Local densities for a class of degenerate diffusions
Alberto Lanconelli Stefano Pagliarani Andrea Pascucci













                                           





                                                        
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    Local densities for a class of degenerate diffusions
        
    Alberto Lanconelli
    Dipartimento di Matematica, Università di Bari Aldo Moro, Bari, Italy.
    e-mail: alberto.lanconelli@uniba.it. Work supported by the Italian  INDAM-GNAMPA Stefano Pagliarani
    DIES, Università di Udine, Udine, Italy.
    e-mail: stefano.pagliarani@uniud.it. Andrea Pascucci
    Dipartimento di Matematica, Università di Bologna, Bologna, Italy.
    e-mail: andrea.pascucci@unibo.it
        This version: December 30, 2023
    ====================================================================================================================================================================================================================================================================================================================================================================================================
    
    
    We study a class of -valued continuous strong Markov processes that are generated, only
    locally, by an ultra-parabolic operator with coefficients that  are regular w.r.t. the intrinsic
    geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is
    a local  Itô formula for functions that are not twice-differentiable in the classical sense, but only
    intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the
    Hörmander condition. The second main contribution, which builds upon the first one, is an
    existence and regularity result for the local transition density. 
    
    
    
    
    
    Keywords: Hörmander condition, intrinsic geometry, intrinsic Hölder spaces, Kolmogorov equations, local densities, strong Feller property.
    
    
    
    
    
    
    
    
    § INTRODUCTION
    
    
    
    
    
    We study an ^d-valued continuous strong Markov process X that is generated, in a way that
    will be specified later, only locally on a domain D⊆^d by the degenerate
    operator
    
        _t:= 1/2∑_i,j=1^p_0_ij(t,x)_x_ix_j+∑_i=1^p_0_i(t,x)_x_i + ⟨ B x, ∇_x ⟩     t∈ [0,T_0[, x∈ D.
    
    Above, p_0≤ d and B is a (d× d)-matrix with constant real entries. In this paper,
    the focus is mainly on the case p_0<d, which implies that no ellipticity condition on _t
    is satisfied (i.e. the second order part is fully degenerate). The main structural assumption on
    the local-generator _t is the following
    
    The matrix B is such that the Kolmogorov operator
    
        := 1/2∑_i=1^p_0^2_x_i + ⟨ B x, ∇_x ⟩ + ∂_t_=:Y,       x∈ℝ^d,
    
    is hypoelliptic on ℝ×ℝ^d. Equivalently, the vector fields
    ∂_x_1,…,∂_x_p_0 and Y satisfy the Hörmander condition
    
        rank Lie(∂_x_1,…,∂_x_p_0,Y)  = d.
    
    Assumption <ref> is the only hypothesis required for the first main result of the paper, namely the instrinsic Itô's formula. The second main result, about the local density of X, is stated under the following additional
    
    There exist N∈_0, α∈ (0,1] and M>0
     such that:
    
    
    
    
    
    
    
    
    
      i)a_ij  ,a_i∈ C^N,α_B(]0,T_0[× D) for any i,j=1,…,p_0, with all the (Lie) derivatives bounded by M;
      
      ii) the following coercivity 
      condition holds on 
      D: 
    
        M^-1 ||^2 ≤∑_i,j=1^p_0
         a_ij(t,x)_i_j≤ M  ||^2 ,     t∈]0,T_0[, x∈
         D,∈ℝ^p_0.
    
    The spaces C^N,α_B(]0,T_0[× D) appearing above are 
    the intrinsic Hölder spaces 
    induced by the vector fields ∂_x_1,…,∂_x_p_0 and Y: their definition
    is recalled, for the reader's convenience, in Section <ref>.
    
    A relevant prototype example that fits Assumptions <ref> and <ref> is the stochastic process X=(X^1,X^2) defined by
    
        X^1_t =X^1_t  W_t, 
         X^2_t = X^1_t  t,
    
    which is generated by the operator
    
        = x^2_1/2∂_x_1x_1+x_1∂_x_2,     (x_1,x_2)∈ℝ_>0^2.
    
    This operator arises in mathematical finance and is related to the valuation of a class of
    path-dependent financial derivatives known as arithmetic Asian options. The process X^1_t is a
    geometric Brownian motion and represents the price of a risky asset, whereas X^2_t represents
    its average. The operator fulfills Assumption <ref> in that the commutator
     
        [∂_x_1 , Y] := ∂_x_1Y - Y∂_x_1=∂_x_2,
    
    and also satisfies Assumption <ref> for any D= ]a,∞[ with a>0. Although more
    sophisticated models, with more flexible  dynamics (local-stochastic volatility) for the price of
    the underlying asset, were proposed to price Asian options, the prototype process
    (<ref>) is complex enough to exhibit some interesting mathematical properties. In
    fact, the problem of analytically characterizing its joint transition density is still partially open, and sharp upper/lower bounds were established only recently in <cit.>. It is easy to recognize in (<ref>) the double degeneracy
    of the generator  that our framework allows for: 
    
    
    on the one hand, the second order part of  is fully degenerate in that 
    the partial derivative _x_2x_2 is missing; 
    
    on the other hand, the coefficient x_1^2 of the second order derivative ∂_x_1 x_1
    also degenerates near zero.
    
    
    More generally (see Proposition <ref> below for the precise statement), the class of stochastic processes that we consider includes locally-integrated diffusions of the form
    
        dX_t=(t,X_t)dt+(t,X_t)dW_t,
    
    with μ:[0,T_0[×^d⟶^d and
    σ:[0,T_0[×^d⟶^d× n such that, for any (t,x)∈ [0,T_0[×
    D, 
    
        μ(t,x)   =(a_1(t,x), ⋯, a_p_0(t,x), 0, ⋯, 0) + B x ,
         
         ^⊤    =[                  A     0_p_0× (d-p_0);     0_(d-p_0)× p_0 0_(d-p_0)× (d-p_0) ],      A=(a_ij(t,x))_i,j=1,⋯,p_0,
    
    and with B and a_ij,a_i satisfying Assumptions <ref> and <ref>
    respectively.
    
    We emphasize that no assumption is required on the generator of X outside the domain D,
    although the process X“lives" on ℝ^d, meaning that its trajectories are allowed to
    go in and out D.
    
    
    
    
     §.§ Main results and comparison with the literature
    
    Here we report and discuss the main results of the paper comparing them to the related
    literature. 
    Granted that precise definitions will be
    given in the sequel, namely in Sections <ref> and <ref>, we
    will provide here a heuristic 
    explanation of all the objects that appear in the
    statements below. 
    
    
    
    
    
    The first main result of this paper 
    is a local intrinsic Itô formula for X. In the following statement, P_t,x represents the
    probability under which the process X starts from the point x at time t with probability one
    and ^t is a filtration to which (X_T)_T≥ t is adapted. Moreover, we denote by 
    the differential operator differential operator
    
        := 
         1/2∑_i,j=1^p_0_ij(t,x)_x_ix_j+∑_i=1^p_0_i(t,x)_x_i + Y
    
    where Y is the vector field as defined in (<ref>). 
    
    Let X be a 
    local diffusion on ^d generated by _t on D (in the sense of Definition
    <ref>) and let Assumption <ref> be in force. Then, for any fixed
    (t,x)∈  ]0,T_0[×^d, ∈  ]0,1] and
    
    f∈ C^2,_B with compact support in ]0,T_0[× D, 
    we have
    
    
        f(T,X_T) = f(t,X_t)+∫_t^T
           f(u,X_u)du + M^t_T,    t≤ T<T_0,
    
    where 
    M^t is a zero-mean
    ^t-martingale under P_t,x, and 
    
        E_t,x[|M^t_T|^2]=E_t,x[ ∫_t^T  ∑_i,j=1^p_0    a_ij(s,X_s)∂_x_i f(s,X_s)∂_x_j f(s,X_s)  s] .
    
    Formula (<ref>) is a local result since no assumption is made on the generator of X outside D.
    Moreover, the Itô formula above is stronger than the classical one as it is proved for a class of functions that are not twice-differentiable in the classical sense, 
    but only with respect to the non-Euclidean geometry induced by 
    the vector fields ∂_x_1,…,_x_p_0 and Y in Assumption <ref>. 
    Roughly speaking, we say that f∈ C^2,_B(]0,T_0[× D)
    if ∂_x_1f,…,_x_p_0f and Yf
    exist on ]0,T_0[× D
    and 
    they are -Hölder continuous with respect to the semi-distance
     
        |T-t|^1/2+|y-e^(T-t)Bx|_B,      (t,x),(T,y) ∈ℝ×ℝ^d,
    
    with |·|_B as in (<ref>).  Note that Yf is meant as a Lie derivative and not as a
    combination of Euclidean derivatives: in principle, ∂_t f and ∂_x_if with
    i>p_0 do not exist; on the other hand, the Euclidean space C^2,α is included in
    C_B^2,α. We also highlight the fact that Assumption <ref> is not required in
    Theorem <ref>.
    
    
    
    
    Just like the classical Itô formula is based on the standard Taylor expansion, 
    the cornerstone of (<ref>) is a non-Euclidean Taylor formula, proved in
    <cit.> and <cit.> for functions in C^2,α_B, that 
    
    roughly states that
    
        f(T,y) = 𝐓^(2)_(t,x)(T,y) +
        O(|T-t|^2+α/2)+O(|y-e^(T-t)Bx|_B^2+α)
          as   (T,y)→ (t,x)∈  ]0,T_0[× D,
    
    where
    
        𝐓^(2)_(t,x)(T,y)    = f(t,x) + (T-t) Y
        f(t,x)+∑_i=1^p_0(y-e^(T-t)Bx)_i_x_if(t,x)
              +1/2∑_i,j=1^p_0(y-e^(T-t)Bx)_i(y-e^(T-t)Bx)_j_x_i,x_jf(t,x).
    
    
    
    
    
    
    
    
    
    
    With (<ref>)-(<ref>) at hand, it is possible to outline the main arguments the proof of Theorem <ref> is built upon. Analogously to the classical case, the key step is proving that 
     
        E_t,x[f(T,X_T)]-f(t,x)/T-t⟶ f(t,x)     as T-t→ 0^+,
    
    uniformly w.r.t. x∈ℝ^d, for any f∈ C_B^2,α  with compact support in
    ]0,T_0[× D. Applying (<ref>) yields 
        E_t,x[f(T,X_T)]-f(t,x)/T-t = E_t,x[
        𝐓^(2)_(t,x)(T,X_T) -f(t,x) ]/T-t_=:g_1(t,T)
        +E_t,x[O(|X_T-e^(T-t)Bx|_B^2+α)]/T-t_=:g_2(t,T)+ O(|T-t|^α/2).
     It is then clear that (<ref>)
    holds true if
    
        g_1(t,T) ⟶ f(t,x),     g_2(t,T) ⟶ 0,    as  T-t→ 0^+,
    
    uniformly w.r.t. x∈ℝ^d. Now, while the proof of the first limit above is quite
    straightforward and stems simply from the fact that X is locally generated by _t on D
    (see Definition <ref>), the second limit is a deeper result and represents the
    main element of novelty in the proof of Theorem <ref>. In particular, we note that
    g_2(t,T) ⟶ 0 is a consequence of the fact that
    
        lim
        _T-t→ 0^+E_t,x[_
        {|X_T -x|<}̣|X_T-e^(T-t)Bx|_B^2+]/T-t
        =0 ,    
        >̣0,
    
    uniformly w.r.t. x∈ H compact subset of D, and we emphasize that the latter is stronger than
    the classical general estimate for diffusion processes (see <cit.> or
    <cit.>)
    
        lim
        _T-t→ 0^+E_t,x[_
        {|X_T -x|<}̣|X_T-x|^2+]/T-t
        =0 ,    
        >̣0,
    
    since the intrinsic quasi-norm |·|_B on ℝ^d is such that |x|=o(|x|_B) as
    x→ 0.
    
