], [28]) shows that the characterization is actually given not only for conitionally positive operators, but also for conitionally positive linear functionals obtaine by fixing the arguments. Namely, it is shown that if the linear functional (Ag)(x) : Cc 2 R is conitionally positive at x, i.e. if Ag(x) 0 whenever a nonnegative g vanishes at x, then Ag(x) has form (1) irrespectively of the properties of Ag(y) in other points y. Now let us choose a partition of unity as a family of n smooth nonnegative functions χ i, i = 1,..., n, such that n i=1 χ i = 1 an each χ i equals one in a neighborhoo of x i (an consequently vanishes in a neighborhoo of any other point 10 (17)

11 x l with l i). By linearity Ω( n i=1 ω iδ xi ) = n i=1 Ω i, with Ω j g = ( n i=1 ω iδ xi )(χ j g).. Clearly each functional Ω i g is conitionally positive at x i in the usual sense, i.e. g(x i ) = 0 implies Ω i g 0 for a non negative function g C 2 c (X). Hence applying a fixe point version of the Courrège theorem to each Ω i, one obtains representation (17). As a corollary we shall obtain now a characterization of the strong conitional positivity. Theorem 2.2 Let X an D be the same as in Theorem 2.1. (i) A linear operator A k : D C sym (X k ) is strongly conitionally positive if an only if A k g(x 1,..., x k ) = k [a k Π j1 (x 1,..., x k ))g(x j ) j=1 + (b k Π j1 (x 1,..., x k )), )g(x j ) (c kπ j1 (x 1,..., x k )), )g(x j ) + (g(x j + y)g(x i ) χ(y)(y, )g(x j ))ν k (Π j1 (x 1,..., x k ); y)], (18) where the operators Π jl in R exchange the coorinates with numbers j an l, each c k (respectively ν j ) is a positive efinite matrix (respectively a Lévy measure), χ is an inicator like in (1), an where the functions a k, b k, c k, ν k are symmetric with respect to the permutations of the variables x 2,..., x k (i.e. all permutations not affecting x 1 ). (ii) A linear operator A k : D C sym (X k ) is strongly conitionally positive if an only if k A k g(x 1,..., x k ) = a k (Π j1 (x 1,..., x k ))g(x j ) + B k g + (x 1,..., x k ), (19) j=1 where a k is the same as in (17) an B k is a conitionally positive (in usual sense) operator in C sym (X k ). (iii) From the continuity of the functions in the image of A it follows that the functions a k, b k, c k, ν k epen continuously on x 1,..., x k (measures ν k are consiere in the weak topology). Proof. (i) From Theorem 2.1 it follows that for arbitrary ifferent x 1,..., x k A k g(x 1,..., x k ) = k [a j k (x 1,..., x k )g(x j ) j=1 + (b j k (x 1,..., x k ), )g(x j ) (cj k (x 1,..., x k ),, )g(x j ) + (g(x j + y) g(x j ) χ(y)(y, )g(x j ))ν j k (x 1,..., x k ; y)], where each c j k is a positive efinite matrix, each νj k is a Lévy measure. Moreover, ue to the symmetry of functions A k g for g Cc 2 (X), the functions a j k, bj k, cj k, νj k epen on x 1,..., x k symmetrically in the sense that, say a j Π jl = a l for all j, l an a j Π lm = a j for 11

