Analysis of the archetypal functional equation in the non-critical case

We study the archetypal functional equation of the form $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure on $\mathbb{R}^2$; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, where $\mathbb{E}$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e., non-constant) bounded continuous solutions is governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,\mu(\mathrm{d}a,\mathrm{d}b)=\mathbb{E}\{\ln|\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K<0$, whereas if $K>0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $\mathbb{P}(\alpha<0)>0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha(x-\beta)$.


The archetypical equation and main results
This paper concerns the archetypical functional equation with rescaled argument [2,7] of the form where µ(da, db) is a probability measure on R 2 . Due to the normalization of the measure µ to unity, such an equation is balanced, in the sense that the total weighted contribution of the (scaled) solution y(·) on the right-hand side of (1) is matched by the non-scaled input on the left-hand side. The integral in (1) has the meaning of expectation with respect to a random vector (α, β) with distribution P{(α, β) ∈ da × db} = µ(da, db); thus, equation (1) can be represented in the compact form The functional equation (1)- (2) is a rich source of various equations specified by a suitable choice of the measure µ, which has motivated its name "archetypical" (see [2]). Examples include many well-known classes of equations with rescaling, both purely functional and functional-differential, such as: equations in convolutions including Choquet-Deny's equation y = y ⋆ σ [5]; equations for self-similar measures of Hutchinson's type [11] including Bernoulli convolutions [20]; two-scale (refinement) equations [6,8]; Schilling's equation [9,18]; the (balanced) pantograph 1 equation [1,7]; Rvachev's equation for the up-function [17], etc. See a more extensive review in Bogachev et al. [2], together with further references therein.
Observing that any function y(x) ≡ const satisfies equations (1)- (2), it is natural to investigate if there are any non-trivial (i.e., non-constant) bounded continuous solutions. Such a question naturally arises in the context of functional and functional-differential equations with rescaling, where the possible existence of bounded solutions (e.g., periodic, almost periodic, compactly supported, etc.) is of major interest in physical and other applications (see, e.g., [4,17,18,22]).
Investigation of the archetypical equation (1)-(2), with a focus on bounded continuous solutions (abbreviated below as b.c.-solutions), was initiated by Derfel [7] (in the case α > 0) who showed that the problem crucially depends on the value More precisely, if K < 0 (subcritical case) then, under some mild technical conditions on the measure µ, there are no b.c.-solutions other than constants, 2 whereas if K > 0 (supercritical case) then a non-trivial b.c.-solution does exist. However, the critical case K = 0 was left open in [7]. Some recent progress in this direction was due to Bogachev et al. [1] who settled the problem for the functionaldifferential (balanced pantograph) equation by showing that if K = i p i ln a i = 0 then there are no non-trivial b.c.-solutions of (4). Note that equation (4) is in fact a particular case of the archetypical equation (1)- (2), where β is chosen to have a unit exponential distribution and α is discrete with atoms P(α = a i ) = p i (see more details in [2,7]). Recently (see [2]) we proved the same result for a general equation (1)- (2) in the critical case subject to an a priori condition of the uniform continuity of y(·). The latter assumption is fulfilled under an L 1 -continuity hypothesis for the density of β conditioned on α, which is satisfied for a large class of examples including the pantograph equation.
The focus of the present work is on the non-critical case K = 0, especially when K > 0 with α possibly taking negative values, aiming to obtain further results including existence, uniqueness and a maximum principle. Under a slightly weaker moment condition on β as compared to [7] we establish the dichotomy of non-existence vs. existence of non-trivial b.c.-solutions in the subcritical (K < 0) and supercritical (K > 0) regimes, respectively.
Let us stress though that in contrast to the subcritical case which is insensitive to the sign of α, for K > 0 we are only able to produce a non-trivial solution under the assumption that α > 0 almost surely (a.s.). Such a solution is constructed, with the help of results by Grintsevichyus [10], as the distribution function F Υ (x) = P(Υ ≤ x) of the random series representing a self-similar measure associated with (α, β), where {(α n , β n )} n≥1 are independent identically distributed (i.i.d.) random pairs with distribution µ each. This solution is unique (up to linear transformations) in the class of functions with finite limits at ±∞ (Theorem 4.3(a)), but the uniqueness in the class of b.c.-solutions may fail to be true: we will present an example of such a solution y(·) oscillating at +∞ (see Remark 4.2).
In the case K > 0 with P(α < 0) > 0, the function F Υ (·) (which is still well defined) is no longer a solution to the archetypical equation More to the point, our results should convince the reader that this case is completely different from the purely positive case, α > 0 (a.s.); for instance, a b.c.-solution y(·) with limits at ±∞ must be constant (Theorem 4.3(b)). This follows from Theorem 4.2 stating that the limits superior at ±∞ coincide (the same is true for the limits inferior). Heuristically, this is a manifestation of "mixing" in (2) for (large) positive and negative arguments of y(·) due to possible negative values of α. Note that Theorem 4.2 is proved with the help of the maximum principle of Theorem 4.1, which is of interest in its own right. This analysis is complemented by uniqueness results in the class of absolutely continuous (a.c.) solutions (using the Fourier transform methods); here, the boundedness is not assumed a priori. Again, we demonstrate a striking difference between the cases α > 0 (a.s.) and P(α < 0) > 0 (see Theorems 4.4 and 4.5, respectively).
Last but not least, certain special cases must be excluded in general considerations (which was tacitly assumed above). Henceforth, we assume that These degenerate cases are treated in full detail in [2].
Layout. The rest of the paper is organized as follows. We start in §2 by introducing an associated Markov chain (X n ) with jumps of the form x α(x − β), and also extend the where τ is a (random) stopping time and E x stands for the expectation subject to the initial condition X 0 = x. Suitable iterations of such a kind will be instrumental throughout the paper. In §3 we prove a stronger version of the aforementioned dichotomy between the cases K < 0 and K > 0 (the latter subject to α > 0). Finally, §4 contains further discussion of the supercritical case, as briefly indicated above.