    The second main result of the paper is the theorem below that states the existence of a local (on
    D) transition density (t,x;T,ξ)  for X, reveals its intrinsic regularity w.r.t. both
    the forward and backward variables and shows that it solves a forward and a backward Kolmogorov
    equation on ]t,T_0[× D and ]0,T[× D, respectively. Before stating the result, we
    need to introduce the last additional assumption, which is only needed to prove the regularity
    w.r.t. the backward variables.
    X is a Feller process on D, i.e. for any T∈  ]0,T_0[ and bounded ϕ∈
    C(^d)
    the function (t,x)↦
    E_t,x[ϕ(X_T)] is continuous on ]0,T[× D.
    
    Note that, since the coercivity condition in Assumption <ref>-ii) 
    only holds on D, the Feller property for the semigroup ϕ↦ E_t,·[ϕ(X_T)] is not ensured. This is due to the fact that the trajectories
    of X are allowed to leave and re-enter the domain D, but no assumption is made on the
    generator of X outside D. Had Assumption <ref> been satisfied for D=ℝ^d,
    the Feller property would stem from PDEs arguments, namely the existence and regularity results
    for the fundamental solution of  on ℝ^d that were proved in <cit.> and
    <cit.> be means of the so-called parametrix method.
    
    
    Let X be a 
    local diffusion on ^d generated by _t on D (in the sense of Definition
    <ref>) and let Assumptions <ref> and <ref> be in force.
    Then:
    
      a)X has a local transition density Γ on D, namely a non-negative measurable 
    function (t,x;T,y) defined for any 0< t<T<T_0 and x∈^d, y ∈ D, such that
    
        p(t,x;T,A) = ∫_A(t,x;T,y)  y,      A∈ℬ(D).
    
    Furthermore, (t,x;T,·) is continuous on D and locally bounded uniformly w.r.t.
    x∈^d;
    
      b) if N≥ 2, then for any (t,x)∈  ]0,T_0[×^d the function Γ(t,x;·,·)∈ C^N,α_B(]t,T_0[× D) and solves the forward Kolmogorov equation
    
        ^* u = 0,     on ]t,T_0[× D,
    
    where ^* is the formal adjoint of ;
    
      c) if Assumption <ref> is also in force, 
    then for any (T,y)∈  ]0,T_0[× D the function Γ(·,·;T,y)∈
    C^N+2,α_B(]0,T[× D) and solves the backward Kolmogorov equation
    
        u = 0,     on ]0,T[× D.
    
    
    This statement partially generalizes <cit.>, Sec. 4, where analogous results were
    obtained under the assumption that the coefficients of the generator are smooth 
    on D. In particular, the main assumption in the latter reference is a sort of
    local hypoellipticity condition for the generator, expressed in terms of Malliavin's matrix, on a
    given domain D of ℝ^d, which reduces to our Assumptions <ref> and
    <ref> with N=∞ when _t is of the form (<ref>). So, if on the one
    hand the framework in <cit.> is more general, on the other hand our results are stronger in this
    particular setting in that they are carried out by assuming the coefficients belonging to the
    intrinsic Hölder space C^N,α_B(]0,T[× D). Again, we stress the fact that
    C^N,α_B(]0,T[× D) not only does contain C^∞(]0,T[× D), but also
    includes the standard Hölder space C^N,α(]0,T[× D). Recent results on the local density assuming standard regularity of the coefficients were proved in <cit.>, under local strong Hörmander-type conditions, and in <cit.>, under local weak Hörmander-type conditions for two-dimensional diffusions.
    
    
    The proof of Theorem <ref> partially relies on some existence and regularity results  (see
    <cit.> and <cit.> among others) obtained in a PDEs' context for the
    fundamental solution and the Green functions of 
    Kolmogorov operators, as well as on some Schauder estimates (see again <cit.>). However,
    it is important to stress that the latter results alone are not enough to prove Theorem
    <ref>. This is due to the fact that our structural assumptions on the generator of X
    only hold on D, whereas there is no assumption on what is the behavior of the process outside
    D. For this reason, it will be necessary to combine the PDE results mentioned above with some
    probabilistic interlacing techniques and a crucial role will be played by the Itô formula of
    Theorem <ref>.
    
    More in detail, we adapt and customize the localization technique introduced in <cit.>, Sec. 4.
    We first prove a Feynman-Kac formula (Lemma <ref>)
    that allows to link the solutions of the Cauchy-Dirichlet problem for  on suitable cylinders
    of ]0,T_0[× D to the semigroup of the stopped diffusion.
    As this step is based on the application of Itô formula to the solution of the Cauchy-Dirichlet
    problem and the latter is not of class C^2,α in the classical sense but only intrinsically, 
    it is clear that the Itô formula needed here is the one in Theorem <ref> and not the
    classical one. The proof of the existence of the local density can be then completed by following
    closely the procedure employed by Kusuoka and Stroock, which makes use of a sequence of stopping
    times that keep track of when the process exits and re-enters the domain D.
    
    Once Part a) is proved, we need to depart from the latter procedure in order to prove part Part b)
    and Part c). In particular, to obtain the intrinsic regularity of the local density
    Γ(t,x;T,y) w.r.t. the forward variables (T,y)∈ D, it will be crucial to employ the
    Schauder internal estimates for the solutions of u=0 proved in <cit.>, combined
    with the Gaussian upper bounds for the Green function of  proved in the same reference.
    Once we have proved that Γ(t,x;·,·)∈ C^N,α_B(]t,T_0[× D), then the
    fact that Γ(t,x;·,·) solves (<ref>) simply follows because the
    latter is satisfied by the transition probability kernel of X in the distributional sense (see
    Remark <ref>).
    
    To prove that Γ(·,·;T,y)∈ C^N+2,α_B(]0,T[× D) and solves
    (<ref>), we first show that the same holds true for the function (t,x)→
    E_t,x[ϕ(X_T)] for any ϕ∈ C_b(ℝ^d). This step is based again on a Feynman-Kac
    formula and a crucial role is played one more time by the intrinsic Itô formula of Theorem
    <ref>. Finally, Part c) follows by proving that the same properties hold true for any
    bounded measurable function ϕ on ℝ^d. We remark that this last step is based on
    the fact that X actually enjoys the strong Feller property, namely the property in Assumption
    <ref> extended to bounded measurable functions, which we prove in Lemma
    <ref> by assuming the standard Feller property. Here we heavily rely again on
    the Schauder estimates in <cit.>. We point out that the latter result, i.e. proving the
    strong Feller property starting from the standard one, might enjoy an independent
    interest as it generalizes some previous results obtained in <cit.> under stronger
    assumptions, basically existence and uniform boundedness of the global transition density.
    
    We conclude this introduction mentioning that, 
    as an application of our results it should be possible to prove sharp  Gaussian upper bounds 
    for the local transition density (t,x;T,y) and its derivatives.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    The bounds would be analogous to those proved in <cit.>, Theor. 4.5, for a wider
    class of local generators, except that they would be explicit and valid under lower regularity
    assumptions on the coefficients.
    
    
    The rest of the paper is organized as follows. In Section <ref> we recall
    the definition of B-quasi-norm, the B-intrinsic Hölder spaces and the related intrinsic
    Taylor formula. In Section <ref> we give the precise definition of -local
    diffusion on ^d and prove Theorem <ref>. In Section <ref> we prove
    Theorem <ref>. In Appendix <ref> we collect some useful PDE results for
    -like operators, and in Appendix <ref> we recall some classic construction
    procedures for Markov processes.
    
    
    § PRELIMINARIES: HÖLDER SPACES AND TAYLOR FORMULA
    
    We recall the following useful characterization of Assumption <ref>, proved in
    <cit.>.
    
    Assumption <ref> is fulfilled if and only if B takes the block form
    
        B=(
        [   ∗   ∗   ⋯   ∗   ∗; B_1   ∗   ⋯   ∗   ∗;   0 B_2   ⋯   ∗   ∗;   ⋮   ⋮   ⋱   ⋮   ⋮;   0   0   ⋯ B_r   ∗ ])
    
    where B_j is a (p_j× p_j-1)-matrix with full rank (equal to p_j) for
    j=1,…,r, the ∗-blocks are arbitrary, p_0≥ p_1≥⋯≥ p_r≥ 1 and p_0 +
    p_1 + ⋯ + p_r = d.
    
    
    
    We introduce the quasi-norm in ^d
    
        |x|_B:=
         ∑_j=0^r ∑_i=p̅_j-1+1^p̅_j |x_i|^2j+1,     
        p̅_j:= ∑_k=0^j p_k,  p̅_-1:=0,
    
    that is homogeneous with respect to the dilations group
    
        D_0(ł)=diag(ł I_p_0,ł^3 I_p_1,…,ł^2r+1 I_p_r),    ł>0.
    
    For any (t,x)∈ and i=1,⋯,p_0, we denote by
    
        e^_x_i(t,x)=(t,x+δ e_i),    
         e^Ỵ(t,x)=(t+δ,e^δ Bx),    δ>0,
    
    the integral curves of the vector fields _x_1,⋯,_x_p_0,Y starting at (t,x).
    Here e_i denotes the i-th element of the canonical basis of ^d.
    Now, let Q be a domain in . For any (t,x)∈ Q we set
      
        _̣(t,x):=sup{∈ ]0,1]| e^_x_1(t,x),⋯,e^_x_p_0(t,x),
          e^Ỵ(t,x)∈ Q for any ∈̣[-,]}.
    
    If V is compactly contained in Q (hereafter we write V⋐ Q), we set
    _̣V=inf_(t,x)∈ V_̣(t,x).
    Note that _̣V∈ ]0,1]. 
    
    Let _1∈ ]0,1], _2∈  ]0,2] and i=1,⋯,p_0. We say that f∈
    C__x_i^α_1( Q) and g∈ C_Y^α_2( Q) if the following semi-norms are
    finite
    
        f_C__x_i^α_1(V):=sup_(t,x)∈ V 0<||̣<_̣V|f(e^δ_x_i(t,x))-
         f(t,x)|/|δ|^_1,
             g_C_Y^α_2(V):=sup_(t,x)∈ V 0<||̣<_̣V|g(e^δ Y (t,x))-
         g(t,x)|/|δ|^_2/2,
    
    for any V⋐ Q.
    