12 m j, l j. Due to this symmetry it is possible to rewrite the above formula as (18) 1 with a k = a 1 k, b k = b 1 k, c k = c 1 k, ν k = νk 1. It remains to notice that ue to the continuity, this representation remains vali even for not necessarily ifferent points x 1,..., x k. (ii) By the Courrège theorem applie to X k, a conitionally positive B k in C (X k ) has form B k f(x 1,..., x k ) = ã(x 1,..., x k )f(x 1,..., x k ) k + ( b j (x 1,..., x k ), xj )f(x 1,..., x k )) ( c(x 1,..., x k ), )f(x 1,..., x k ) j=1 + (f(x 1 + y 1,..., x k + y k ) f(x 1,..., x k ) k (χ(y i )(y i, xi )f(x 1,..., x k )) ν(x 1,..., x k ; y 1 y k ). (20) Applying this to f = g + an comparing with (17) yiels the require result. (iii) By one an the same proceure, one proves first the continuity for ifferent x 1,..., x k, then for two of them coinciing, etc. If all x 1,..., x k are ifferent, one can use the ecomposition from the proof of Theorem 2.1 reucing the problem to proving the continuity for the operator A k (χ j g)(x 1,..., x k ). Choosing g to be a constant, one proves the continuity of the coefficients a k for this operator. Then choosing g to be a constant in a neighborhoo of a certain point x one proves the continuity of ν k. Then choosing g to be linear aroun x one gets the continuity of b k, an at last the continuity of c k follows. Thus operators A k in (4) are strongly conitionally positive if an only if they have form A k g = B k g + /k! with some conitionally positive B k in C (X) (up to some multiplication operators) an thus if they correspon to a kinetic equation for some Markov moel of k-ary interacting particles. Theorem 2.3 Suppose again that X = R an D contains Cc 2 (X). A linear mapping (15) is conitionally positive if an only if each of the operators A k has form (18) with the same symmetry conition on its coefficients as in (18) an with c k an ν k being not necessarily positive but only such that the matrix K m m k ω i1 ω ik 1 c k (x, x i1,..., x ik 1 ) (21) k=1 i 1 =1 i k 1 =1 is positive efinite an the measure K m m k k=1 i 1 =1 i k 1 =1 i=1 ω i1 ω ik 1 ν k (x, x i1,..., x ik 1 ) (22) is positive for any m, any collections of positive numbers ω 1,..., ω m an points x, x 1,..., x m (with all ν k satisfying (2), of course). In particular, A k have form (19) with B k given by (20) but with not necessarily positive c an ν. Proof. Step 1. Here we shall prove that all A k have form (17) with n = k, but possibly without positivity of c k an ν k. For arbitrary fixe x 1,..., x m the functional of g given by K m m ω i1 ω ik A k g(x i1,..., x ik ) (23) k=1 i 1 =1 i k =1 12

13 is x = (x 1,..., x m )- conitionally positive an consequently has form (17) (with m instea of n), as follows from Theorem 2.1. Using ɛω j instea of ω j, iviing by ɛ an then letting ɛ 0 one obtains that m i=1 A 1g(x i ) has the same form. As m is arbitrary, this implies on the one han that A 1 g(x) has the same form for arbitrary x (thus giving the require structural result for A 1 ) an on the other han that K k=2 ɛ k 2 m i 1 =1 m ω i1 ω ik A k g(x i1,..., x ik ) i k =1 has the require form. Again letting ɛ 0 yiels the same representation for the functional m m ω i1 ω i2 A 2 g(x i1, x i2 ). i 1 =1 i 2 =1 As above this implies on the one han that ω 2 1A 2 g(x 1, x 1 ) + 2ω 1 ω 2 A 2 g(x 1, x 2 ) + ω 2 2A 2 g(x 2, x 2 ) has the require form an hence also A 2 g(x 1, x 1 ) has this form for arbitrary x 1 (put ω 2 = 0 in the previous expression), an hence also A 2 g(x 1, x 2 ) has the same form for arbitrary x 1, x 2 (thus giving the require structural result for A 2 ), an on the other han that K k=3 ɛ k 3 m i 1 =1 m ω i1 ω ik A k g(x i1,..., x ik ) i k =1 has the require form. Following this proceure inuctively yiels the claim of