Associated Markov chain and harmonic functions
The archetypical equation (2) admits an important interpretation via an associated Markov chain (X n ) on R determined by the recursion where {(α n , β n )} n≥1 are i.i.d. random pairs with the same distribution as a generic copy (α, β). Transition operator T of the Markov chain (6) is given by where the index x indicates the initial condition ; hence, according to (7) solutions of equation (2) are equivalently described as harmonic functions.

Iterations and stopping times
Equation (2) can be expressed as y(x) = E x {y(X 1 )}, and by iteration y(x) = E x {y(X n )} (n ∈ N). Explicitly, For n ∈ N 0 := {0} ∪ N, let F n := σ{X i , i ≤ n} be the σ-algebra generated by events {X i ∈ B} (with Borel sets B ∈ B(R)); the increasing sequence (F n ) n≥0 is referred to as the (natural) filtration of (X n ). A random variable τ with values in N ∪ {+∞} is called a stopping time with respect to filtration (F n ) if it is adapted to (F n ) (i.e., {τ = n} ∈ F n , n ∈ N 0 ) and τ < ∞ a.s. We shall systematically use the following simple fact. (Note that the continuity of y(·) is not required.) Lemma 2.1. Let τ be a stopping time with respect to filtration F α n := σ{α 1 , . . . , α n } ⊂ F n , n ∈ N 0 . If y(·) is a bounded T -harmonic function then Proof. Clearly, τ is adapted to the filtration F α,β n := σ{(α i , β i ), i ≤ n} ≡ F n . Using (6) it is easy to check that E{y(X n ) | F n−1 } = y(X n−1 ) (a.s.), and hence (y(X n )) is a martingale [16, p.
Lemma 3.1. Let assumption (5) be in force, and also assume that Then the random series converges a.s., and its distribution function F Υ (x) := P(Υ ≤ x) is continuous on R.
Recall that the parameter K is defined in (3). The next two results (for K < 0 and K > 0, respectively) were obtained by Derfel [7] in the case α > 0 (a.s.) under a more stringent condition E{|β|} < ∞; but his proofs essentially remain valid in a more general situation as described below. Proof. Applying Lemma 2.1 with τ ≡ n ∈ N, we obtain (see (8), (9))

Canonical solution in the supercritical case with α > 0
The next theorem provides a non-trivial b.c.-solution to the archetypical equation (2) in the case of positive α. Recall that Υ is the random series (14) and F Υ (x) is its distribution function (see Lemma 3.1). (5) is satisfied, along with conditions (13), and also assume that α > 0 a.s. Then y = F Υ (x) is a b.c.-solution of the archetypical equation (2).
We will refer to y = F Υ (x) as the canonical solution of equation (2).