    We can now define the so-called B-Hölder spaces.
    We point out that slightly different versions of such spaces were previously adopted in several works (see 
    <cit.> and 
    <cit.>
    
    among others). Here we use the definition given <cit.>, which is basically  the one required
    in order to prove an intrinsic Taylor formula where the remainder is in terms of the intrinsic
    quasi-norm.
    
    Let Q a domain of  and let ∈ ]0,1], then:
    
      i)f∈ C^0,_B( Q) if f∈ C^_Y( Q) and f∈ C^__x_i( Q) for any i=1,…,p_0;
      
      ii)f∈ C^1,_B( Q) if f∈ C^1+_Y( Q) and _x_if∈ C^0,_B( Q) for any
      i=1,…,p_0;
      
      iii)forn∈ℕ with n≥2, f∈ C^n,_B( Q) if Yf∈ C^n-2,_B( Q) and
      _x_if∈ C^n-1,_B( Q) for any
      i=1,…,p_0.
    
    Moreover, for f∈ C^n,_B( Q) and V⋐ Q, we set
    
        f_C^n,_B( V):={   f_C^_Y( V)+∑_i=1^p_0f_C^__x_i( V),       n=0
           f_C^+1_Y( V)+∑_i=1^p_0∂_x_if_C^0,_B( V),        n=1
           Y f_C^n-2,_B( V)+∑_i=1^p_0∂_x_if_C^n-1,_B( V),       
        n≥ 2.
        .
    
    If f∈ C^n,_B(Q) and has compact support then we write f∈
    C_0,B^n,α(Q).
    The next result was proved in <cit.> in the particular case when the ∗-blocks  in
    (<ref>) are null and then extended to the general case in <cit.>.
    
    Let Q be a domain of , α∈  ]0,1] and n∈. If f∈ C^n,_B( Q)
    then we have:
    
      1) there exist 
    
        Y^k ∂_x^βf∈ C^n-2k-[β]_B,α_B( Q),
             0≤ 2 k + [β]_B ≤ n,
    
    where []̱_B denotes the height of the multi-index $̱ defined as
    
        []̱_B:  =   ∑_j=0^r ∑_i=p̅_j-1+1^p̅_j (2 j +1) _̱i;
    
      2)
    for any(t_0,x_0)∈ Q, there exist two bounded domainsU,V, such that(t_0,x_0)∈ U⊆ V⊆ Qand
    
        |f(t,x)-𝐓^(n)_(s,y) f(t,x)|≤ c_B,U
         f_C^n,_B( V
         )
         ( |s-t|^1/2+|y-e^(s-t)Bx|_B)^n+,     (t,x),(s,y) ∈ U,
    
    wherec_B,Uis a positive constant 
    and𝐓^(n)_(s,y)is the n-th order intrinsic Taylor polynomial of f centered at
    (s,y) given by 
    
    
    
    
        𝐓^(n)_(s,y) f(t,x)=  ∑_k∈_0, β∈_0^d
        0≤ 2 k + [β]_B ≤ n1/k! β!
         ( Y^k ∂_y^βf(s,y)) (t-s)^k(x-e^(t-s)By  )^β,    
         (t,x)∈.
    
    If f∈ C^n,_0,B( Q), then (<ref>) holds true with U=
    supp(f) and V= Q.
    
    
    
    
    § LOCAL DIFFUSIONS AND INTRINSIC ITÔ FORMULA
    
    For a givenT_0>0we consider a continuous^d-valued strong Markov processX=(X_t)_t∈[0,T_0[(in the sense of <cit.> as it is recalled in Appendix <ref>) with transition probability functionp=p(t,x;T,d), defined on a space(Ø,,(_T^t)_0≤ t≤ T< T_0,(P_t,x)_0≤ t< T_0,x∈^d).For any bounded Borel measurable functionϕ, we denote by
    
        E_t,x[ϕ(X_T)]:=(_t,Tϕ)(x):=∫_^dp(t,x;T,d)ϕ(),     0≤ t<T≤
           T_0, x∈^d,
    
    theP_t,x-expectation and the semigroup associated with the transition probability functionp, respectively (cf. Chapter 2.1 in <cit.>). Hereafter 
    we also fix a domainD, that is an open and connected subset of^d.
    
    
    For a given function f(t,T) with T∈  ]0,T_0[ and t∈[0,T[, we set
    
        lim_T-t→ 0^+ f(t,T)    := lim_h→ 0^+ f(t,t+h) =lim_h→ 0^+ f(t-h,t),      t∈[0,T_0[,
    
    when the second and the third limits exist and coincide with each other.
    
    
    The following two sets of limits will be used to give the definition of local diffusion. 
    
    
    
    
    
      [ Lim-i)]
    For any t∈[0,T_0[, >̣0, 
      and H compact subset of D, there exist the limits
    
        lim_T-t→ 0^+∫_{|-x|>}̣∩ Hp(t,x;T,d)/T-t
             =0  ,      uniformly w.r.t. x∈^d, 
        
           lim_T-t→ 0^+∫_|-x|>p(t,x;T,d)/T-t
             =0,     uniformly w.r.t. x∈ H.
    
    
    
    
       [Lim-ii)]
    For any t∈[0,T_0[, >̣0, 
      and H compact subset of D, 
      and for any i=1,⋯,d, there exist the limits
      
        lim_T-t→ 0^+∫_|-x|<(-x)_ip(t,x;T,d)/T-t
             =  _i(t,x)+(Bx)_i    if i=1,⋯,p_0
         
        
        (Bx)_i    if  i=p_0+1,⋯,d 
          ,
        lim_T-t→ 0^+∫_|-x|<(-x)_i(-x)_jp(t,x;T,d)/T-t
             = _ij(t,x)    if  i,j = 1,⋯, p_0 
        
        
        0    if i or j= p_0+1,⋯,d
          ,
    
    uniformly w.r.t. x∈ H, for some B as in (<ref>) and a_ij,a_i∈
    L^∞_ loc([0,T_0[× D).
     
    
    Let 
    _t an operator as in (<ref>). We say that X is a local diffusion 
    generated by _t on D (an -local diffusion
     in short) if
    [Lim-i)] and [Lim-ii)] hold. In case 
    they hold with D = ^d then we call X a global diffusion 
    generated by _t (an
    _t-global diffusion
     in short).
    
    The following proposition is useful for the applications because several models are defined in
    terms of solutions to stochastic differential equations. It shows that (stopped) solutions of SDEs
    are local diffusions in the sense of Definition <ref>. 
    
    Since we are dealing with stopping times, we point that we did not impose any right-continuity assumption on the filtrations ^t. 
    This is justified by the fact that, in the next proposition as well as in the rest of the paper, we will only consider hitting times of closed sets, which appear to be stopping times even if the the filtration is not right-continuous (see <cit.>, Theorem 2.2 p. 25).
    
    Let (X_t)_t∈[0,T_0[ be a continuous Markov process defined as
    X_t=X̂_t∧, where:
    
      i)X̂ is a solution of the SDE
    
        dX̂_t=(t,X̂_t)dt+(t,X̂_t)dW_t
    
    where W is a n-dimensional Brownian motion and the coefficients  and  are continuous
    and as in (<ref>)-(<ref>);
    
    
    
    
    
    
    
    
    
    
    
    
    
      
      ii) is the first exit time of X̂ from a domain D' containing D.
    
    Then X is a -local diffusion on ^d in the sense of Definition
    <ref>.
    
    The statement is a particular case of Lemma 2.3 in <cit.>, which proves that [Lim-i)] and [Lim-ii)] hold for the kernel of X.
    
    We have the first key result.
    
    Let Assumptions <ref> be in force. Then X is an -local diffusion on ^d
    if and only if [Lim-i)] holds and for any t∈[0,T_0[, >̣0, 
      and H compact subset of D, we have
    
        lim_T-t→ 0^+∫_|-x|<(-e^(T-t)Bx)_ip(t,x;T,d)/T-t
          =_i(t,x),
           lim_T-t→ 0^+∫_|-x|<(-e^(T-t)Bx)_i(-e^(T-t)Bx)_jp(t,x;T,d)/T-t
          =_ij(t,x),
           lim sup_T-t→ 0^+∫_|-x|<
        |  -e^(T-t)Bx|^2_B p(t,x;T,d)/T-t
          <∞,
    
    for any i,j=1,⋯,p_0, uniformly w.r.t. x∈ H, for some B as in (<ref>) and a_ij,a_i∈
    L^∞_ loc([0,T_0[× D).
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    The following lemma 
    formalizes the fact that_tas in (<ref>) is the generator ofXon[0,T_0[× D.
    
    Let X be a -local diffusion on ^d and Assumption <ref> be in force. Then,
    for any ϕ∈ C_0(]0,T_0[× D) and 
    f∈ C_0,B^2
    (]0,T_0[× D)
    we have 
    
        lim_T-t→ 0^+
         sup_x∈| (_t,Tϕ(T,·))(x)-ϕ(t,x)|
         =0,
           lim_T-t→ 0^+
         sup_x∈| (_t,Tf(T,·))(x)-f(t,x)/T-t-
          f(t,x)|
         =0,
    
    for any t∈]0,T_0[. Moreover, for any 0< t< T< T_0 and x∈^d, it holds that
    
        d/dT(_t,Tf(T,·))(x)=_t,T(
          f(T,·))(x).
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    In <cit.> it was already proven a weaker form of Proposition <ref> and Theorem <ref>, which applies to a wider class of local generators _t. When applied to this particular framework, it yields that if 
    X is a -local diffusion on ^d, then 
    (<ref>), (<ref>), (<ref>) and thus also (<ref>) and (<ref>), hold true for a smaller class of functions, namely f∈ C^2_0(]0,T_0[× D).
    
    By Proposition <ref> it is clear that, for any (t,x)∈  ]0,T_0[×^d,
    p(t,x;·,·) satisfies
    
        ^*  u = 0     on ]t,T_0[×D,
    
    in the sense of distributions, with ^* being the formal adjoint of , i.e.
    
        ∫_t^T_0∫_Dp(t,x;T,d)
           f(T,)d T = 0,      f∈ C_0^∞(]t,T_0[× D).
    
    
     §.§ Proofs 
    
    
    
    In this section we prove Proposition <ref>, Proposition <ref> and Theorem <ref>, respectively, the proof of each being based on the previous one.
    
    
      §.§.§ Proof of Proposition <ref>
    
    The proof of Proposition <ref> is preceded by the following
    
    Under the hypothesis [Lim-i)], 
    
    for any H⋐ D and for any 0≤ t < T<T_0, <̣:= dist(H, D), we
    have
    
        lim_T-t→ 0^+∫_{|-x|≥}̣∩ Hp(t,x;T,d)/(T-t)^m
          =0,     m≥ 1,
    
    uniformly w.r.t. x∈ H.
    
    Let ϕ^(x)∈ C_0^∞( D) be a family of functions with partial derivatives uniformly bounded w.r.t. x∈ H, and such that ϕ^(x)(x)=0 and
    
        ϕ=ϕ^(x) = 1  on H∖ B(x,)̣.
    
