Article Contents
Article Contents

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

• We study the archetypal functional equation of the form $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(da,db)$ ($x\in\mathbb{R}$), 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 $K:=\iint_{\mathbb{R}^2}\ln|a|\,\mu(da,db) =\mathbb{E}\{\ln|\alpha|\}$; 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 $x ⇝ \alpha(x-\beta)$.
Mathematics Subject Classification: Primary: 39B05; Secondary: 34K06, 39A22, 60G42, 60J05.

 Citation:

•  [1] L. Bogachev, G. Derfel, S. Molchanov and J. Ockendon, On bounded solutions of the balanced generalized pantograph equation, in Topics in Stochastic Analysis and Nonparametric Estimation (eds. P.-L. Chow et al.), Springer-Verlag, New York, 2008, pp. 29-49. [2] L. V. Bogachev, G. Derfel and S. A. Molchanov, On bounded continuous solutions of the archetypal equation with rescaling, Proc. Royal Soc. A, 471 (2015), 20150351, 1-19. [3] B. van Brunt and G. C. Wake, A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model, European J. Appl. Math., 22 (2011), 151-168. [4] A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc., 93 (1991), no. 453. [5] G. Choquet and J. Deny, Sur l'équation de convolution $\mu=\mu\star\sigma$, (French) [On the convolution equation $\mu=\mu\star\sigma$], C. R. Acad. Sci. Paris, 250 (1960), 799-801. [6] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992. [7] I. Daubechies and J. C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22 (1991), 1388-1410. [8] G. A. Derfel, Probabilistic method for a class of functional-differential equations, Ukrainian Math. J., 41 (1989), 1137-1141 (1990). [9] G. Derfel, N. Dyn and D. Levin, Generalized refinement equations and subdivision processes, J. Approx. Theory, 80 (1995), 272-297. [10] G. Derfel and A. Iserles, The pantograph equation in the complex plane, J. Math. Anal. Appl., 213 (1997), 117-132. [11] G. Derfel and R. Schilling, Spatially chaotic configurations and functional equations with rescaling, J. Phys. A, 29 (1996), 4537-4547. [12] A. K. Grintsevichyus, On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines, Theor. Probab. Appl., 19 (1974), 163-168. [13] J. E. Hutchinson, Fractals and self similarlity, Indiana Univ. Math. J., 30 (1981), 713-747. [14] A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. [15] T. Kato and J. B. McLeod, The functional-differential equation $y'(x)=a y(\lambda x)+b y(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937. [16] J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. London A, 322 (1971), 447-468. [17] D. Revuz, Markov Chains, $2^{nd}$ edition, North-Holland, Amsterdam, 1984. [18] V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications, Russian Math. Surveys, 45 (1) (1990), 87-120. [19] R. Schilling, Spatially chaotic structures, in Nonlinear Dynamics in Solids (ed. H. Thomas), Springer-Verlag, Berlin, 1992, pp. 213-241. [20] A. N. Shiryaev, Probability, $2^{nd}$ edition, Springer-Verlag, New York, 1996. [21] B. Solomyak, Notes on Bernoulli convolutions, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1 (eds. M. L. Lapidus and M. van Frankenhuijsen), Amer. Math. Soc., Providence, RI, 2004, pp. 207-230. [22] N. Steinmetz and P. Volkmann, Funktionalgleichungen für konstante Funktionen, (German) [Functional equations for constant functions], Aequationes Math., 27 (1984), 87-96. [23] G. Strang, Wavelets and dilation equations: A brief introduction, SIAM Rev., 31 (1989), 614-627.
Open Access Under a Creative Commons license