In recent years, the picture of discrete space time has been studied in the context of stochastic theory. There are a number of ramifications, which are briefly examined. We argue that the causality of physiics has its roots in the analyticity within the two dimensons of a fractal quantum path and further show how this picture has convergence wth quantum superstrings.

ractal geometry provides a basis for modF eling the infinite detail found in nature. Fractal methods are quite popular in computer graphics for modeling natural phenomena such as mountains, clouds, and many kinds of plants. Linear fractal models such as the iterated function system (IFS), recurrent iterated function system (RIFS), and Lindenmeyer systern (L-system) concisely describe complex ob...

Global asymptotic stability conditions for discrete nonlinear scalar stochastic systems with state delay are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov-Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of ...

This paper is concerned with the stability analysis for uncertain stochastic systems with interval time-varying delay. Improved delay-dependent robust stability criteria of uncertain stochastic systems with interval time-varying delay are proposed without ignoring any terms by considering the relationship among the time-varying delay, its upper bound, and their difference, and using both Itô′s ...

The following full text is a preprint version which may differ from the publisher's version. Abstract: We study the critical probability p c (M) in two-dimensional M-adic fractal percolation. To find lower bounds, we compare fractal perco-lation with site percolation. Fundamentally new is the construction of an computable increasing sequence that converges to p c (M). We prove that p c (2) > 0....

We consider the nonlinear stochastic heat equation in one dimension. Under some conditions on the nonlinearity, we show that the "peaks" of the solution are rare, almost fractal like. We also provide an upper bound on the length of the "islands", the regions of large values. These results are obtained by analyzing the correlation length of the solution.

We prove finite speed of propagation for stochastic porous media equations perturbed by linear multiplicative space-time rough signals. Explicit and optimal estimates for the speed of propagation are given. The result applies to any continuous driving signal, thus including fractional Brownian motion for all Hurst parameters. The explicit estimates are then used to prove that the corresponding ...

in this paper, a theorem is derived for the existence of a common quadratic Lyapunov function for stability analysis of TSK fuzzy model. This paper proposed a new method based on stochastic stability. In this paper we study the stochastic stability properties of certain TSK Fuzzy system has been studied.

A converse Lyapunov theorem is established for discrete-time stochastic systems with non-unique solutions. In order to prove the result, mild regularity conditions are imposed on the set-valued mapping that characterizes the update of the system state. It is shown that global asymptotic stability in probability implies the existence of a continuous Lyapunov function, smooth outside of the attra...

Wemake the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and ...