    Note that Remark <ref> gives
    
        ϕ(X_T)  =ϕ(X_t)+ ∫_t^T_sϕ(X_s)ds + M^t_T,
    
    where M^t is an ^t-martingale under P_t,x with 
    
        E_t,x[ |M^t_T|^2 ] = E_t,x[ ∫_t^T   ∑_k,l=1^p_0
         a_kl(s,X_s)  (∂_x_kϕ(X_s))(∂_x_lϕ(X_s))   s].
    
    Thus we obtain
    
        ∫_{|-x|≥}̣∩ Hp(t,x;T,d)   ≤ E_t,x[ ( ϕ(X_T) )^2(m+1)]  ≤ 2^2m+1
         E_t,x[ | ∫_t^T_sϕ(X_s)ds |^2(m+1) + | M^t_T |^2(m+1)]
           ≤ 2^2m+1(
         E_t,x[ | ∫_t^T_sϕ(X_s)ds |^2(m+1)]+ ( 2(m+1)/2m + 1)^2(m+1) E_t,x[ 
         |M^t_T|^2   ]^m+1),
    
    where we used <cit.> and Jensen's inequality in the last step. 
    Eventually, (<ref>) stems from (<ref>) combined with the fact that ϕ∈ C_0^∞( D) and the coefficients of  are in L^∞_ loc([0,T_0[× D). 
    
    
    We are now in the position to prove Propostion <ref>.
    
    
    Part 1: if.
    We assume [Lim-i)] to be satisfied together with the limits (<ref>)-(<ref>)-(<ref>), and prove that [Lim-ii)] holds. Let t∈[0,T_0[, >̣0, 
    and H compact subset of D be fixed. All the following limits are uniform w.r.t. x∈ H.
    
    We first prove (<ref>).
    
    For any i=1,...,d, it holds that : 
    
    
    
    
    
    
        lim_T-t→ 0^+∫_|x-|<δ(-x)_ip(t,x;T,d)/T-t   =lim_T-t→ 0^+∫_|x-|<δ( -e^(T-t)Bx +e^(T-t)Bx -x )_ip(t,x;T,d)/T-t
        
           =lim_T-t→ 0^+∫_|x-|<δ(-e^(T-t)Bx)_ip(t,x;T,d)/T-t+(Bx)_i.
    
    Here we used the property
    
        lim_T-t→ 0^+∫_|x-|<δp(t,x;T,d)=1,
    
    
    
    
    
    which follows from 
    [Lim-i)]. Now, for i=1,⋯,p_0, (<ref>) stems from (<ref>) and (<ref>). On the other hand, for i=p_0+1,⋯,d, note that condition (<ref>) is equivalent to
    
        lim sup_T-t→ 0^+∫_|-x|<
        |  (-e^(T-t)Bx)_i|^2/q_ip(t,x;T,d)/T-t
          <∞
    
    for a certain q_i≥ 3. Therefore, fixing ε>0, for any ρ∈  ]0,δ]
    we obtain
    
        lim sup_T-t→ 0^+∫_|-x|<|  (-e^(T-t)Bx)_i|^2/q_i+εp(t,x;T,d)/T-t
           = lim sup_T-t→ 0^+∫_|-x|<ρ|  (-e^(T-t)Bx)_i|^2/q_i+εp(t,x;T,d)/T-t + ∫_ρ≤ |-x|<|  (-e^(T-t)Bx)_i|^2/q_i+εp(t,x;T,d)/T-t
           ≤lim sup_T-t→ 0^+∫_|-x|<ρ|  (-e^(T-t)Bx)_i|^2/q_i+εp(t,x;T,d)/T-t +( +̣ |x| |I_d-e^(T-t)B|)^2/q_i+ε∫_ρ≤ |-x|<p(t,x;T,d)/T-t(by (<ref>))   = lim sup_T-t→ 0^+∫_|-x|<ρ|  (-e^(T-t)Bx)_i|^2/q_i+εp(t,x;T,d)/T-t
           ≤lim sup_T-t→ 0^+ ( ρ + |x| |I_d-e^(T-t)B|)^ε∫_|-x|<ρ|  (-e^(T-t)Bx)_i|^2/q_ip(t,x;T,d)/T-t
           ≤ρ^εlim sup_T-t→ 0^+∫_|-x|<|  (-e^(T-t)Bx)_i|^2/q_ip(t,x;T,d)/T-t,
    
    
    
    
    
    
    
    which in turn implies
    
        lim sup_T-t→ 0^+∫_|-x|<|  (-e^(T-t)Bx)_i|^2/q_i+εp(t,x;T,d)/T-t=0,
    
    and thus, as  is arbitrary,
    
        lim_T-t→ 0^+∫_|x-|<δ(-e^(T-t)Bx)_ip(t,x;T,d)/T-t=0.
    
    The latter, combined with (<ref>), proves (<ref>) for i=p_0+1,⋯,d.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    We now prove (<ref>). By using (<ref>) it is straightforward to show that
    
        lim_T-t→ 0^+∫_|-x|<(-x)_i(-x)_jp(t,x;T,d)/T-t =lim_T-t→ 0^+∫_|-x|<(-e^(T-t)Bx)_i(-e^(T-t)Bx)_jp(t,x;T,d)/T-t.
    
    Now, for i,j=1,...,p_0,(<ref>) simply stems from (<ref>).
    
    
    
    
    
    In the case i>p_0 or j>p_0, 
    by the Cauchy-Schwarz inequality we get
    
        ∫_|-x|<| (-e^(T-t)Bx)_i(-e^(T-t)Bx)_j| p(t,x;T,d)/T-t   ≤( ∫_|-x|<(-e^(T-t)Bx)^2_ip(t,x;T,d)/T-t)^1/2
             ×( ∫_|-x|<(-e^(T-t)Bx)^2_jp(t,x;T,d)/T-t)^1/2,
    
    and the limits of the right-hand side integrals are both finite, at lest one of each being zero. In fact, by (<ref>) and (<ref>) we obtain
    
        ∫_|-x|<(-e^(T-t)Bx)^2_kp(t,x;T,d)/T-t =
        a_kk(t,x)    if  k=1,⋯,p_0 
        
        0       if  k=p_0+1, ⋯, d.
    
    Therefore, we can conclude that for i>p_0 or j>p_0
        lim_T-t→ 0^+∫_|-x|<(-e^(T-t)Bx)_i(-e^(T-t)Bx)_jp(t,x;T,d)/T-t=0,
    
    which, combined with (<ref>), yields (<ref>).
    
    
    Part 2: only if. We now assume 
    
    
    [Lim-i)]-[Lim-ii)] and prove the limits (<ref>)-(<ref>)-(<ref>) to be true. 
    Fix t∈[0,T_0[, >̣0, 
    and H compact subset of D. Again, all the following
    limits are uniform w.r.t. x∈ H.
    
    The limits (<ref>) and (<ref>) stem again from (<ref>)-(<ref>) with i,j=1,⋯,p_0. Thus to conclude we only need to prove (<ref>).
    
    
    We remark that, by (<ref>), 
    it is not restrictive to assume <̣ dist(H, D). By (<ref>) and by Jensen's inequality, it is sufficient to prove
    
        lim sup_T-t→ 0^+1/T-t E_t,x[|(X_T-e^(T-t)Bx)_j|
        _{|X_T -x| <δ}]^2/2i+1 <∞,
    
    for any fixed i∈{ 1,⋯,r} and 
    j∈{∑_k=0^i-1p_k+1,⋯,d}. We prove (<ref>) in two different steps:
    
    Step 1. We prove that
    
        lim sup_T-t→ 0^+1/T-tE_t,x[|   (X_T-e^(T-t)Bx)_j|
        _{|X_T -x| <δ}]^2/2i+1
           ≤lim sup_T-t→ 0^+1/T-t(   ∫_t^T  E_t,x[ | ⟨ B^(j), X_s - e^(s-t)B x⟩|  _{|X_s -x| <δ}]   
         s   )^2/2i+1  ,
    
    for any i≥ 1 and j={p_0+1,⋯,d}. Here and further on, B^(j) denotes the j-th row of B. 
    Let ϕ^(t,x)_j ∈ C_0^∞([t,T_0[× D), j=p_0+1,⋯,d, be a family of
    functions with partial derivatives uniformly bounded w.r.t. (t,x)∈ [0,T_0[× H and such
    that
    
    
        ϕ_j(s,ξ)=ϕ^(t,x)_j(s,) = _j  -(e^(s-t)Bx)_j,    
        | - x|<.̣
    
    Note that we have
    
        ∂_s ϕ_j(s,) = - ⟨ B^(j), e^(s-t)Bx⟩
        ∂_ξ_jϕ_j(s,) = 1
        ∂_ξ_kϕ_j(s,) = 0    for  k≠ j
        ∂_ξ_k ξ_lϕ_j(s,)= 0
        ,     s∈ [t,T_0[, |ξ - x|<.̣
    
    
    
    
    
    
    
    
    
    By Remark <ref> we have
    
    
    
        ϕ_j(T,X_T)     =ϕ_j(t,X_t)+ ∫_t^T (∂_s + _s) ϕ_j(s,X_s)ds + M^t_T
        (by (<ref>)-(<ref>))    =
        ϕ_j(t,X_t)+ ∫_t^T⟨ B^(j), X_s -e^(s-t)Bx ⟩_{|X_s -x|<δ
        } ds + ∫_t^T (∂_s+_s) ϕ_j(s,X_s)_{|X_s -x|≥δ
        }  ds + M^t_T,
    
    where M^t is an ^t-martingale under P_t,x with 
    
        E_t,x[ |M^t_T|^2 ]= E_t,x[ ∫_t^T   ∑_k,l=1^p_0
         a_kl(s,X_s)  (∂_x_kϕ_j(s,X_s))(∂_x_lϕ_j(s,X_s))   s].
    
    Therefore, for any i≥ 1 and j=p_0+1,⋯,d one obtains
    
        E_t,x[|(X_T-e^(T-t)Bx)_j| _{|X_T -x| <δ}]^2/2i+1≤
        E_t,x[|ϕ_j(T,X_T) |]^2/2i+1≤
        ∑_k=1^3 I^2/2i+1_k(t,x;T) ,
    
    
    
    where
    
        I_1(t,x;T)    = E_t,x[|   ∫_t^T (∂_s + _s)
        ϕ_j(s,X_s)_{|X_s -x|≥δ}  ds  |],      I_2(t,x;T) =
        E_t,x[|M^t_T  |], 
         I_3(t,x;T)    = E_t,x[|  ∫_t^T⟨
        B^(j), X_s -e^(s-t)Bx ⟩_{|X_s -x|<δ} s  |] ,
    
    where we used that ϕ_j(t,X_t) = 0P_t,x-almost surely.
    Since 
    the coefficients of  are locally bounded on [0,T_0[× D and ϕ_j∈ C_0^∞([t,T_0[× D), we obtain
    
        I_1(t,x;T)    ≤
        C  ∫_t^T E_t,x[  _{|X_s -x|≥δ}_{X_s∈supp (ϕ_j)}]   s  =  C  ∫_t^T (s-t)^2i+1/2∫_(| -x|≥δ) ∩supp(ϕ_j) 
          p(t,x;s,d)/(s-t)^2i+1/2 s
          (by (<ref>) with m=2i+1/2)   ≤  C   ∫_t^T (s-t)^2i+1/2 s ≤ C (T-t)^2i+1/2+1,
    
    for any i≥ 1, 
    which proves
    
        lim_T-t→ 0^+1/T-t  I^2/2i+1_1(t,x;T) = 0,     i≥ 1.
    
