Probability: Algorithms, Stochastic Analysis, & Applications Part 1

Speaker(s)

David Nualart (University of Kansas)
Juan Jose Viquez (Purdue University)
Richard Eden (Purdue University)
Michael Röckner (University of Bielefeld)
Oana Mocioalca (Kent State University)
Luis Duffaut Espinosa (University of New South Wales)
Cheng Ouyang (University of Illinois, Chicago)
Jonathon Peterson (Purdue University)

Description

The theory of probability and stochastic processes is as diverse as there are ways to apply it. Probability is at the core of all models of uncertainty, while stochastic processes include the additional component of time evolution for phenomena which would otherwise seem hopelessly unpredictable. A hallmark of continuous-time stochastic evolutions is the erratic character of their paths. A recent revolutionary development in this respect is the study of "rough paths", at the confines of functional analysis and probability, while more classical treatments of such models, based on Brownian motion and semi-martingales, continue to be widely used, and discrete probability models present a distinct set of challenges and opportunities. This Session will bring together early-career and internationally recognized experts to explore how these fruitful tools might be applied to a variety of topics in areas outside of probability, including engineering questions in applied control theory, stochastic PDEs in large-scale fluid dynamics such as global circulation and climate models, and the analysis of algorithms in computational complexity.

Schedule

Sat, June 23 - Location: STEW 314

Time	Speaker	Title

8:30-9:25AM	David Nualart	Weak symmetric integrals and change-of-variable formulas for Gaussian processes
	Abstract: The purpose of this talk is to present some recent results on the ItÃ´ formula for a class of Gaussian processes that includes the fractional Brownian motion with Hurst parameter H = 1/6 and H = 1/4 . This type of change-of-variable formula involves a weak symmetric integral defined as the limit in distribution of midpoint Riemann sums (case H =1/4 ), or Riemann sums based on the trapezoidal approximation (case H =1/6 ). Unlike the classical ItÃ´ formula, the correction term is a stochastic integral with respect to an independent Brownian motion. The proof of these results is based on a central limit theorem for multiple Skorohod integrals, which can be considered as an extension of the classical martingale central limit theorem.
9:30-9:55AM	Group Discussion
10:00-10:30AM	Break
10:30-11:00	Juan Jose Viquez	Nourdin-Peccati analysis on Wiener and Wiener-Poisson space and the second order PoincarÃ© inequality
	Abstract: Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet-Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. We obtained from this bound a second order PoincarÃ©-type inequality for the Normal case. We made a survey of these results on the Wiener space, the Poisson space, and the Wiener-Poisson space, and showed several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields, Poisson functionals in the first Poisson chaos restricted to "small" jumps (particularly fractional LÃ©vy processes) and the product of two Ornstein-Uhlenbeck processes; and non central limit theorems obtained for bilinear functionals of Gaussian subordinated fields where the underlying process is a fractional Brownian motion with Hurst parameter bigger than 1/2.
11:00-11:30AM	Richard Eden	General tail estimates using Nourdin-Peccati analysis, and necessary conditions for convergence in a fixed Wiener chaos
	Abstract: For a centered random variable X in a Wiener space, which is assumed to be differentiable in the sense of Malliavin, the functional G is defined using operators from the Malliavin calculus applied to X. It has been shown that comparing G with the constant 1 allows one to show Gaussian-type lower and upper bounds on the tail P[X>z]. We are then able to extend proofs of these results to derive Pearson-type tail bounds, this time assuming linear or quadratic bounds on G. Our techniques rely on a strategy developed by I. Nourdin and G. Peccati, as we relate G with Stein's equation relative to indicator functions. If time allows, we will consider known results on when a sequence in a fixed Wiener chaos converges to a Normal or Gamma distribution. Most notable is the "fourth moment theorem" (a central limit theorem) recently revealed by Peccati and Nualart in Wiener chaos. We will show that similar results are impossible for other Pearson distributions (beyond Normal and Gamma), and perhaps more, because they fail necessary conditions for limits in distribution. These conditions are stated in terms of cumulants, fractional exponential moments, and the functional G. These are joint works with F. Viens.
11:30-11:55AM	Group Discussion
12:00-1:30PM	Lunch
1:30-2:25PM	Michael Röckner	Regularization of ordinary and partial differential equations by noise