Step 1. Step 2. From the obtaine representation for A k it follows that K m k=1 i 1 =1 m ω i1 ω ik A k g(x i1,..., x ik ) = i k =1 K k k=1 K m l=1 i 1 =1 m i k 1 =1 ω l ω i1 ω ik 1 [ a k (x l, x i1,..., x ik 1 )g(x l ) + (b k (x l, x i1,..., x ik 1 ), )g(x l ) (c k(x l, x i1,..., x ik 1 ), )g(x l ) ] + (g(x j + y) g(x j ) χ(y)(y, )g(x j ))ν k (x l, x i1,..., x ik 1 ; y). As this functional has to be strongly conitionally positive, the require positivity property of c an ν follows from Theorem 2.1. Corollary 1 In the case K = 2 the notions of conitional positivity an strong conitional positivity for (15) coincie. Proof. In case K = 2 the positivity of (21) reas as the positivity of the matrix c 1 (x) + 2 m ω i c 2 (x, x i ) (24) i=1 for all natural m, positive numbers ω j an points x, x j, j = 1,..., m. Hence c 1 is always positive (put ω j = 0 for all j ). To prove strong conitional positivity one has to prove 13

14 that c(x, y) is positive efinite for all x, y. But if there exist x, y such that c(x, y) is not positive efinite, then by choosing large enough number of points x 1,..., x m near y, one woul get a matrix of form (24) that is not positive efinite (even for all ω j = 1). This contraiction completes the proof. Taking into account explicitly the symmetry of generators in (6) (note that in (6) or (8) the use of nonsymmetric generators or their symmetrizations specifies the same equation) allows to get a useful equivalent representation for (5), (7). Namely, the following statement is obvious. Corollary 2 (i) If A is conitionally positive an hence by Theorem 2.2 its components have form (18), one can rewrite (6) as t (g, µ t) = K k=1 k with A 1 k : C (X) C sym (X k ) being efine as A 1 kg(x, y 1,..., y k 1 ) = a k (x, y 1,..., y k 1 )g(x) (A 1 kg)(x, y 1,..., y k 1 )µ t (x)µ t (y 1 ) µ t (y k 1 ), (25) + (b k (x, y 1,..., y k 1 ), g(x)) (c k(x, y 1,..., y k 1 ), )g(x) + Γ k (y 1,..., y k 1 )g(x), where is, of course, the graient operator with respect to the variable x R an where Γ k (y 1,..., y k 1 )g(x) = (g(x + z) g(x) χ(z)(z, )g(x))ν k (x, y 1,..., y k 1 ; z). (ii) If A k are given by (11) (an hence (6) has form (10)), then A 1 k = (1/k!)B kπ, where the lifting operator π is given by πg(x 1,..., x k ) = g(x 1 ). (iii) The strong form of equation (25)-(27) for measures µ t having ensities with respect to Lebesgue measure, i.e. having form µ t (x) = f t (x)x with some f t L 1 (X) is t f t(x) = where K k=1 (26) (27) [ 1 k 2 (c k(x, y 1,..., y k 1 ), )f t (x) + (( c k b k )(x, y 1,..., y k 1 ), f t (x)) +(a k b k (, c k))(x, y 1,..., y k 1 )f t (x) + Γ k(y 1,..., y k 1 )f t (x) ( c k (x, y 1,..., y k 1 )) j = (, c k )(x, y 1,..., y k 1 ) = i=1 i,j=1 x i c ij (x, y 1,..., y k 1 ), 2 x i x j c ij (x, y 1,..., y k 1 ), ] k 1 an where Γ k (y 1,..., y k 1 ) is the ual operator to the integral part (27) of (26). 14 f t (y l )y l, l=1 (28)