    Similarly, Jensen's inequality 
    yields 
    
        I_2(t,x;T)   ≤ E_t,x[|M^t_T  |^2]^1/2
        (by (<ref>) combined with (<ref>))   =E_t,x[
         ∫_t^T  _{|X_s -x|≥δ}∑_k,l=1^p_0
         a_kl(s,X_s)  (∂_x_kϕ_j(s,X_s))(∂_x_lϕ_j(s,X_s))
         s ]^1/2(by using again the local boundedness of a_kl on
        [0,T_0[× D and ϕ_j∈ C_0^∞([0,T_0[× D))   ≤ C E_t,x[ ∫_t^T
        _{|X_s -x|≥δ}_{X_s∈supp (ϕ_j)} s
        ]^1/2,
    
    and by proceeding as we did to estimate I_1(t,x;T) it easily follows that
    
        lim_T-t→ 0^+1/T-t  I^2/2i+1_2(t,x;T) = 0,     i≥ 1.
    
    Finally, 
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        lim sup_T-t→ 0^+1/T-t  I^2/2i+1_3(t,x;T) ≤lim sup_T-t→ 0^+1/T-t(   ∫_t^T  E_t,x[ | ⟨ B^(j), X_s - e^(s-t)B x⟩|_{|X_s -x| <δ}]    
         s   )^2/2i+1
    
    yields (<ref>) for any i≥ 1.
    Step 2. We prove (<ref>) by induction on i. To start the inductive
    procedure, set i=0 and prove (<ref>) for any j∈{ 1,⋯,d  }. If
    j∈{1,…,p̅_0}, then (<ref>) stems from (<ref>) by applying
    Jensen's inequality. If j∈{p̅_0+1,⋯,d}, then (<ref>) follows trivially
    from (<ref>) by observing that | ⟨ B^(j), X_s - e^(s-t)B
    x⟩| _{|X_s -x| <δ} is uniformly bounded.
    
    Set now i̅∈{0,⋯,r-1}, assume (<ref>) true for i=i̅ and j∈{∑_k=0^i̅-1 p_k+1,⋯, d} and prove it true for i=i̅+1 and j∈{∑_k=0^i̅ p_k+1,⋯, d}. Now, by the block structure of B(<ref>),
    we obtain
    
        ∫_t^T E_t,x[ | ⟨ B^(j), X_s - e^(s-t)B x⟩|
        _{|X_s -x| <δ}]  s   ≤ C ∑_k=∑_k=0^i̅-1 p_k+1^d ∫_t^T
        E_t,x[ |( X_s - e^(s-t)B x)_k |  _{|X_s -x| <δ}]
         s (by inductive hypothesis)   ≤ C ∫_t^T (s-t)^2i̅+1/2 s
        ≤ C (T-t)^2(i̅+1)+1/2,
    
    which, combined with (<ref>), yields exactly (<ref>) for i=i̅+1 and concludes the proof.
    
    
    
    
    
      §.§.§ Proof of Proposition <ref>
    [Proof of Proposition <ref>] The proof of (<ref>) is identical to the proof of
    <cit.> as it is only based on the limits (<ref>) and
    (<ref>). The same goes for (<ref>), which is a corollary of (<ref>)-(<ref>) and whose proof coincides with the proof of <cit.>.
    
    
    Therefore, we only need to prove (<ref>). Setf∈ C_0,B^2,α(]0,T_0[× D)whose support is contained in the interior of]0,T_0[× H, withH⊂ Da compact subset. 
    
    We have_t,Tf(T,x)-f(t,x)/T-t=I_t,T,1(x)+I_t,T,2(x)where
    
        I_t,T,1(x)= ∫_Hp(t,x;T,d)f(T,)-f(t,x)/T-t,    
         I_t,T,2(x)=-f(t,x)/T-t∫_^d∖ Hp(t,x;T,d),
    
    First note that, by (<ref>), ifx∉Hit holds that
    
        I_t,T,1(x) + I_t,T,2(x) = I_t,T,1(x) ⟶ 0   as T-t→ 0^+,     unif. w.r.t.  x∈ℝ^d∖ H.
    
    We now consider the casex∈ H. By (<ref>) the termI_t,T,2(x)is negligible in the limit. As forI_t,T,1(x), 
    
    the intrinsic Taylor formula of Corollary <ref> yields
    
        f(T,)-f(t,x)=   (T-t) Y f(t,x)+∑_i=1^p_0(-e^(T-t)Bx)_i_x_if(t,x)
           +1/2∑_i,j=1^p_0(-e^(T-t)Bx)_i(-e^(T-t)Bx)_j_x_i,x_jf(t,x) + R(t,x;T,).
    
    withRsuch that
    
        |R(t,x;T,)|≤ c_H,B( |T-t|^1/2+|-e^(T-t)Bx|_B)^2+,
             (t,x),(T,ξ)∈  ]0,T_0[× H.
    
    Next we prove that
    
        lim_T-t→ 0^+∫_H
            |-e^(T-t)Bx|_B^2+p(t,x;T,d)/T-t = 0,    unif. w.r.t. x∈ H.
    
    For anyx∈ Hand>̣0suitably small we have
    
    
    
    
        ∫_H
          |-e^(T-t)Bx|_B^2+p(t,x;T,d)/T-t   ≤  C ∫_H∖ D_(t,x,T)p(t,x;T,d)/T-t
              + 
         ^̣α∫_D_(t,x,T)
         |-e^(T-t)Bx|_B^2p(t,x;T,d)/T-t,
    
    whereD_(t,x,T)={∈^d|
     |-e^(T-t)Bx|_B≤}̣andCis a positive
    constant. By (<ref>)
    and (<ref>) we obtain
    
        lim sup_T-t→ 0^+∫_H
         |-e^(T-t)Bx|_B^2+p(t,x;T,d)/T-t≤ C_1δ^α ,    unif. w.r.t. x∈ H.
    
    This proves (<ref>) since
    $̣ is arbitrary.
    
    Eventually, (<ref>) follows by plugging (<ref>) into (<ref>) and passing to the
    limit using (<ref>), (<ref>), (<ref>) and (<ref>)-(<ref>).
    This concludes the proof.
    
    
      §.§.§ Proof of Theorem <ref>
    
    We are now ready to prove the intrinsic Itô formula of Theorem <ref>.
    
    
    First observe that, integrating (<ref>), we get the identity
    
        (_t,Tf(T,·))(x)-f(t,x)=∫_t^T_t,(
          f(,·))(x)d,      T∈  ]t,T_0[.
    
    Note that the integrand in (<ref>) is bounded, as a function of τ, because of Assumption <ref> and since f∈ C^2,α_0,B(]0,T_0[× D) and _t,τ is a contraction. Consider now the process M^t defined through (<ref>). For ∈[t,T] we have
    
        E_t,x[M^t_T|^t_]   =M^t_+E_t,x[f(T,X_T)-f(,X_)-∫_^T
           f(u,X_u)du|^t_]
        
        =M^t_+Φ(,X_)
    
    where, by the Markov property,
    
        Φ(τ,x)   =E_,x[f(T,X_T)-f(,x)-∫_^T
         f(u,X_u)du]
        (by Fubini's theorem)   =(_,Tf(T,·))(x)-f(,x)-∫_^T_,u(
           f(u,·))(x)du
    
    which is 0 by (<ref>).
    
    
    
    
    
    
    
    
    
    
    To conclude we need to show that
    
        Y^t_T := (M^t_T)^2 - ∫_t^T  ∑_i,j=1^p_0    a_ij(s,X_s)∂_x_i f(s,X_s)∂_x_j f(s,X_s)  s,
    
    has null P_t,x-expectation. First note that
    
        ∑_i,j=1^p_0    a_ij(s,X_s)∂_x_i f(s,X_s)∂_x_j f(s,X_s)=  f^2(s,X_s) - 2 f(s,X_s)  f(s,X_s),
    
    which implies
    
        Y^t_T = Y^1,t_T +Y^2,t_T +Y^3,t_T,
    
    with
    
        Y^1,t_T     =   f^2(T,X_T)  -∫_t^T
         f^2(u,X_u)du + f^2(t,X_t) ,
        
        Y^2,t_T     = 2  f(t,X_t) ( ∫_t^T
         f(u,X_u)du  -  f(T,X_T)  )   ,
        
        Y^3,t_T     = - 2 ∫_t^T (    f(T,X_T)  -f(u,X_u)    )  
         f(u,X_u)du  +( ∫_t^T
         f(u,X_u)du  )^2     .
    
    Now, by applying the first part of Theorem <ref> to f^2 and f respectively, it is clear that Y^1,t and Y^2,t are martingales with null P_t,x-expectation. Finally, the identity
    
        ( ∫_t^T
         f(u,X_u)du  )^2 = 2  ∫_t^T( ∫_u^T 
         f(s,X_s) d s ) 
         f(u,X_u) d u
    
    along with (<ref>) yields
    
        Y^3,t_T    =  - 2 ∫_t^T (    M^t_T  -M^t_u    )  
         f(u,X_u)du,
    
    which 
    shows that Y^3,t has null P_t,x-expectation and thus concludes the proof.
    
    
    
    § LOCAL DENSITIES
    
    
    
    
    
    This section is devoted to the proof of Theorem <ref>. We will adapt and customize a localization procedure first introduced in <cit.>. 
    
    
    
    From now on, throughout the rest of this section, we will assume that the coefficients a_ij,a_i satisfy Assumption <ref> for some N, α, M.
    
    
    
     §.§ Proof of Theorem <ref>, Part a),b)
    
    
    We start by observing that the nature of Theorem <ref> a),b) is strictly local for what
    concerns the existence of the transition density and its regularity w.r.t. the forward
    space-variable. In other words, it is enough to prove it for (t,x;T,ξ) defined for any 0<
    t<T<T_0 and x∈^d, ξ∈ D' and for any sub-domain D' ⋐ D. 
    Therefore, it is not restrictive to assume that there exists a family
    _ij  ,_i:]0,T_0[×^d→, i,j=1,⋯,p_0, such that (_ij ,_i)
    coincide with (a_ij  ,a_i) on ]0,T_0[× D and:
    
    
    
    
    
    
    
     
      (ĩ)_ij  ,_i∈ C^N,α_B(]0,T_0[×^d) for any i,j=1,…,p_0, with all the (Lie) derivatives bounded by M;
    
      (ĩĩ) the following coercivity condition holds on ^d
    
        M^-1 ||^2 ≤∑_i,j=1^p_0
         _ij(t,x)_i_j≤ M  ||^2 ,     t∈]0,T_0[, x∈^d, ∈ℝ^p_0.
    
