On Cauchy-dirichlet Problem in Half-space for Linear Integro-differential Equations in Weighted H ¨ Older Spaces

We study the Cauchy-Dirichlet problem in half-space for linear par-abolic integro-differential equations. Sufficient conditions are derived under which the problem has a unique solution in weighted Hölder classes. The result can be used in the regularity analysis of certain functionals arising in the theory of Markov processes.

In many problems arising in the theory of Markov processes it is important to study the smoothness properties of the functionals v(s, x) = E τ s,x ∧T 0 f (t, X s,x t ) dt, where τ s,x is the first exit time of the process X s,x t from a domain D ⊆ R d and E denotes the mathematical expectation.If v is a sufficiently smooth function, it is a solution to Cauchy-Dirichlet problem where [u(t, x + c(t, x, z)) − u(t, x) − ∂ i u(t, x)c i (t, x, z)1 U1 (z)] π(t, x, dz), a ij (t, x) = (1/2)σ(t, x)σ * (t, x), 1 U1 is the indicator function of U 1 and the implicit summation convention over repeated indices is assumed.
The problem of such type (including the case of nonlinear equations) was considered by a number of authors (see e.g.[1], [4], [2], [5], [7] and references therein) in Sobolev and Hölder spaces under certain restrictive assumptions on the behavior of the function c(t, x, z).We show for D = {x ∈ R d : x d > 0} that these assumptions can be relaxed by considering the problem in a suitably chosen weighted spaces of functions that are Hölder continuous with respect to the space variable x (no regularity in t is assumed).These spaces are deterministic versions of the spaces used in [6] for the analysis of stochastic partial differential equations.A time-independent version of such spaces was considered in [8].In [2], the equation (1.3) was solved in similar weighted spaces of functions that are Hölder continuous not only in x but in t as well.Besides that, the conditions in [2] imposed on π(t, x, dz) and c(t, x, z) are rather restrictive: only finite number of jumps can occur outside of D in finite time, Theorem 3.9 of Chapter II in [2] covers only a bounded variation jump part.The results in [2] do not apply for the equation (1.2) which is obviously covered by our Theorem 1 (see Remark 1 below).We discuss more precisely the differences with [2] immediately after Theorem 1 in the next section.
The results of the paper can be extended to more general situations, including the case of random coefficients and additional stochastic terms, which correspond to the stochastic processes mentioned above in a random environment (see e. g. [6] for the case c = 0).The results can be also used in the analysis of non-linear equations arising in the optimal control theory of Markov processes (see e.g.[5] for the case of fully nonlinear integro-differential equations in Hölder spaces).
The main result of the paper is presented in Section 2 and proved in Section 3.

Notation and main result
Let R d be a d-dimensional Euclidean space with points x = (x 1 , . . ., x d ), Let B loc (H) be the space of locally bounded Borel functions on H, i.e.
If u is a function on H, we denote its partial derivatives as follows: is the gradient of u with respect to x and ∂ 2 u = (∂ 2 ij u) is the matrix of the second order partial derivatives of u with respect to x.
Let C m (H) be the class of all functions u ∈ B loc (H) such that u is m-times continuously differentiable in x and its derivatives Let C m ( H) be the class of all u ∈ C m (H) whose derivatives up to the order m have continuous extensions to H, the closure of H, and the norm where The summation convention that repeated indices indicate summation from 1 to d is followed here as it will throughout.The functions a ij , b i , r ∈ B(H), a ij (t, x) = a ji (t, x) for each (t, x) ∈ H and i, j = 1, . . ., d, r ≥ 0, π(t, x, dy) is a measure on Borel subsets of R d such that Throughout he paper C = (•, . . ., •) denotes constants depending only on quantities appearing in parentheses.In a given context the same letter will be used to denote different constants depending on the same set of arguments.
Let us consider the Cauchy-Dirichlet problem Let us introduce the following assumptions.
A1. (i) The functions a ij , b i , r, i, j = 1, . . .d, belong to the space C β ( H); (ii) There is a constant κ > 0 such that for each (t, x) ∈ H and ξ ∈ R d and for each t ∈ (0, T ) There exists a sequence of measurable subsets U n ⊆ U 1 , n ≥ 2, with the following properties: (i) There are positive constants ν(n), k(n) such that ν(n) → 0, as n → ∞, and for all x = (t, x) ∈ H Remark 1.Typical measures π(t, x, dz) arising in the theory of stable Markov processes are as follows: where α ∈ (0, 2), z ∈ U = R d .As can be easily seen, A2 and A3 are satisfied with 0 and the function p satisfies Hölder condition in x with exponent β uniformly with respect to t and y (in the case of (1.2), p(t, x, z) = 1).Assumptions A2 and A3 are also satisfied if the exponent α is a sufficiently smooth function α = α(t, x) in x, the case corresponding to the stable measures of varying order.
The main result of the paper is the following theorem.
Then for each f ∈ N β (H) there exists a unique solution u ∈ S2+β (H) to the problem (2.1).Moreover, there is a constant C not depending on u and f such that and for all s ∈ (0, T ) where In [2], the existence and uniqueness in weighted spaces of functions that are Hölder continuous in space and time variables was proved.Theorem 3.9 in Chapter II of [2] (see Section 1 of Chapter II in [2] as well) requires the following assumptions (i)-(v) for π and c: (i) The set U 1 = ∅, and there is a measure π(dz) on U 0 such that there is a measurable function j γ (z) and a constant K > 0 such that for all x = (t, x) ∈ H, we have |c(x, z)| ≤ j γ (z) and (iv) There exists a constant c 1 ∈ (0, 1] such that the determinant where I is d × d identity matrix; (v) There is a constant M such that for all (t, x), (s, ξ) ∈ H, For an application of our Theorem 1 in a similar case, we need the following condition instead: A. The set U 1 = ∅, and there is a constant K such that [2] without a proof that the case γ ∈ (1, 2] can be considered under additional and very technical condition (see (3.25) in Chapter II of [2]).If γ > 0, the jumps outside D are not allowed.Only finite number of them can occur in finite time when γ = 0.