15 As a sie result, we can give now the structure of positivity preserving evolutions (5). Theorem 2.4 Suppose X an D are as in Theorem 2.1. Let A be a mapping from a ense subspace D of M(X) containing finite combinations of Dirac measures to linear operators in C (X) with omains containing D. Let the r.h.s. of (5) be conitionally positive an let A be continuous in the sense that if µ n µ weakly, µ, µ n D an g D, then A(µ n )g A(µ)g in C(X). Then A(µ) has form (7). Proof. From Theorem 2.1 an the efinition of conitional positivity it follows that A(µ) has form (7) for µ being finite linear combinations of Dirac measures. For arbitrary µ = lim µ n D with µ n being finite combinations of Dirac measures, A(µ n )g converges to a continuous function for arbitrary g D. This efines the coefficients a, b, c, ν in (7) as continuous functions for arbitrary µ D by first proving the continuity of a by choosing g = 1, then the continuity of ν by choosing g to be constant in an open set, an then the continuity of b by choosing g to be linear in an open set. Some remarks an examples to the obtaine results are in orer. Remark 1. It is clear that if only two components, say A i an A j, o not vanish in the mapping (15), an A is conitionally positive, then each non-vanishing component A i an A j is conitionally positive as well (take ɛω j instea of ω j in the efinition an then pass to the limits ɛ 0 an ɛ ). In case of more than two non-vanishing components in the family A, the analogous statement is false (in particular, Corollary 1 can not be extene to K > 2). Namely, if A is conitionally positive, the bounary operators A 1 an A K are conitionally positive as well (the same argument), but the intermeiate operators A k nee not to be, as shows alreay a simple example of the operator A = (A 1, A 2, A 3 ) with A i g(x 1,..., x i )) = a i ( g(x 1 ) + + g(x i )), i = 1, 2, 3, with a 1 = a 3 = 1 an with a 2 being a small enough negative number. Of course, it is easy to write an explicit solution to equation (4) in this case. Remark 2. We gave our results for X = R, but using localization arguments (like in linear case, see [9]) the same results can be easily extene to close manifols. It is seemingly possible to characterize in the same way the corresponing bounary conitions (generalizing also the linear case from [9]), though this is alreay not so straightforwar. Remark 3. Basic conitions of positivity of (21), (22) can be written in an alternative integral form. Namely, the positivity of matrices (21) is equivalent (at least for boune continuous functions c) to the positivity of the matrices K k=1 k c k (x, y 1,..., y k 1 )µ(y 1 ) µ(y k 1 ) (29) for all non-negative Borel measures µ(y). Conitions (21), (22) actually represent moifie multi-imensional matrix-value or measure-value versions of the usual notion of positive efinite functions. For instance, in case k = K = 3 an = 1, (29) means that c 3 (x, y, z)ω(y)ω(z)yz is a non-negative number for any non-negative integrable function ω. The usual notion of a positive efinite function c 3 (as a function of the last two variables) woul require the same positivity for arbitrary (not necessarily positive) integrable ω. 15

16 Remark 4. As shows Proposition 1.1 the statement of Corollary 1 oes not hol for iscrete X. In case of continuous state space X one can start feeling the ifference between strictly conitionally positive an conitionally positive operators only with k = 3. A simple example of a conitionally positive, but not a strictly conitionally positive operator represents the operator efine as A 3 g(x 1, x 2, x 3 ) = cos(x 2 x 3 ) g(x 1 ) + cos(x 1 x 3 ) g(x 2 ) + cos(x 1 x 2 ) g(x 3 ). (30) Equation (28) in this case takes the form t f t(x) = f t (x) f t (y)f t (z) cos(y