Liouville Brownian motion

We construct a stochastic process, called the Liouville Brownian motion, which is the Brownian motion associated to the metric $e^{\gamma X(z)}\,dz^2$, $\gamma<\gamma_c=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to be related to the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner. The construction amounts to changing the speed of a standard two-dimensional Brownian motion $B_t$ depending on the local behavior of the Liouville measure "$M_{\gamma}(dz)=e^{\gamma X(z)}\,dz$". We prove that the associated Markov process is a Feller diffusion for all $\gamma<\gamma_c=2$ and that for all $\gamma<\gamma_c$, the Liouville measure $M_{\gamma}$ is invariant under $P_{\mathbf{t}}$. This Liouville Brownian motion enables us to introduce a whole set of tools of stochastic analysis in Liouville quantum gravity, which will be hopefully useful in analyzing the geometry of Liouville quantum gravity.Comment: Published at http://dx.doi.org/10.1214/15-AOP1042 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the heat kernel and the Dirichlet form of Liouville Brownian Motion

In \cite{GRV}, a Feller process called Liouville Brownian motion on $\R^2$ has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field $e^{\gamma X}$ and is the right diffusion process to consider regarding 2d-Liouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following essentially \cite{fuku} and the techniques introduced in \cite{GRV}. Then we carry out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of the Liouville heat kernel via a non-trivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in \cite{GRV} was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. One possible approach was to use the theory developed for example in \cite{stollmann,sturm1,sturm2}, whose aim is to capture the "geometry" of the underlying space out of the Dirichlet form of a process living on that space. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide an intrinsic metric which is interpreted as an extension of Riemannian geometry applicable to non differential structures. We prove that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of the metric aspect of Dirichlet forms.Comment: 31 page

Forecasting volatility in the presence of Leverage Effect

We define a simple and tractable method for adding the Leverage effect in general volatility predictions. As an application, we compare volatility predictions with and without Leverage on the SP500 Index during the period 2002-2010.

User-Extensible Sequences in Common Lisp

Common Lisp is often touted as the programmable programming language, yet it sometimes places large barriers in the way, with the best of intentions. One of those barriers is a limit to the extensibility by the user of certain core language constructs, such as the ability to define subclasses of built in classes usable with standard functions: even where this could be achievable with minimal penalties. We introduce the notion of user-extensible sequences, describing a protocol which implementations of such classes should follow. We show examples of their use, and discuss the issues observed in providing support for this protocol in a Common Lisp, including ensuring that there is no performance impact from its inclusion

Using Lisp Implementation Internals: Unportable but fun

We present a number of developer tools and language extensions that are available for use with Steel Bank Common Lisp, but which are perhaps not as well-known as they could be. Our motivation is twofold: firstly, to introduce to a developer audience facilities that can make their development or deployment of software more rapid or efficient. Secondly, in the context of the development of the Common Lisp language itself, we offer some observations of patterns of use of such extensions within the development community, and discuss the implications this has on future evolution of the language

Using Lisp-based pseudocode to probe student understanding

We describe our use of Lisp to generate teaching aids for an Algo-rithms and Data Structures course taught as part of the undergrad-uate Computer Science curriculum. Specifically, we have made use of the ease of construction of domain-specific languages in Lisp to build an restricted language with programs capable of being pretty-printed as pseudocode, interpreted as abstract instructions, and treated as data in order to produce modified distractor versions. We examine student performance, report on student and educator reflection, and discuss practical aspects of delivering using this teaching tool

David Temperley, Music and Probability

review of David Temperley's "Music and Probability". Cambridge, Massachusetts: MIT Press, 2007, ISBN-13: 978-0-262-20166-7 (hardcover) $40.00