Bounds coming from infinity
The next result is akin to the maximum principle for the usual harmonic functions. The continuity of y(·) is not presumed.
where the same + or − sign should be chosen on both sides of each equality. Then where m := min{m + , m − }, M := max{M + , M − }.
The case where α may take on negative values has an interesting general property as follows (note that conditions (13) are not needed here).

Uniqueness for solutions with limits at infinity
We can now prove the following uniqueness result (again, the continuity of solutions is not presumed). Note that the cases α > 0 (a.s.) and P(α < 0) > 0 are drastically different.
Proof. (a) Denote the right-hand side of (22) by y * (x). By linearity of (2) and according to Theorem 3.3, y * (x) satisfies equation (2), and it has the same limits L ± at ±∞ as the solution y(x). Hence, y(x) − y * (x) is also a solution, with zero limits at ±∞. But  In the case P(α < 0) > 0, Theorem 4.3(b) holds true if just one of the limits L ± is assumed (due to (20), the other limit exists automatically). [13, p. 923, Theorem 9(iii)] showed inter alia that the pantograph equation y ′ (x) + y(x) = y(αx) with α = const > 1 has a family of C ∞solutions on the half-line x ∈ [0, ∞) such that y(x) = g(ln x/ ln α) + O(x −θ ) as x → +∞, where g(·) is any 1-periodic function, Hölder continuous with exponent 0 < θ ≤ 1. Noting from the equation that y ′ (0) = 0, such solutions can be extended to the entire line R by defining y(x) := y(0) for all x < 0. It is known (see [2,7]) that y(·) automatically satisfies the archetypical equation (2) (with the same α > 1 and exponentially distributed β), thus furnishing an example of (a family of) bounded continuous (even smooth) solutions that do not have limit at +∞.

Uniqueness via Fourier transform
Here, we obtain uniqueness results in the class of a.c. solutions with integrable derivative. In what follows, abbreviation "a.e." stands for "almost everywhere" (with respect to Lebesgue measure on R). Note that the boundedness of solutions is not presumed. It is convenient to state and prove these results separately for positive and negative α (see Theorems 4.4 and 4.5,respectively). Recall that Υ is the random series (14).
(b) To identify z(·) from its Fourier transform (23), it is convenient to integrate both parts of equation (23) against a suitable class of test functions. Consider the Schwartz space S(R) of smooth functions ϕ(x) with finite support and such that their Fourier transform ϕ(s) = R e isx ϕ(x) dx is integrable; by the inversion formula, ϕ(x) = (2π) −1 R e −isx ϕ(s) ds. With this at hand, we can write Similarly, Thus, thanks to equation (23), from (28) and (29) we obtain Since S(R) is dense in both L 1 (R; z(x) dx) and L 1 (R; dF Υ (x)), equation (30) extends to indicator functions of any intervals, yielding (by the continuity of F Υ (·)) A counterpart of Theorem 4.4 for α with possible negative values is strikingly different (cf. Theorem 4.3). Theorem 4.5. Let q := P(α < 0) > 0, and let a solution y(·) be a.e. differentiable, with y ′ ∈ L 1 (R). Then y ′ = 0 a.e. If in addition y(·) is a.c. then y ≡ const.
Iterating as before, we obtain for each n ∈ N z(s) = (−1) n E{e is Υnẑ (Ã −1 n s)}, s ∈ R, where due to (32) we have a.s.Ã −1 n = n i=1α −1 i → 0, Υ n = n i=1β iÃ −1 i−1 → Υ. Hence, the expectation in (34) converges toẑ(0) E{e is Υ }; however, due to the sign alternation the limit of (34) does not exist unlessẑ(0) = 0, in which caseẑ(s) = 0 for all s ∈ R. By the uniqueness theorem for the Fourier transform, this implies that z(x) = y ′ (x) ≡ 0 a.e. Finally, if y(·) is a.c. then it follows that y(x) ≡ const. Thus, the limits of y(x) at ±∞ exist, and the rest immediately follows from Theorem 4.3. However, the uniqueness results for the derivative y ′ , contained in Theorems 4.4 and 4.5, cannot be obtained along these lines.