z)yz. (31) The nonlinear iffusion coefficient is not strictly positive here. Namely, since f(y)f(z) cos(y z)yz = 1 2 ( ˆf(1) 2 + ˆf( 1) 2 ), where ˆf(p) is the Fourier transform of f, this expression o not have o be strictly positive for all non-vanishing non-negative f. However, one can fin the explicit solution to the Cauchy problem of (31): with f t = 1 2πωt } (x y)2 exp { f 0 (y)y, 2ω t ω t = ln(1 + t( ˆf 0 (1) 2 + ˆf 0 ( 1) 2 ). This is easily obtaine by passing to the Fourier transform of equation (31) that has the form t ˆf t (p) = 1 2 p2 ( ˆf t (1) 2 + ˆf t ( 1) 2 ) ˆf t (p), an which is solve by observing that ξ t = ˆf t (1) 2 + ˆf t ( 1) 2 solves the equation ξ t = ξt 2 an consequently equals ξ t = (t + ξ0 1 ) 1. Remark 5. There is a natural ecomposable class of operators, for which (6) reuces straightforwarly to a linear problem. Namely, suppose k!a k g = B k g + = ( B k g) + for all k = 1,..., K with some B k in C (X) that generate Markov processes their (i.e. Bk 1 = 0). Then (25) takes the form t (g, µ t) = K k=1 1 (k 1)! ( B k g, µ t ) µ t (k 1), which is a linear equation epening on µ t = µ 0 as on a parameter. For conclusion of this section let us iscuss the lifting of (6), (10) to linear evolutions on the multi-particle state space X which is of importance for the corresponing particle systems escribing their propagation of chaos property. In fact equations (34), (35) below are erive in [7] from a scaling limit of the moment measures of interacting particle systems in the spirit of e.g. [30] or [56]. For a finite subset I = i 1,..., i k of a countable setj, we enote by I the number of elements in I, by Ī its complement J\I, by x I the collection of the variables 16

17 x i1,..., x ik an by x I the measure x i1 x ik. Clearly each f B sym (X ) is efine by its components f k on X k so that for x = (x 1,..., x k ) X k X, say, one can write f(x) = f(x 1,..., x k ) = f k (x 1,..., x k ) (the upper inex k at f is optional an is use to stress the number of variables in an expression). Similar notations are for measures. In particular, the pairing between C sym (X ) an M(X ) can be written as (f, ρ) = f(x)ρ(x) = f 0 ρ 0 + (x 1,..., x n )ρ(x 1 x n ), n=1 f C sym (X ), ρ M(X ), (32) so that ρ = (1, ρ) for ρ M + (X ). To an arbitrary Y (x) M(X) there correspons a measure Y M sym (X ) efine by its components (Y ) n (x 1 x n ) = 1 n! Y n (x 1 x n ) = 1 n! Y (x 1) Y (x n ). (33) To each g C sym (X ) there correspons a analytic functional (g, Y ) on M(X) (also calle sometimes the generating functional for g). Such a functional will be calle a polynomial on M(X), if g C sym fin (X ), i.e. if only a finite number of the components of g o not vanish. Theorem 2.5 equation t νl t(x 1 x l ) = (i) If µ t satisfies (10), then ν t = (µ t ) M sym (X ) satisfies the linear l K j=1 k=1 C l l+k 1 (B j,l+1,...,l+k 1 k x l+1,...,x l+k 1 ) ν k+l 1 t (x 1 x l+k 1 ), (34) where Cm l are the usual binomial coefficients, Bk is the ual to B k an (Bk I) ν t (x 1 x m ) means the action of Bk on the variables with inexes from I {1,..., m}. (ii) If the evolution of t ν t M sym (X ) is specifie by (34), then the ual evolution on C sym (X) is given by the equation ġ(x 1,..., x l ) = (L B g)(x 1,..., x l ) = (B j,i I +1 gī )(x 1,..., x l ), (35) I {1,...,l} j I where g I (x 1,..., x l ) = g(x I ) an B j 1,...,j k k j 1,..., j k. In particular, means the action of B k on the variables (L B g 1 )(x 1,..., x l ) = (B l (g 1 ) + )(x 1,..., x l ). Proof. Observe that the strong form of (10) is µ t (x) = K k=1 1 (k 1)! B k(µ t µ t )(xy 1 y k 1 ), 17