    Let us denote by _t the operator defined as
    
        _t:= 1/2∑_i,j=1^p_0_ij(t,x)_x_ix_j+∑_i=1^p_0_i(t,x)_x_i + ⟨ B x, ∇_x ⟩     t∈ [0,T_0[, x∈^d,
    
    where B is as in (<ref>). We consider an auxiliary ^d-valued continuous strong Markov
    process =(_t)_t∈[0,T_0[ defined on a space
    
    (,,(_T^t)_0≤ t≤ T< T_0,(_t,x)_0≤ t<
    T_0,x∈^d),
    
    with transition probability function =(t,x;T,d), such that  is an
    _t-global diffusion on ^d in the sense of Definition <ref>; in
    Appendix <ref> we briefly recall the standard construction of such .
    
    In particular, it will result in (t,x;T,d) having a density (t,x;T,ξ), which coincides with the fundamental solution of the operator
    
        = _t + ∂_t = 1/2∑_i,j=1^p_0_ij(t,x)_x_ix_j+∑_i=1^p_0_i(t,x)_x_i + Y,     t∈ [0,T_0[, x∈ℝ^d.
    
    In Lemma <ref> in Appendix <ref>, some previous results are reported
    about the existence of , its regularity and some sharp Gaussian upper bounds for  and its
    derivatives.
    
    
    
    
    
    
    
    
    
    
    
    
    For any x∈^d, t<T and ∈  ]0,1[, we set the cylinder
     
        H_(t,x;T) :=  ]t,T[× S_(x),     S_(x) := B_1(x- e_1) ∩ B_1(x+ e_1),
    
    where B_r(x)⊂^d is the Euclidean (open) ball with radius r centered at x and
    e_1=(1,0,⋯,0) is the first vector of the canonical basis of ^d. We define
    the lateral boundary and parabolic boundary of H_(t,x;T), respectively, as
     
        ∂_Σ H_(t,x;T) :=  ]t,T[ ×∂ S_ (t,x),    ∂_P H_(t,x;T) :=  ∂_Σ H_(t,x;T)∪({ T }×S_(x)).
    
    
    
    
    
    
    We also denote by G=G(t,x;T,ξ) the Green function of (∂_t+_t) for
    H_(0,x_0;T_0), which is defined for any 0< t<T<T_0 and
    x,ξ∈S_(x_0) and enjoys the properties listed in Lemma <ref>. In
    the latter, we report some preliminary existence (and uniqueness) and regularity results for the
    solution of the Cauchy-Dirichlet problem on H_(0,x_0;T_0) and in particular for the Green
    function G.
    
    
    Roughly speaking, the following result shows that, prior to the exit time from S_(x_0),
    X and X̃ have the same law whose density coincides with the Green function G. The
    proof is based on the crucial fact that the Itô formula (<ref>) is valid for functions that
    are C^2 in the intrinsic sense, i.e. C^2_B, and not only for functions the are C^2 in
    the Euclidean sense.
    
    
    
    
    Let x_0∈ D and ∈  ]0,1[ such that S_(x_0)⊂ D. Then, for any
    (t,x)∈ H_(0,x_0;T_0) we have
    
        P_t,x(X_T∈ A ,τ^(t) > T ) = ∫_A G(t,x;T,ξ) ξ =
        _t,x(_T∈ A , ^(t) > T ) ,      T∈  ]t,T_0[,  
        A∈ℬ(S_(x_0)),
    
    where τ^(t) and ^(t) are the t-stopping times defined, respectively, as
    
        τ^(t) := inf{ s≥ t : X_s ∉ S_(x_0)   },      
        
        ^(t) := inf{ s≥ t : X̃_s ∉ S_(x_0)   }.
    
    Before to prove Lemma <ref> we want to stress the following
    
    By (<ref>) we obtain
    
        ∫_A G(t,x;T,ξ) ξ≤_t,x(_T∈ A  ) = ∫_A(t,x;T,ξ) ξ ,    
        A∈ℬ(S_(x_0)),
    
    which implies
    
        G(t,x;T,ξ)≤ (t,x;T,ξ),       0< t<T<T_0,    x,ξ∈ S_(x_0).
    
    
    
    
    
    
    
    
    
    Throughout the proof we will set τ:=τ^(t) to shorten notation.
    Note that (<ref>) is equivalent to
    
        E_t,x[ϕ(X_T) _τ > T]= ∫_S_(x_0) G(t,x;T,ξ)  φ(ξ)  ξ = Ẽ_t,x[ϕ(_T) _ > T] ,     
        ϕ∈ C^∞_0( S_(x_0) ).
    
    Denote by f the unique solution in 
    
    
    
     
     
    
    
    
    C^N+2,α_B(H_(0,x_0;T))  ∩  C((H_∪∂_PH_)(t,x_0;T))
    (see Lemma <ref>) of
    
        f=0       on  H_(0,x_0;T), 
        
            f=0    on ∂_Σ H_(0,x_0;T),
        
            f(T,·) = φ   on  S_(x_0)
    
    which is given by 
        f(t,x) = ∫_S_(x_0) G(t,x;T,ξ)  φ(ξ)  ξ.
     Now, by
    Corollary <ref> combined with Optional Sampling Theorem, the process M^t_·∧τ with M^t as in (<ref>) and f as in (<ref>), is an
    ^t-martingale under P_t,x: we notice explicitly that, even if f is not defined on
    [0,T_0[× D as required by Theorem <ref>, a standard extension-truncation argument can
    be employed. 
    
    
    Thus
    
        E_t,x[ϕ(X_T) _T<τ]= E_t,x[ f(T∧τ,X_T∧τ)   ]=f(t,x).
    
    On the other hand, since = on ]0,T_0[× D, 
    
     is also an -local diffusion on 
    D and thus 
    
    it holds
    
        Ẽ_t,x[ϕ(_T) _T<]= Ẽ_t,x[ f(T∧,X_T∧)   ]=f(t,x),
     which proves (<ref>) and
    concludes the proof.
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
     Fix (t,x)∈  ]0,T_0[×^d and
    x_0∈ D and let ∈  ]0,1[ such that S_(x_0)⊆ D. 
    Let also U and V be two non-empty open subsets such that x_0∈
    U⋐V⋐S_(x_0).
    
    
    
    
    
    
    
    
    
    Define now τ^(t)_0≡ t and the families (σ^(t)_n)_n∈ and
    (τ^(t)_n)_n∈ through the following recursion:
     
        σ^(t)_n:= inf{  s≥τ^(t)_n-1 : X_s∈V},
    
    and τ^(t)_n: = τ^(^(t)_n) according to notation (<ref>), which is
      
        τ^(t)_n= inf{  s≥σ^(t)_n : X_s∉ S_(x_0) }.
    
    Hereafter, whenever it is clear from the context, we will drop the suffix (t) in τ^(t)_n,^(t)_n to ease the notation.
    
    
    Part a): note that for any T∈  ]t,T_0[ the event X_T∈
    U is included in the disjoint union ∪_n∈(σ^(t)_n <T < τ^(t)_n ).
    Therefore, for any A∈ℬ(U) one obtains
    
        p(t,x;T,A)    = ∑_n=1^∞ P_t,x(  X_T∈ A,  σ_n <T < τ_n ) =
        ∑_n=1^∞ P_t,x(  X_T∈ A,  σ_n <T < τ^(_n)) 
            =
        ∑_n=1^∞ E_t,x[ P_t,x( X_T∈ A,  σ_n <T < τ^(_n)|
        __n)] 
           =   ∑_n=1^∞ E_t,x[  P_t,x( X_T∈ A,  T <
        τ^(_n)| __n)  _σ_n <T] (by the strong Markov
        property)
            = ∑_n=1^∞ E_t,x[  
        P_s,y(X_T∈ A ,τ^(s) > T )|_s=σ_n,y=X_σ_n_σ_n
        <T ]
        
        
        = ∑_n=1^∞ E_t,x[  ∫_A(_n,X__n;T,ξ) ξ _σ_n <T ] ,
    
    where we used (<ref>) in the last equality. Our derivation of (<ref>) follows
    closely the original argument by <cit.> even if here we go a step further using
    the representation in terms of the Green kernel (<ref>), which is crucial in the
    subsequent study of the regularity properties of the local density of X.
    
    From (<ref>) and since x_0 is arbitrary, it follows that p(t,x;T,·) is
    absolutely continuous w.r.t. the Lebesgue measure on D and therefore admits a density
    (t,x;T,ξ). Moreover, for any x_0∈ D and ∈  ]0,1[ such that
    S_(x_0)⊂ D, we have the local representation
    
        (t,x;T,ξ) =  ∑_n=1^∞  E_t,x[  
         (_n,X__n;T,ξ) _σ_n <T 
         ],      T∈  ]t,T_0[, ξ∈ S_(x_0).
    
    Assume now that there exists C_1>0, independent of t, x and T, such that
    
        ∑_n=1^∞ P_t,x(σ^(t)_n <T) ≤ C_1,      T∈  ]t,T_0[.
    
    Then, by the continuity of G(t,x;T,·) on S_(x_0) combined with
    (<ref>) and the estimates (<ref>), it follows that
    (t,x;T,·) is continuous and bounded on S_(x_0), uniformly w.r.t. x∈^d.
    Therefore, to conclude the proof of Part a) we only need to prove (<ref>).
    
    
    
    
    
    
    
    
    
    
    
    Start by observing that
    
        ∑_n=2^∞ P_t,x( σ_n <T  ) ≤∑_n=1^∞ P_t,x( τ_n <T ),
    
    and that, by classical maximal estimates (e.g. <cit.>, p. 296), it holds that
    
        P_s,y( τ^(s) < T  ) ≤ C e^-1/C(T-s) ,        0<s<T<T_0,    y∈∂ V,
    
    where C>0 only depends on T_0 and _t, but not on s,T and y.
    
    Therefore, for any n≥ 1 we have
    
        P_t,x( τ_n < T  )    = E_t,x[ E_t,x[  _τ^(_n) <T  | __n
        ] ] = E_t,x[ E_s,y[  _τ^(s) <T  ]|_s=_n,y=X__n]
        (by (<ref>))   ≤ C e^-1/C(T-t) P_t,x( σ_n
        < T  )≤ C e^-1/C(T-t) P_t,x( τ_n-1 < T  ),
    
    which 
    yields P_t,x( τ_n < T  )≤(C e^-1/C(T-t))^n. 
    This combined with (<ref>) proves (<ref>) for T-t≤ T^* and for a positive T^* suitably small only dependent on T_0 and _t. To prove (<ref>) for a generic T∈  ]t,T_0[ consider a partition t=t_0<t_1<⋯<t_N = T, such that t_k+1-t_k< T^*. 
    Define i_k:=inf{ n∈ : ^(t)_n ≥ t_k }. We first observe that
    
        ∑_n=1^∞_t_k≤σ^(t)_n <t_k+1    =∑_n=i_k^∞_σ^(t)_n <t_k+1 = ∑_m=0^∞_σ^(t)_i_k+m <t_k+1
        (since we have  ^(t)_i_k∈{^(t_k)_1, ^(t_k)_2} and thus, by induction, ^(t)_i_k+m≥^(t_k)_1+m)
        
        
           ≤∑_m=1^∞_σ^(t_k)_m <t_k+1,      k=0,⋯,N-1.
    
