Estimates of perturbation series for kernels

For integral kernels on space-time we indicate a class of nonnegative Schr\"odinger perturbations which produce comparable integral kernels.


Introduction
Schrödinger operators ∆ + q were studied for the Laplacian ∆, e.g., in [9,8,14,19]. Local integral smallness of the function q, defined as a Kato-type condition ( [8,19]) played an important role in these considerations. Similar Schrödinger operators based on the fractional Laplacian ∆ α/2 were studied in [6,1,2] (see also [7]), with focus on comparability of the resulting Green functions. The corresponding estimates for general transition densities were then studied in [3] under the following integrability condition on q, t s X p(s, x, u, z)|q(u, z)|p(u, z, t, y)dzdu ≤ [η + β(t − s)]p(s, x, t, y), (1) where p is a finite jointly measurable transition density, β and η are fixed nonnegative numbers, while times s < t and states x, y are arbitrary. Given from (E × E) to [0, ∞] is a kernel on E, called the composition of K and L, and denoted KL. Composition of kernels is associative ( [10]). We write q ∈ E + if q : E → [0, ∞] and q is E-measurable. We will denote by the same symbol the kernel q(x, A) = q(x)1 A (x). Here 1 A is the indicator function of A. We let K n = (Kq) n K, n = 0, 1, . . .. Associativity yields the following. Lemma 1. K n = K n−1−m qK m for all n ∈ N and m = 0, 1, . . . , n − 1.
We will consider the perturbation of K by q, defined as the kernel Of course, K ≤K. In what follows we will prove upper bounds forK under additional conditions on K and K 1 = KqK.

Estimates for kernels on space-time
In what follows we consider a set X (the state space) with σ-algebra M, the real line R (the time) equipped with the Borel sets B R , and E = R × X (the space-time) with the product σ-algebra E = B R × M. We also fix q ∈ E + , a number η ∈ [0, ∞) and a function Q : R×R → [0, ∞) satisfying the following condition of super-additivity: Let K be a kernel on E. We will assume that K is a forward kernel, i.e.
Theorem 1 has two fine or pointwise variants, which we will state under suitable conditions. We fix a (nonnegative) σ-finite, non-atomic measure dt = µ(dt) on (R, B R ) and a function k(s, x, t, A) defined for s < t, x ∈ X, A ∈ M, such that (s, x, t) → k(s, x, t, A) ∈ [0, ∞) is jointly measurable. We will call k a transition kernel if it satisfies the Chapman-Kolmogorov conditions, see (26). For instance, if p is a transition probability, and we let k(s, x, t, A) = p s,t (x, A), then k is a transition kernel, provided it is jointly measurable. We let k 0 = k, and for n = 1, 2, . . ., we define Proof. If m = 0, then the equality (12) holds by the definition of k n . In particular, this proves our claim for n = 1. If n ≥ 1 is such that (12) holds for all m < n, then so for every m = 1, 2, . . . , n, we obtain We definek We will assume that for all or k 1 (s, x, t, dy) ≤ [η + Q(s, t)]k(s, x, t, dy). Thus, (14) is a fine version of (6).
For the finest variant of Theorem 1, we fix a σ-finite measure on (X, M). We will consider function κ(s, x, t, y) defined for s < t and x, y ∈ X, such that (s, We will call such κ a (forward) kernel density, because For instance, we may take k(s, x, t, y) = p s,t (x, y), if measurable and finite, where p is a transition probability density function. We define κ 0 (s, x, t, y) = κ(s, x, t, y), Lemma 3. For all n = 1, 2, . . ., m = 0, 1, . . . , n − 1, s, t ∈ R and x, y ∈ X, Proof. The result was stated in [3, Lemma 3] under stronger conditions, so for the comfort of the reader we repeat the arguments of [3]. If m = 0, then the equality (19) holds by the definition of κ n . In particular, this proves our claim for n = 1. If n ≥ 1 is such that (19) holds for all m < n, then for every m = 1, 2, . . . , n, by Fubini we indeed obtain The Schrödinger perturbation of κ by q is defined as follows, We will assume that for all s < t ∈ R and x, y ∈ X, t s X κ(s, x, u, z)q(u, z)κ(u, z, t, y)dzdu ≤ [η + Q(s, t)]κ(s, x, t, y), (21) . This is a fine analogue of (6) and (14). The following is a fine version of Theorem 1 and 2. We note that (23, 24, 25), but not (22), were first proved in [12] by involved combinatorics.
Proof. We proceed as in the proof of Theorem 1, using Lemma 3 and (21).

Discussion and Applications
The proofs of Theorem 1, 2 and 3 indicate that our estimates are rather tight. The observation is supported by the exact formulas for Schrödinger perturbations of transition densities by Dirac measures (not directly manageable by the methods of the present paper), see [4]. We like to note that the iterated integrals defining K n , k n and κ n exhibit similarity to the expectations of powers of the additive functional in Khasminski's lemma ( [8], [1]), to Wiener chaoses and the multiple integrals in the theory of rough paths ( [15]). In fact, our results offer a far-reaching extension and strengthening of Khasminski's lemma for transition kernels and densities. On a formal level, a unique feature of our estimates is the combinatorics triggered by η, Q and the assumptions (6), (14), (21). As we will see below, the presence of η is quite convenient in applications, and Q is often chosen linear.

Example 1.
Let k(s, x, t, dy) ≥ 0 be a (jointly measurable) transition kernel, so that the following Chapman-Kolmogorov identity holds for all A ∈ M, x ∈ X and s < u < t, If du is the linear Lebesgue measure and q ∞ := sup |q(u, z)| < ∞, then Theorem 2, Q(s, t) = q ∞ (t − s) and η = 0 yield the well-expected bound, k(s, x, t, dy) ≤ k(s, x, t, dy)e q ∞(t−s) .
By Theorem 3, an analogous pointwise version of (27) also holds.

Remark 2.
Let κ be a (forward) kernel density. We will say that q is of relative Kato class for κ, if inf{c : by Example 2, when κ satisfies the 3P Theorem, the Kato condition implies the relative Kato condition. Accordingly, the two are equivalent for the transition density of the fractional Laplacian ∆ α/2 with 0 < α < 2, but not α = 2, because 3P fails for the Gaussian kernel. The details and further references are given in [3] for transition densities, see also [4] for the special case of Schrödinger perturbations of the Cauchy transition density.
We will make a connection to Schrödinger operators analogous to ∆+q, as aforementioned in Introduction. Consider a kernel K on E, function q ∈ E + and real-valued E-measurable functions φ and ψ on E such that Kψ = −φ.
Here we assume absolute integrability: K|ψ| < ∞. Then, provided the integrals are absolutely convergent for all arguments.
For forward kernels we can give rather explicit sufficient conditions for the absolute integrability. We will say K is locally finite in time if for all real s < t, u ∈ R and z ∈ X, we have K1 (s,t) (u, z) = K(u, z, (s, t) × X) < ∞.
As a rule, if K is a left inverse of an operator L on space-time, thenK is a left inverse of L + q. Namely, if for some function φ, then we consider ψ = Lφ, and obtain under the assumptions of Lemma 4. This is quite satisfactory if L is local in time, because if φ is compactly supported in time, then so is ψ, and the boundedness of ψ may usually be secured by appropriate assumptions on φ, see, e.g., [3,5].
If L is nonlocal in time, then more flexible conditions on K may be needed.
The result follows, since K 2 1 (a,b) < ∞ for finite a < b.