18 which implies (34) by straightforwar manipulations. From (34) it follows that t (g, ν t) = = l=0 K k=1 I {1,...,l+k 1}, I =k 1 j I m=0 I {1,...,m} j I B j,i k gī (x 1,..., x l+k 1 )ν t (x 1 x l+k 1 ) B j,i I +1 gī (x 1,..., x m )ν t (x 1 x m ), which implies (35). From the uality between (34) an (35) one conclues that whenever one has the well poseness for the Cauchy problem of equations (35) an (10) for some ense subspaces of initial continuous functions g on X an initial measures µ 0 on X, one has the uality relation (g t, µ 0 ) = (g 0, µ t ) (36) implying the invariance of the corresponing space of analytic functionals of the type (g, Y ) on M(X ) uner the action of the semigroup T t F (µ) = F (µ t ) on C(M(X)). In many cases one can prove that this space provies a core for the generator of this semigroup, with the r.h.s. L B of (35) specifying the form of the generator on this core. 3 A well poseness result Suppose X = R. Our strategy of solving (28) will be to look for a fixe point of a mapping from u t C([0, t 0 ], L 1 (X)) to the solution F t C([0, t 0 ], L 1 (X)) of the linear equation t F t(x) = K k=1 [ 1 k 2 (c k(x, y 1,... y k 1 ), )F t (x) + (( c k b k )(x, y 1,..., y k 1 ), F t (x)) +( b k 1 2 (, c k) + a k )(x, y 1,..., y k 1 )F t (x) + Γ k(y 1,..., y k 1 )F t (x) ] k 1 u t (y l )y l. Theorem 3.1 Suppose the coefficients of (37) as functions of x belong to C 2 (X) with all bouns being uniform with respect to other variables (erivatives of the Lévy measures in the integral operators Γ k are taken in the weak sense). Suppose for any t 0 an any u t S 0 C([0, t 0 ], L + 1 (X)) (the subset in C([0, t 0 ], L + 1 (X)) of functions having fixe L 1 -norm u 0 at all times) the resolving operator (or the propagator) U t,s ([u]), 0 s t t 0 to the Cauchy problem (37) in L 1 (X) is well efine. More precisely suppose that U t,s ([u]) is the family of positivity preserving linear isometries in L 1 (X) (to have isometries one usually assumes that all a k vanish in (37)) epening strongly continuously on t, s an such that (i) U t,t ([u]) is the ientity operator for all t, (ii) the semigroup ientity U t,s ([u])u s,r ([u]) = U t,r ([u]) hols for all t s r, (iii) the function g t = U t,s ([u])g s belongs to C 2 (X) an is the unique solution to (37) with the initial conition g s = g whenever g C 2 c (X), 18 t=1 (37)

19 (iv) the space W 2 (X) is preserve by U t,s ([u]) an hols with C 2 (t 0 ) epening only on u 0. U s,r ([u])g W 2 (X) C 2 (t 0 ) g X 2 (X) (38) Then for an arbitrary nonnegative f 0 W 2 (X) C 2 c (X) there exists a unique non negative classical solution f t W 2 (X) C 2 (X) of (28) with the initial conition f 0 (i.e. it satisfies (28) for all t 0 an coincies with f 0 at t = 0) an the mapping f 0 f t extens to a strongly continuous semigroup of (nonlinear) operators in (W 2 (X)) + that preserve the L 1 -norm of f 0 an yiel a solution to the Cauchy problem of (25). Proof. We are going to show that in case u 0 W 2 (X) there exists a unique fixe point of the mapping u t F t = U t,0 ([u])u 0. First observe that ifferentiating