In this work, we study the large deviation properties of random walk in a random environment on Z with d ≥ 1. We start with the quenched case, take the point of view of the particle, and prove the large deviation principle (LDP) for the pair empirical measure of the environment Markov chain. By an appropriate contraction, we deduce the quenched LDP for the mean velocity of the particle and obta...

Consider a stochastic process {X,},,, where JI is either Iw + = [0, cc) or N (the set of natural numbers). For some compact metric space D, let PO, 8 ED, be a family of probability measures on the trajectory (‘canonical’) space of ( X,},EJ, equipped with the appropriate topology and the corresponding Bore1 u-field. Assume that the ‘true’ probability measure is PO,, for some 19, ED. Pe7-, T E ,I...

We consider optimal stopping problems for ambiguity averse decision makers with multiple priors. In general, backward induction fails. If, however, the class of priors is time–consistent, we establish a generalization of the classical theory of optimal stopping. To this end, we develop first steps of a martingale theory for multiple priors. We define minimax (super)martingales, provide a Doob–M...

The results established in contemporary statistical physics indicating that, on very small space and time scales, the entropy production rate may be negative, motivate a generalization of continuum mechanics. On account of the fluctuation theorem, it is recognized that the evolution of entropy at a material point is stochastically (not deterministically) conditioned by the past history, with an...

We examine the question of determining the "best" linear filter, in an expected squared error sense, for a signal generated by stochastic linear differential equation on a Hilbert space. Our results, which extend the development in Kalman and Bucy (1960), rely heavily on the integration theory for Banach-space-valued functions of Dunford and Schwartz (1958). In order to derive the Kalman-Bucy f...

Abstract We generalize the usual Doob maximal operator as well fractional and introduce $$M_{\gamma ,s,\alpha }$$ M γ , s α , a new for martingales. prove that under log-Hölder continuity conditio...

Abstract The infinitesimal generator of a one-dimensional strictly $$\alpha $$ α -stable process can be represented as weighted sum (right and left) Riemann-Liouville fractional derivatives order one obtains the Laplacian in case symmetric stable processes. Using this relationship, we compute inverse on Lizor...

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Nearest neighbor random walks in the quarter plane that are absorbed when reaching the boundary are studied. The cases of positive and zero drift are considered. Absorption probabilities at a given time and at a given site are made explicit. The following asymptotics for these random walks starting from a given point (n0, m0) are computed : that of probabilities of being absorbed at a given sit...