This study focuses on the spectral set of the sensitive noise associated with the interface stochastic differential equation (ISDE) defined on the metric star graph with three or more legs. The main assertion is that the supremum of the Hausdorff dimension of the spectral set is equal to (1−α)∕2. Here, α=4arccot(N−1)∕π, determined by the number N of legs of the graph, is the well-known characteristic number governing the coalescing tendency of the associated two-point motion, a self-joining of the solution process: A lower coalescing tendency brings more various solution processes, and hence a more complicated measurable structure, which is reflected in a larger spectral set. For the proof, the spectral set is first examined in a discrete-time setting, because then the probabilistic structure reduces to successive coin flips inside legs and dice rolls at the origin, which appear to represent the stable and sensitive structure separately. We define a subset of the spectral set derived from dice, which turns out to be generated by the coalescences of the discretized two-point motion. From this viewpoint, the tail distribution of the excursion length of the discretized two-point motion from (0,0) plays a crucial role and is also governed by α: An application of a complex analytical method, which has been frequently used for random walks in the quarter plane, reveals the transmission of the coalescing tendency from the continuous-time to the discrete-time setting.
We establish the unique weak and strong solvability of a class of time-inhomogeneous stochastic differential equations with drift belonging to subcritical Lebesgue–Hölder space Lp([0,T];Cbβ(Rd;Rd)), driven by α-stable processes with α∈(0,2). Weak well-posedness is obtained for β∈(0,1), α+β>1 and p>α∕(α+β−1) by combining Prohorov’s theorem, Skorohod’s representation and regularity estimates for solutions of a class of fractional Kolmogorov equations. Pathwise uniqueness and Davie’s type uniqueness are shown for β>1−α∕2 via Itô–Tanaka’s trick. Furthermore, we construct a counterexample to pathwise uniqueness in the supercritical Lebesgue–Hölder regime for α∈(0,2), demonstrating the sharpness of our results.
Let {vt:t≥0}, with values in TM, denote the linearization along trajectories of a stochastic differential equation on a Riemannian manifold M, and let λt=t−1log‖vt‖ denote the associated finite time Lyapunov exponent. The rate function for large deviations of λt from its almost sure limit λ is related via the Gärtner-Ellis theorem to the moment Lyapunov exponent Λ(p)=limt→∞t−1logE[‖vt‖p]. When M is compact there is a characterization of Λ(p) in terms of an eigenvalue problem for an associated differential operator acting on functions on the unit sphere bundle SM. We formulate growth conditions which, together with standard assumptions of hypoellipticity and positivity, ensure that the eigenvalue characterization of Λ(p) remains valid on a suitable function space. There is a central limit theorem for λt, and there are moderate deviation estimates for λt, both involving the second derivative Λ″(0). Examples are given showing the various different growth rates for Λ(p) as p→∞ which may happen when M is not compact.
The present paper focuses on studying the well-posedness of infinite horizon backward stochastic differential equations driven by G-Brownian motion (G-BSDEs) with quadratic generators. To this end, we provide a full construction of explicit solutions of linear G-BSDEs with unbounded coefficients and develop the linearization method under the quadratic assumption. In addition, comparison theorems for both finite and infinite horizon G-BSDEs are established.
We study the limiting laws of a recurrent one-dimensional L & eacute;vy process conditioned to avoid a set A up to a parametrized random time tending to infinity. For A, we consider finite sets of points under several clocks, a bounded F sigma-set under an exponential clock, and an integer lattice under an exponential clock.