    Hence
    
        ∑_n=1^∞ P_t,x(σ^(t)_n <T)
             =∑_k=0^N-1∑_n=1^∞  P_t,x(t_k≤σ^(t)_n <t_k+1)  ≤∑_k=0^N-1∑_m=1^∞  P_t,x(σ^(t_k)_m <t_k+1)
            = ∑_k=0^N-1∑_m=1^∞ E_t,x[ P_t,x(σ^(t_k)_m <t_k+1| ^t_t_k) ] =  ∑_k=0^N-1 E_t,x[ ∑_m=1^∞ P_t_k,y(σ^(t_k)_n <t_k+1)|_y=X_t_k],
    
      which proves (<ref>).
    Part b):
    we 
    only prove the statement for N=2, the general case being analogous. 
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    By combining the representation of G in <cit.>, p. 36, with the internal estimates in the same reference that are reported in Theorem <ref> below, it follows that
    
        |∂_ξ_i(s,y;T,ξ)| + |∂_ξ_iξ_j(s,y;T,ξ)| + |Y_T,ξ(s,y;T,ξ)| ≤ C_2,       0< s < T<T_0,   y∈∂ V,   ξ∈ U,
    
    for any i,j=1,⋯,p_0. 
    
    
    
    
    
    
    
     
    
     
    
    
    
    
    
    
    
    
    
    This and (<ref>) allow us to employ bounded convergence theorem and differentiate twice under the sign of expectation 
    the right-hand side of (<ref>) w.r.t. ξ. For any T∈  ]t,T_0[, ξ∈ U we obtain:
    
        ∂_ξ_i(t,x;T,ξ)    =  ∑_n=1^∞  E_t,x[  
        ∂_ξ_i(_n,X__n;T,ξ) _σ_n <T 
         ],
         
        ∂_ξ_iξ_j(t,x;T,ξ)    =  ∑_n=1^∞  E_t,x[  
        ∂_ξ_iξ_j(_n,X__n;T,ξ) _σ_n <T 
         ],
    
    for 
    i,j=1,⋯,p_0. As for Y (t,x;·,·), 
    we have
    
        Y_T,ξ(t,x;T,ξ)    = lim_h→ 0(t,x;T+h,e^hBξ) - (t,x;T,ξ)/h
            = lim_h→ 01/h∑_n=1^∞( E_t,x[  
        (_n,X__n;T+h,e^hBξ) _σ_n <T+h 
         ] -
         E_t,x[  
        (_n,X__n;T,ξ) _σ_n <T 
         ] ) 
            = lim_h→ 01/h∑_n=1^∞  E_t,x[  
        (_n,X__n;T+h,e^hBξ) _T≤σ_n <T+h _=:I_1,n(h)
         ] 
              + lim_h→ 01/h∑_n=1^∞  E_t,x[  
        (  (_n,X__n;T+h,e^hBξ) - (_n,X__n;T,ξ) )
        _σ_n <T _I_2,n(h)
         ].
    
    Remark <ref> together with 
    (<ref>) on one hand, and mean value theorem on the other, yield
    
        1/hI_1,n(h)    ≤ C _T≤σ_n <T+h , 
        1/hI_2,n(h)    = Y_T,ξ(_n,X__n;T+h̃,e^h̃ Bξ) _σ_n <T ,   with    |h̃|≤ h.
    
    Here the low index in Y_T,ξ is meant to stress that Y is computed with respect to the variables (T,ξ).
    By (<ref>) and (<ref>) we can apply bounded convergence theorem and obtain
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
        Y_T,ξ(t,x;T,ξ) =  ∑_n=1^∞  E_t,x[  _σ_n <T 
        Y_T,ξ(_n,X__n;T,ξ) 
         ],      T∈  ]t,T_0[,  ξ∈ U.
    
    Proceeding analogously, by employing again the Schauder estimates 
    reported in Theorem <ref>, one also proves
    
    
        ∂__i(t,x;·,·)∈ C^1+α_Y(]t,T_0[ × U),   ∂__iξ_j(t,x;·,·),Y(t,x;·,·) ∈ C^0,α_B(]t,T_0[ × U),      i,j=1,⋯,p_0,
    
    which is (t,x;·,·)∈ C^2,α_B(]t,T_0[ × U).
    
    Eventually, (t,x;·,·)∈ C^2,α_B(]t,T_0[ × D) follows from the fact that x_0 is arbitrary. The fact that (t,x;·,·) solves (<ref>) is now a straightforward consequence of Remark <ref>, by integrating by parts the left-hand side of (<ref>) and since f is arbitrary.
    
    
    
    
    
    
     §.§ Proof of Theorem <ref>, Part c)
    
    Hereafter throughout this section we asssume 
    Assumption <ref> to be in force as well.
    
    
    For any T∈  ]0,T_0[ and any ϕ bounded and Borel-measurable on ^d (in short
    ϕ∈ mℬ_b), let u_φ,T:]0,T[× D → be the function defined as
    u_φ,T(t,x) :=  (_t,Tφ)(x).
    
    Let 
    T∈  ]0,T_0[ and φ∈ mℬ_b such that u_φ,T∈ C( ]0,T[× D ).
    
    Then, u_φ,T
    ∈ C^N+2,α_B( ]0,T[× D )
    and solves the backward Kolmogorov equation (<ref>).
    
    
    
    
    Again, the proof is based on the crucial fact that the Itô formula (<ref>) is valid for
    functions that are C^2 in the intrinsic sense, i.e. C^2_B, and not only for functions the
    are C^2 in the Euclidean sense.
    
    Let δ>0, x_0∈ D, ∈  ]0,1[ such that S_(x_0)⊂ D and
    denote by f the unique solution in C^N+2,α_B(H_(0,x_0;T-δ))  ∩ 
    C((H_∪∂_PH_)(0,x_0;T-δ)) (see Lemma <ref>) of
    
        f=0       on  H_(0,x_0;T-δ), 
        
            f=u_ϕ,T   on ∂_P H_(0,x_0;T-δ).
    
    For any t∈  ]0,T-δ[ let now τ=τ^(t) be the t-stopping time as defined in
    (<ref>). By Theorem <ref> combined with Optional Sampling Theorem, the
    process M^t_·∧τ, with M^t as in (<ref>) and f as in
    (<ref>), is an ^t-martingale under P_t,x. Thus
    
        f(t,x)   = E_t,x[ u_ϕ,T((T-)̣∧τ,X_(T-)̣∧τ)   ]
        = E_t,x[ E_s,y[ϕ(X_T) ]|_s=(T-)̣∧τ , y=X_(T-)̣∧τ]
        (by Strong Markov property)   = E_t,x[   E_t,x[  ϕ(X_T) | _(T-)̣∧τ ]   ] = E_t,x[  ϕ(X_T) ] = u_ϕ,T(t,x).
    
    Since x_0 and $̣ are arbitrary, thenu_φ,T
    ∈ C^N+2,α_B( ]0,T[× D )and solves 
    (<ref>).
    
    For any T∈  ]0,T_0[ and any φ∈ mℬ_b, u_φ,T∈ C( ]0,T[×
    D ).
    
    First note that, 
    by Assumption <ref> combined with Lemma <ref>, u_ψ,T∈ C^N+2,α_B( ]0,T[× D ) and solves the backward Kolmogorov equation (<ref>), for any ψ∈ C_b(^d). 
    Thus, by the internal Schauder estimates 
    reported in Theorem <ref>,
    for any 
    bounded domain 
    V⋐  ]0,T[× D  we have
    
        u_ψ,T_C^0,α_B(V)≤ C sup_]0,T[× D  |u_ψ,T| ≤ C 
         ψ_∞,
    
    where C is a positive constant independent of ψ. In particular, by Theorem 
    <ref>, for any (t_0,x_0)∈  ]0,T[× D there exists a neighborhood U_(t_0,x_0)
    that
    
        | u_ψ,T(t,x) - u_ψ,T(t',x') | ≤ C ϕ_∞ (t',x')^-1∘ (t,x) _B ,      (t,x),(t',x')∈ U_(t_0,x_0),
    
    for any ψ∈ C_b(^d) such that ψ_∞≤ϕ_∞. 
    
    
    
    
    Therefore, in order to prove that u_ϕ,T is continuous in (t_0,x_0) it suffices to prove that, for any 
    (t,x)
    ∈ U_(t_0,x_0), there exist a sequence 
    of functions ψ_n∈ C_b(^d) with ψ_n _∞≤ϕ_∞ such that
    
        u_ψ_n,T(t,x) ⟶ u_ϕ,T(t,x),    u_ψ_n,T(t_0,x_0) ⟶ u_ϕ,T(t_0,x_0)     as  n→∞.
    
    To see this, let μ be the measure on ℬ(^d) defined as
    
        μ( z) = p(t,x;T, z) + p(t_0,x_0;T, z).
    
    Note that we have
    
        p(t,x;T, z),  p(t_0,x_0;T, z) ≪μ ( z).
    
    Moreover, μ is a finite measure and thus, by Proposition 3.16 in <cit.>, there exists a sequence of ψ_n∈ C_b(^d) with ψ_n _∞≤ϕ_∞ such that
    
        ψ_n -ϕ_L^1(ℬ(^d),μ)⟶ 0    as n→∞.
    
    Therefore, by (<ref>), ψ_n→ϕ both p(t,x;T, z)- and
    p(t_0,x_0;T, z)-almost everywhere. Thus bounded convergence theorem yields
    (<ref>) and concludes the proof.
    
    
    
    For any T∈  ]0,T_0[ and any ϕ∈ mℬ_b, u_φ,T
    ∈ C^N+2,α_B( ]0,T[× D )
    and solves the backward Kolmogorov equation (<ref>).
    
    It is an immediate consequence of Lemmas <ref> and <ref>.
    
    For any 0< t <T<T_0 and ξ∈ D,
    
    the function (t,·;T,ξ)∈ mℬ_b, and 
    
        (t,x;T,ξ) =  E_t,x[(s,X_s;T,ξ)] = (_t,s(s,·;T,ξ) )(x) 
        ,    
        t<s<T,  
        x∈^d
        .
    
    The boundedness is already contained in Part a) of Theorem <ref>.
    
     
    We first 
    prove the measurability of (t,·;T,ξ).
    Let (ϕ_n)_n∈ a family of functions in C_0(D) such that ϕ_n→_̣ξ, i.e.
    
        ∫_D f(y) ϕ_n(y)  y → f(ξ)  as n→∞ ,      f∈ C(D).
    
    Therefore, since Γ(t,x;T,·)∈ C(D) (again by Part a) of Theorem <ref>), we have 
    
    
    
    
    
        Γ(t,x;T,ξ) = lim_n→∞∫_Dϕ_n(y) Γ(t,x;T,y)   y = lim_n→∞ u_ϕ_n,T(t,x),      x∈^d,
    
    
    and since u_ϕ_n,T(t,·) is continuous (by Assumption <ref>), 
    (t,·;T,ξ) is measurable as it is the pointwise limit of a sequence of measurable functions.
    