Proof of Theorem 1
In order to prove Theorem 1 we need the following auxiliary statements.Lemma 1.For each ε > 0 there is a constant C ε such that for all u ∈ S 2+β (H) and i = 1, . . ., d This estimate, together with the well known inequality that for each ε > 0 there is a constant C ε such that for all u ∈ S 2+β (H) , d, yields the assertion of the lemma.Lemma 2. Let Assumptions A2 and A3 be satisfied.Then for each ε > 0 there is a constant C ε such that for all u ∈ S2+β (H) a) First, we shall prove that for each ε > 0 there is a constant C ε such that for all u ∈ S2+β (H) For x = (t, x) we have Let us estimate I 1 u(x).The inclusion z ∈ Γ(x, n) implies the inequalities Thus, according to Assumption 2, where ν(n) → 0, as n → ∞.
Let us estimate I 2 u(t, x).Since u(t, •) satisfies Lipschitz condition on R d uniformly with respect to t, we have Hence, according to Assumption 2 The estimates (3.2)-( 3.4), together with Lemma 1, enable us to conclude that for each γ > 0 there is a constant C γ such that By Assumption 2, ν(n) → 0, as n → ∞.Hence, choosing first a sufficiently large n ≥ 2 and then a sufficiently small γ > 0, we get assertion (3.1).
b) Let us prove that for each ε > 0 there is a constant C ε such that for all u ∈ S2+β (H) H .These estimates, together with assertion (3.1), enable us to conclude that in order to prove assertion (3.5)it is sufficient to consider the case |x − ξ| < ξ d ∧ δ 0 for arbitrary small δ 0 > 0, which implies that We have Obviously, Let us estimate J 1 u(t, x, ξ) .
The following statement is essentially a partial case of Theorem 2 in ( [6]) for stochastic partial differential equations.Theorem 2. Let λ > 0 and let the assumption A1 be satisfied.Then for each f ∈ N β (H) the Cauchy -Dirichlet problem , has a unique solution u ∈ S 2+β (H).Moreover, there is a constant C not depending on u, f and λ and a bounded function δ(λ) not depending on u and f such that where δ(λ) → 0, as λ → ∞.
Theorem 2 in [6] has a multiplier s 1/4 in (3.34), which appears because of the influence of stochastic terms in the problem.Analyzing the proof of Lemma 11 in [6] (the case g = 0, h = 0), we conclude that (3.34) holds with the multiplier s 1/2 .Using Lemma 2 and apriori estimate (2.2) we get estimate (2.3).
We finish the proof using apriori estimate (2.2) and applying a standard procedure of extension by a parameter.Let According to Theorem 2, T 0 is an onto map.By Theorem 5.2 in [3], all the T τ are onto maps.The theorem is proved.
1), be the class of all functions u ∈ C m ( H) such that Hence the derivative ∂ t u is defined at all Lebesgue points of the function (Lu + Iu + f )(•, x).