
The Random Conductance Model (RCM) is a model of a reversible or symmetric random walk in a random environment. i.i.d. weights m_{e} are assigned to the edges in Z^{d}. A random walk X is then run, which makes its jumps with probabilities proportional to the edge weights.
This model is now quite well understood in the special cases when either the law of m_{e} is concentrated on [0,1] or 1,¥). I will discuss what happens in these cases, and in particular in the case when E (m_{e}) = ¥.
We consider a Brownian Motion W = (W^{i})_{i Î N} and an adapted process of dimension n (X_{t})_{t ³ 0}. We define t_{R} = inf{t : X_{t}  x_{t}  ³ R_{t}}, where x_{t}, t ³ 0 is a deterministic differentiable curve in R^{n} and R_{t} > 0, t ³ 0 a ratio that depends on time. Assume that until t_{R} the process X is a solution of the equation


We obtain lower bounds of the form:
 (1) 
We apply the results to obtain lower bounds for option prices for stochastic volatility models.
Joint work with V. Bally and A. Meda.
In this talk some of the main parallels between weak convergence and large deviations will be explained, particularly, weak convergence of stochastic integrals (in the sense of KurtzProtter) and large deviations for sequences of stochastic integrals. Conditions will be given under which a sequence of stochastic processes (X_{n},Y_{n}) implies large deviations for òX_{n} dY_{n}.
In this talk we present the qualitative properties of the Asymmetric Exclusion Quantum Dynamical Semigroup: its qualitative properties, their invariant states, convergence to the equilibrium, and characterization of the domain attractions of each of the invariant states.
This is a joint work with Roberto Quezada Batalla and Lepoldo Pantaleon Martinez.
In this talk we will introduce two new statistics A_{p}^{n} and T_{n} defined for random samples of size n, of a pair of continuous random variable (X,Y) with copula C. The statistics measure the associativity and symmetry of the samples respectively, that is, if the copula satisfies


We will study the properties of the new statistics, and we will include some applications.
We deal with the dynamic maximization of a robust utility function which penalize the possible probabilistic models. The context will be of a market model with prices determined by an external factor which is driven by a Lévy stochastic integral. We characterize first the classes of measures (densities) related to such a market. Once it is established the relation of the penalty associated to the utility function with a convex risk measure, we are able to use duality theory recently developed for an optimal investment in an risk and ambiguity averse setting.
Stochastic control problems with longrun average criteria (also known as ergodic criteria) were introduced by Richard Bellman (1957) in the context of a manufacturing process, and nowadays play a predominant role in control applications to queueing systems, telecommunication networks, and economic and financial problems, to name a few.
This talk presents some recent advances on discrete and continuous time stochastic control systems with longrun average criteria, including overtaking (or catchingup) optimality, bias optimality, discountsensitive criteria, and the existence of average optimal strategies with minimum variance.
Suppose that the vertices of Z^{d} are assigned random colours via a finitary factor of i.i.d. random vertexlabels. That is, the colour of vertex v is determined by a rule which examines the labels within a finite (but random and perhaps unbounded) distance R of v, and the same rule applies at all vertices. We investigate the tail behaviour of R if the coloring is required to be proper (that is, adjacent vertices receive different colours). Depending on the dimension and the number of colours, the optimal tail is either power law or superexponential.
The looperased random walk [^(S)]^{n} is the process obtained by running a random walk in Z^{d} from the origin to the first exit time of the ball of radius n and then chronologically erasing its loops. If we let M_{n} denote the number of steps of [^(S)]^{n} then the growth exponent a is defined to be such that E[M_{n}] grows like n^{a}. The value of a depends on the dimension d. In this talk we'll focus on d=2 where it's been shown that a = 5/4. We will establish a second moment result and use it to get estimates for the probability that M_{n} is close to its mean. Namely, we show that there exists 0 < p < 1 such that for all n and l large, P(M_{n} < l^{1} E[M_{n}]) < p^{l1/6}.
This is joint work with Martin Barlow.
We obtain bounds at the expected order for the variance of the logarithm of the solution of the stochastic heat equation and correlation functions of its derivative, which are understood as the solutions of the KardarParisiZhang and Stochastic Burgers equations.
Joint work with Marton Balazs and Timo Seppalainen.
Let X be a superBrownian motion (SBM) defined on R^{n} and (X_{D}) be its exit measures indexed by subdomains of R^{d}. We pick a bounded subdomain D, and condition the superbrownian motion inside this domain on its "boundary statistics", random variables defined on an auxiliary probability space generated by sampling from the exit measure X_{D}. Among these, two particular examples are conditioning on a Poisson random measure with intensity bX_{D}, and X_{D} itself. We find the conditional laws as htransforms of the original SBM law using Xharmonic functions.
The Xharmonic function H^{n} corresponding to conditioning on X_{D}=n is of special interest, as it can be thought as the analogue of the Poisson kernel. An open problem is to show that H^{n} is extreme at least for some n when D is a smooth domain. An equivalent problem is to show that the tail sigma field of SBM in D is trivial with respect to P^{n}. We prove a weaker version of this result using an approximation, first by conditioning on a Poisson random measure with intensity n X_{D} and then letting n go to infinity. We show that for any A in the tail sigma field of X, P^{XD}(A)=0 or 1 almost surely.
The topological entanglements of polygons confined to a lattice tube and under the influence of an external tensile force f will be examined. The tube constraint allows us to prove a pattern theorem via transfer matrix arguments for any arbitrary fixed value of f. The resulting stretched polygon pattern theorem can then be used to show that the knotting probability of an nedge stretched polygon confined to a tube goes to one exponentially as n®¥. Thus as n®¥ when polygons are influenced by a force f, no matter its strength or direction, topological entanglements, as defined by knotting, occur with high probability.
This is joint work with M. Atapour and S. G. Whittington.
The stochastic heat equation is the heat equation driven by white noise. We consider its numerical solutions using the finite difference method. Its true solutions are Hölder continuous with parameter ([ 1/2]e) in the space variable, and ([ 1/4]e) in the time variable. We show that the numerical solutions share this property in the sense that they have nontrivial limiting quadratic variation in x and quartic variation in t. These variations are discontinuous functionals on the space of continuous functions, so it is not automatic that the limiting values exist, and not surprising that they depend on the exact numerical schemes that are used; it requires a very careful choice of scheme to get the correct limiting values. In particular, part of the folklore of the subject says that a numerical scheme with excessively long timesteps makes the solution much smoother. We make this precise by showing exactly how the length of the timesteps affects the quadratic and quartic variations.
This is joint work with Yuxiang Chong.