Francesco Caravenna

Bio: Francesco Caravenna is an academic researcher from University of Milan. The author has contributed to research in topics: Random walk & Phase transition. The author has an hindex of 21, co-authored 86 publications receiving 1267 citations. Previous affiliations of Francesco Caravenna include University of Milano-Bicocca & University of Padua.

Polynomial chaos and scaling limits of disordered systems

Abstract: Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1+1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction

Abstract: We consider a random field ϕ:{1, …, N}→ℝ as a model for a linear chain attracted to the defect line ϕ=0, that is, the x-axis. The free law of the field is specified by the density exp(−∑iV(Δϕi)) with respect to the Lebesgue measure on ℝN, where Δ is the discrete Laplacian and we allow for a very large class of potentials V(⋅). The interaction with the defect line is introduced by giving the field a reward ɛ≥0 each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity ɛ of the pinning reward varies: both in the pinning (a=p) and in the wetting (a=w) case, there exists a critical value ɛca such that when ɛ>ɛca the field touches the defect line a positive fraction of times (localization), while this does not happen for ɛ<ɛca (delocalization). The two critical values are nontrivial and distinct: 0<ɛcp<ɛcw<∞, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at ɛ=ɛcp is delocalized. On the other hand, the transition in the wetting model is of first order and for ɛ=ɛcw the field is localized. The core of our approach is a Markov renewal theory description of the field.

Universality in marginally relevant disordered systems

Abstract: We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2+1), the long-range directed polymer model with Cauchy tails in dimension (1+1) and the disordered pinning model with tail exponent 1/2. We show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit: a log-normal random field with a multi-scale correlation structure, which undergoes a phase transition as the disorder strength varies. As a by-product, we show that the solution of the two-dimensional Stochastic Heat Equation, suitably regularized, converges to the same limit. The proof, which uses the celebrated Fourth Moment Theorem, reveals an interesting chaos structure shared by all models in the above class.

Polynomial chaos and scaling limits of disordered systems

Abstract: Inspired by recent work of Alberts, Khanin and Quastel, we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1+1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

Invariance principles for random walks conditioned to stay positive

Abstract: Let {Sn} be a random walk in the domain of attraction of a stable law Y, i.e. there exists a sequence of positive real numbers (an) such that Sn/an con- verges in law to Y. Our main result is that the rescaled process (S⌊nt⌋/an, t ≥ 0), when conditioned to stay positive, converges in law (in the functional sense) to- wards the corresponding stable Levy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.

Table Of Integrals Series And Products

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Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Lévy processes and infinitely divisible distributions

Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Applied Probability and Queues

Abstract: (1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

Forecasting Multifractal Volatility

Abstract: This paper develops analytical methods to forecast the distribution of future returns for a new continuous-time process, the Poisson multifractal. Out model captures the thick tails and volatility persistence exhibited by many financial time series. We assume that the forecaster knows the true generating process with certainty, but only observes past returns. The challenge in this environment is long memory and the corresponding infinite dimension of the state space. We show that a discretized version of the model has a finite state space, which allows an analytical solution to the conditioning problem. Further, the discrete model converges to the continuous-time model as time scale goes to zero, so that forecasts are consistent. The methodology is implemented on simulated data calibrated to the Deutschemark/US Dollar exchange rate. Applying these results to option pricing, we find that the model captures both volatility smiles and long-memory in the term structure of implied volatilities.