	Abstract: It is a well- known phenomenon that an ordinary differential equation becomes "more regular", if one adds a noise term, as e.g. a stochastic differential given by a Brownian motion. On the level of the associated Fokker-Planck-Kolmogorov equations (FPKE), whose solutions are just the transition probabilities of the resulting solution process, this becomes more or less obvious, since the FPKE becomes elliptic, if the noise is not degenerate. From a purely analytic point of view, this regularizing property of the noise is most impressively manifested by the fact that noise can "produce" (existence and, in particular) uniqueness of solutions . Indeed, e.g. a classical result of A. Yu. Veretennikov (see [1] and the references therein), tells us that, given an initial condition, any two corresponding solutions of an ordinary differential equation in ddimensional Euclidean space given by a just measurable bounded vector field and perturbed by the differential of a d-dimensional Brownian path, coincide for almost every such path. In contrast to this, in the deterministic case, neither existence nor uniqueness of solutions hold in such a case. The purpose of this talk is to present recent results of the same type, but for partial differential equations perturbed by noise, i.e. for the infinite dimensional analogue of the situation described above. References [1] N.V. Krylov, M. Roeckner, Strong solutions of stochastic equations with singular time-dependent drift, Probab. Theory Rel. Fields 131, No. 2, pp.154-196 (2005). [2] G. Da Prato, F. Flandoli, E. Priola, M. Roeckner, Strong uniqueness for stochastic evolution equations in Hilbert spaces with bounded measurable drift, CRC-701-Preprint and arXive 1109.0363, pp. 42 (2011), conditionally accepted by Annals of Probability. Key words: ordinary and partial differential equations, stochastic ordinary and partial differential equations, regularization by noise, evolution equations Subject classification: 60H15, 35R60, 60H10, 34F05
2:30-2:55PM	Oana Mocioalca	Exit Times under Fractional Brownian Motion
	Abstract: Unlike Brownian motion fractional Brownian motion exhibits long- range dependence. It has been argued that phenomena like financial asset prices, information transportation as well as many natural phenomena show long range dependence, and thus fBm has been proposed as a better model than Bm for describing such processes. In this talk we will present a few properties of the exit times of fBm, the time when fBm reaches a certain level from its maximum.
3:00-3:30PM	Break
3:30-3:55PM	Luis Duffaut Espinosa	Cascade of nonlinear systems driven by rough paths
	Abstract: It was recently shown that the lack of a suitable probabilistic characterization of the input process for a system of interconnected analytic nonlinear input-output maps is an obstacle to well-posedness. For example, the cascade connection of two such systems is only known to be well-posed when a certain independence property is preserved by the first system in the connection. Hence, it appears that some alternative characterization of an input process is needed in this setting. One possibility is to employ T. Lyons' construction of a rough path. This concept employs the p -variation of a path and Chen's identity in order to extend the notion of integration with respect to paths having finite p -variation larger than 1. The primary advantage of such an approach in the context of system interconnections is that independence is no longer needed for producing well-posed cascaded analytic nonlinear systems.
4:00-4:25PM	Cheng Ouyang	Concentration property, Sobolev inequality and Gaussian upper bound for SDEs driven by fractional Brownian motions
	Abstract: Concentration property and Log-Sobolev inequalities for stochastic differential equations (SDE) are usually discussed under a Markovian setting for the underlying semi-group of the system. In this talk, we present some results on this direction for some SDEs driven by fractional Brownian motions, which are known not to be Markovian systems. In particular, based on our concentration property, we derive a global Gaussian upper bound for the density function of solution to such SDEs. This talk is based on a joint work with Fabrice Baudoin and Samy Tindel.
4:30-4:55PM	Jonathon Peterson	The contact process on the complete graph with random, vertex-dependent, infection rates
	Abstract: The contact process is an interacting particle system that is a very simple model for the spread of an infection or disease on a network. Traditionally, the contact process was studied on homogeneous graphs such as the integer lattice or regular trees. However, due to the non-homogeneous structure of many real-world networks, there is currently interest in studying interacting particle systems in non-homogeneous graphs and environments. In this talk, I consider the contact process on the complete graph, where the vertices are assigned (random) weights and the infection rate between two vertices is proportional to the product of their weights. This setup allows for some interesting analysis of the process and detailed calculations of phase transitions and critical exponents.