the semigroup ientity for U t,s ([u]) one gets that s U t,s([u])g = U t,s ([u])l(s)g (39) for g W 2 (X), where L(s) enote the generators (epening on u) at time s (the operators on the r.h.s. of equation (37)). Let u 0 be fixe an let u 1, u 2 coincie with u 0 at time zero an belong to the sphere S 0 C([0, t 0 ]), L + 1 (X)). Due to (39) one has F t ([u 1 ]) F t ([u 2 ]) = = t 0 t 0 s U t,s([u 2 ])U s,0 ([u 1 ])u 0 s U t,s ([u 2 ])(L 1 (s) L 2 (s))u s,0 ([u 1 ])u 0 s, (40) where L 1 (s), L 2 (s) enote the generators at times s of the equation (37) with u 1 an u 2 respectively. Clearly (L 1 (s) L 2 (s))g L1 (X) C 1 u 1 (s) u 2 (s) L1 (X) g W 2 (X) (41) where C 1 epens on u 0 L1 (X)an the bouns for the erivatives of the coefficients of equation (37). Consequently (38), (40), (41) imply that sup F s ([u 1 ]) F s ([u 2 ]) L1 (X) tc 1 C 2 (t 0 ) u 0 W 2 (X) sup u 1 (s) u 2 (s) L1 (X). (42) 0 s t 0 s t Hence the mapping u t to F t in the subset of S 0 C([0, t 0 ], L 1 (X)) obtaine by fixing u 0 is a contraction whenever tc 1 C 2 (t 0 ) u 0 W 2 (X) < 1. (43) Consequently one obtains a unique solution to (39) on the interval [0, t 1 ] for any t 1 t 0 satisfying (43), an moreover sup u s W 2 (X) C 2 (t 0 ) u 0 W 2 (X). (44) 0 s t 1 If t 1 < t 0 one can repeat this proceure starting from the time t 1 obtaining a unique continuation of the solution to the interval [t 1, t 2 ], where (t 2 t 1 )C 1 (C 2 (t 0 )) 2 u 0 W 2 (X) < 1. (45) 19

20 A key observation now is that as this new solution again solves (37) one still has the estimate (44) with t 1 replace by t 2 (i.e. C 2 o not have to be replace by C2 2 here), which allows to attain the time t 0 by using this proceure p times with p being the minimal integer exceeing t 0 C 1 (C 2 (t 0 )) 2 u0 W 2 (X). This completes the proof of the Theorem. We are going to illustrate the general result obtaine by applying it to the case of equations with a strictly non-egenerate iffusion part such that K k=1 k c k (x, y 1,..., y k 1 )µ(y 1 ) µ(y k 1 ) µ m (46) for all x an (non-negative) µ an some fixe real m. Theorem 3.2 Suppose all a k vanish in (25), (26), c k satisfy (46) an the measures ν k in (27) have ensities with respect to Lebesgue measure: ν k (x, y 1,..., y k 1 ; z) = ψ k (x, y 1,..., y k 1 ; z)z (47) such that ( ) C ψ k (x, y 1,..., y k 1 ; z) max z, C β z β 2+ for all k, x, z, y j with some constants β 1 (0, 2), β 2 > 0, C > 0 an with sup ψ (x + z, y) x y 2 <, sup 2 ψ (x + z, y) x2 y 2 <. y 1 z 1 y 1 z 1 (48) Suppose that b C 3 (X) an c C 4 (X) with all bouns being uniform with respect to other variables. Then all the conitions (an hence the conclusions) of Theorem 3.1 are satisfie. Proof. Observe first that the ual Γ k to operator (27) has form Γ k(y 1,..., y k 1 )g(x) = [g(x z)ψ(x z, y 1,..., y k 1 ; z) g(x)ψ(x, y 1,..., y k 1 ; z) + ( g(x), z)χ(z)ψ(x, y 1,..., y k 1 ; z) + g(x)χ(z)( x ψ(x, y 1,..., y k 1 ; z), z)] z which can be written in the form (g(x z) g(x) ( g(x), z))χ(z))ψ(x z; z) z + g(x) (ψ(x z; z) ψ(x; z) + χ(z)( x ψ(x; z), z)) z + ( g(x), zχ(z)(ψ(x, z) ψ(x z, z)) z) (49) (where we omitte the arguments y 1,..., y k 1 of the function ψ), i.e. it has the same form (47) plus a first orer ifferential operator with boune coefficients (the latter being ue to our assumptions on ψ). The well poseness of the Cauchy problem for 20