    
    
    
    
    
    We now prove (<ref>). 
    
    By (<ref>) along with Markov property, it holds that  
    
    
    
    
    
        Γ(t,x;T,ξ)     =lim_n→∞ E_t,x[ϕ_n(X_T)] =lim_n→∞ E_t,x[ E_t,x[ϕ_n(X_T)|^t_s] ] =lim_n→∞ E_t,x[ E_s,y[ϕ_n(X_T)]|_y=X_s]  
            = lim_n→∞ E_t,x[ u_ϕ_n,T(s,X_s) ]=  E_t,x[(s,X_s;T,ξ)] ,
    
    where, in the last equality, we employed again (<ref>) with t=s and x=X_s along with bounded convergence theorem (it is not restrictive to assume ϕ_n_L^1(D) = 1 and thus, since (s,x;T,·) is locally bounded on D uniformly w.r.t. x∈^d, u_ϕ_n,T(s,X_s) bounded uniformly w.r.t. n). 
     It is a straightforward consequence of Proposition <ref> and Lemma <ref>.
    
    
    § PRELIMINARY PDE RESULTS
    
    
    In this appendix we collect some useful results about the operatorsin (<ref>) andin (<ref>), under the Assumptions <ref> and <ref> and with the coefficients_ij  ,_i:]0,T_0[×^d→,i,j=1,⋯,p_0, satisfying the conditions (ĩ)-(ĩĩ) at the beginning of Section <ref>. 
    There exists a unique fundamental solution 
    for , namely a continuous
    non-negative function =(t,x;T,ξ) defined for any 0< t<T<T_0 and x,ξ∈^d enjoying the following properties:
    
      a) for any (T,ξ)∈  ]0,T_0[×^d, the function (·,·;T, ξ)∈
    C^N+2,α_B(]0,T[×^d) and is a solution of
    
        u=0       on  ]0,T[×^d, 
        
             u(T,·) = δ_ξ,
    
    where the terminal condition is in the distributional sense, i.e.
      lim_
      t→ T^-∫_^d(t,x;T,ξ)ϕ(x)dx=ϕ(ξ),    ϕ∈ C_0(^d);
      b) if N≥ 2, then for any (t,x)∈  ]0,T_0[×^d, the function (t,x;·,·)∈ C^N,α_B(]t,T_0[×^d) and is  solution to
    
        ^* u=0       on  ]t,T_0[×^d, 
        
             u(t,·) = δ_x,
    
    where ^* is the formal adjoint of .
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
      c) for any α,β∈_0^d and p,q∈_0 with 2p+[α]_B≤ N+2 and 2q + [β]_B≤
    N _[2,∞[(N), we have
    
        |Y^p_t,x∂_x^α(t,x;T,ξ) |    ≤ C (T-t)^-
        2p+[α]_B/2Γ̅_M (t,x;T,ξ),
        
        | Y^q_T,ξ∂_ξ^β(t,x;T,ξ) |    ≤ C (T-t)^-
        2q+[β]_B/2Γ̅_M (t,x;T,ξ),
    
    for any 0< t<T<T_0 and x,ξ∈^d, where C is a positive constant that only depends on
    B,M,T_0 and N and where
    
        Γ̅_M (t,x;T,ξ) = 1/√((2π)^d (T-t))exp(-1/2⟨^-1(T-t) (y - e^(T-t)Bx),
            (y -e^(T-t)Bx)⟩),
    
    with
    
        (s)= ∫_0^s e^r B[            M I_p_0     0_p_0× (d-p_0);     0_(d-p_0)× p_0 0_(d-p_0)× (d-p_0) ]e^r B^* d r.
    
    See Theorem 1.4 in <cit.>.
    
    
    In the next result, the setsS_(x_0)⊆are as defined in Notation
    <ref>.
    
    Let x_0∈ D and ∈  ]0,1[ such that S_(x_0)⊆ D. Then, for
    any T∈  ]0,T_0[ and h∈ C(∂_PH_(0,x_0;T)), there exists a unique
    solution in C^N+2,α_B(H_(0,x_0;T))  ∩ 
    C((H_∪∂_PH_)(0,x_0;T)) to
     
        f=0,       on  H_(0,x_0;T), 
        
            f=h,    on ∂_P H_(0,x_0;T).
    
    Moreover, if h|_∂_ΣH_(0,x_0;T)≡ 0, then the following representation holds:
    
        f(t,x) = ∫_S_(x_0) G(t,x;T,ξ)  h(T,ξ)  ξ,      (t,x)∈(H_∪∂_PH_)(0,x_0;T),
    
    where G denotes the Green function of  for H_(0,x_0;T_0), namely a continuous
    non-negative function G(t,x;T,ξ) defined for any 0< t<T<T_0 and x,ξ∈S_(x_0) enjoying the following properties:
    
      a) for any (T,ξ)∈ H_(0,x_0;T_0),
    the function G(·,·;T,ξ)∈ C^N+2,α_B(H_(0,x_0;T))  ∩ 
    C((H_∪∂_ΣH_)(0,x_0;T)) and solves
     
        f=0       on 
           H_(0,x_0;T)
          , 
        
            f= 0    on ∂_Σ H_(0,x_0;T), 
        
               f(T,·) = δ_ξ   on  S_(x_0);
    
      b) if N≥ 2, then G is also the Green function of the
    formal adjoint ^* for H_(0,x_0;T_0). In particular, for any
    (t,x)∈ H_(0,x_0;T_0), the function G(t,x;·,·)∈
     C^N,α_B(H_(t,x_0;T_0))  ∩  C((H_∪∂_ΣH_)(t,x_0;T_0)) and solves
     
        ^* f=0       on H_(t,x_0;T_0), 
        
           f= 0    on ∂_Σ H_(t,x_0;T_0), 
        
               f(t,·) = δ_x   on  S_(x_0).
    
    See <cit.> and <cit.>.
    
    Let Q⊂ be a bounded domain, 
    and u be a bounded function in C^2+α_B (Q) such that u = 0 on Q. Then, for any domain Q_0
    ⋐ Q, there exists c>0 such that
    
        sup_Q_0( ∑_i=1^p_0 |∂_x_i u |  + ∑_i,j=1^p_0 |∂_x_i x_j u | + |Y u| ) + u_C^0,_B(Q_0)+u_C^1,_B(Q_0)+u_C^2,_B(Q_0)≤ c sup_Q u .
    
    It is a particular case of <cit.>.
    
    
    § FUNDAMENTAL SOLUTIONS AND MARKOV PROCESSES
    
    We recall some basic notions about Markov processes as given in <cit.> and
    <cit.>. A transition distribution is a kernelp(t,x;T,·)that
    satisfies:
    
      1)p(t,x;T,·) is a probability measure on (ℝ^d,ℬ(ℝ^d)) for all 0≤ t<T< T_0 and x∈^d;
    
      2)p(t,·;T,A) is ℬ(ℝ^d)-measurable for any 0≤ t<T< T_0 and A∈ℬ(ℝ^d);
    
      3) if 0≤ t<s<T< T_0, x∈ℝ^d and A∈ℬ(ℝ^d), the following Chapman-Kolmogorov identity holds:
    
        p(t,x;T,A)=∫_ℝ^dp(s,ξ;T,A)p(t,x;s,dξ).
    
    A Markov process with transition distribution p is a stochastic processX=(X_t)_0≤
    t< T_0defined on the quartet(Ø,,(_T^t)_0≤ t≤ T< T_0,(P_t,x)_0≤ t<
    T_0,x∈^d)such that:
    
      (a)(Ø,) is a measurable space and (_T^t)_0≤ t≤ T< T_0 is a family of
    filtrations satisfying _T^t⊆_T'^t' for t'≤ t,T≤ T' and
    =^0_T_0 (i.e.  is the smallest σ-algebra containing all _T^t);
    
    
      (b)(X_T)_t≤ T<T_0 is adapted to ^t for any t∈[0,T_0[;
    
    
      (c) for any (t,x)∈[0,T_0[×, P_t,x is a probability measure on
    (Ø,^t_T_0) satisfying
    
        P_t,x(X_t =x)   = 1,   
        
        P_t,x(X_T ∈ A | ^t_s)   = p(s,X_s;T,A) ,     t≤ s < T< T_0,   A∈ℬ(ℝ^d).
    
    Theorem 2.2.2 in <cit.> guarantees that for any transition distributionpthere
    exists a Markov processX=(X_t)_t∈[0,T_0[havingpas transition distribution.
    
    A transition distribution can be defined from a differential operatorof the
    form (<ref>) satisfying conditions
    (ĩ)-(ĩĩ) at the beginning of Section <ref>.
    Indeed, ifΓ̃denotes the fundamental solution ofin Theorem <ref>
    then
    
        p̃(t,x;T,A):=∫_A Γ̃(t,x;T,ξ)dξ,     0≤ t<T<T_0, x∈ℝ^d,  A∈ℬ(ℝ^d)
    
    defines a transition distribution.
     In virtue of the properties ofΓin Theorem <ref>, ifp̃is as in
    (<ref>) then the associated Markov process admits a continuous version and
    is an _t-global diffusion on ^d in the sense of Definition <ref>. 
    This is a consequence of Kolmogorov continuity theorem (see, for instance, Theorems 2.1.6 and
    2.2.4 in <cit.>) and the estimate given in the following
    
    Let X̃=(X̃_t)_t∈[0,T_0[ be a Markov process with transition distribution p̃ in
    (<ref>). Then for any q≥ 1 there exists a positive constant C such that
    
        E_t,x[|X̃_T-X̃_s|^q]≤ C(1+|x|^q)|T-s|^q/2,    
          t≤ s<T< T_0, x∈^d.
     We recall the definition ofD_0in (<ref>) and notice that
    
        |z|=|D_0(ł)D_0(ł^-1)z|≤
         C ł|D_0(ł^-1)z|,     z∈^d, 0<ł<1.
    
    Now we have
    
        E_t,x[|X̃_T-X̃_s|^q]   =E_t,x[E_s,X̃_s[|X̃_T-X̃_s|^q]
         ]
         
         =∫_^d(t,x;s,)∫_^d(s,;T,y)|y-|^qdy d
           ≤ C∫_^dΓ̅_M(t,x;s,)∫_^dΓ̅_M(s,;T,y)(|y-e^(T-s)B|^q+
         |(e^(T-s)B-I)|^q)dy d(applying estimate (<ref>) with z=y-e^(T-s)B, ł=(T-s)^q/2 and
        by Proposition 3.5 in <cit.>, for some M'>M)   ≤ C (T-s)^q/2∫_^dΓ̅_M(t,x;s,)∫_^dΓ̅_M'(s,;T,y)dy
         d
           + C(T-s)^q∫_^dΓ̅_M(t,x;s,)∫_^d||^qΓ̅_M(s,;T,y)dy d
    
    that yields (<ref>).
    We finally observe that Theorem <ref> also implies thatX̃is a Feller process on D in the sense of Assumption <ref>, and as such it is a strong Markov process (see <cit.>, Corollary 2.6, p. 28).
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
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