2016 Stochastic Analysis Workshop at Korea University
Program
Overlaps and Pathwise Localization in the Anderson Polymer Model
(Michael Cranston)
Exponential rates in Wasserstein distances for SDEs with Levy noises via refined basic couplings(Jian Wang)
We consider large time behavior of typical paths under the Anderson polymer measure. If is the measure induced by rate , simple, symmetric random walk on started at , this measure is defined as
where is a field of standard, one-dimensional Brownian motions, and the normalizing constant. We establish that the polymer measure gives a macroscopic mass to a small neighborhood of a typical path as , for parameter values outside the perturbative regime of the random walk, giving a pathwise approach to polymer localization, in contrast with existing results. The localization becomes complete as in the sense that the mass grows to 1. The proof makes use of the overlap between two independent samples drawn under the Gibbs measure , which can be estimated by the integration by parts formula for the Gaussian environment. Conditioning this measure on the number of jumps, we obtain a canonical measure which already shows scaling properties, thermodynamic limits, and decoupling of the parameters. This talk is based on joint work with Francis Comets.
We establish the exponential contractivity with respect to Wasserstein distance and the total variation for the semigroup corresponding to SDEs with additive Levy noises, when the drift term is dissipative outside some compact set. The method is based on the refined basic coupling for Levy jump processes. As a byproduct, we obtain sufficient conditions for the strong ergodicity of SDEs with jumps.
AN Lp-THEORY FOR STOCHASTIC ANOMALOUS DIFFUSION EQUATIONS(Ildoo Kim)
In this talk, we introduce an -theory for stochastic anomalous diffusion equations. Especially, we present an -theory for the timefractional quasi-linear stochastic partial differential equations (SPDEs) of type
where and and denote the Caputo derivative of order and respectively. The processes are independent one-dimensional Wiener processes defined on a probability space is a second order operator of either divergence or non-divergence type, and are linear operators of order up to two. The coefficients of the equations depend on and are allowed to be discontinuous. This talk is based on works with Kyeong-Hun Kim, Panki Kim, and Sungbin Lim.
Intermittency and Multi-fractality(Kunwoo Kim)
In this talk, we consider two types of SPDEs: (i) a nonlinear variant of a parabolic Anderson model and (ii) a stochastic heat equation with additive noise. We investigate the two dimensional sets of points (t,x) at which the solutions to those SPDEs exceed certain tall peaks and show that both SPDEs exhibit multi-fractal behavior even though (i) is intermittent and (ii) is not intermittent. This is based on on-going work with Davar Khoshnevisan and Yimin Xiao.
Intrinsic ultracontractivity of symmetric jump processes on unbounded
open sets(XIN CHEN)
We will study some criterions of the intrinsic ultracontractivity for a large class of symmetric jump process killed on exiting an unbounded open set, including the stable process and truncated stable process killed on exiting a hornshaped region. We will provide some examples to show that our criterions are sharp in some sense, and for the horn-shape region, a two-side estimate for the associated ground state will also be given. The talk is based on a joint work with Panki Kim and Jian Wang.
On the chaotic character of stochastic (fractional) heat equation
(Shang-Yuan Shiu)
We consider the following stochastic (fractional) heat equation:
with initial data where is a Lipschitz continuous function and is a Gaussian random noise. It can be shown, in variant cases, that there exist two universal constants and such that for all
here and depend on and the noise In the case of parabolic model (when is linear and is the space-time white noise), is realized as a solution to KPZ equation and the fluctuation exponent is . These chaotic results can be obtained by scaling and localization that will be discussed in this talk.
Modeling: random interruptions to Heat diffusion(Kijung Lee)
In the inhomogeneous heat equation
The term models the interruptions to the heat diffusion along time and on space locations. Especially, the effect of in time direction is more troublesome and the regularity of is subject to the regularity of . In this talk we discuss a type of modeling of in the form where is a random noise process which can be whiter than the white noise. We also discuss a regularity relation between and .
Implementation Problem and Cameron-Martin-Girsanov Theorem(Un-Cig Ji)
Within the framework of quantum white noise calculus, we introduce the notions of quantum white noise derivatives for white noise operators. Then the implementation problems for white noise operators are equivalent to certain differential equations associated with the quantum white noise derivatives, and then we study the implementation problems in terms of quantum white noise differential equations. As an application, the displacement operators are characterized by implementation problems which are equivalent to linear differential equations associated with the quantum white noise derivatives. Then the displacement operators are applied to study a generalization of the Cameron--Martin--Girsanov theorem. More precisely, we prove that the affine transform, with an isometric dilation and a regular drift, of a Brownian motion is again a Brownian motion with respect to a new probability measure which is derived explicitly in terms of the displacement operators. The talk is based on a series of joint works with Nobuaki Obata.
Time changes of stochastic processes on fractals(Takashi Kumagai)
In recent years, interest in time changes of stochastic processes according to irregular measures has arisen from various sources. One typical example of such time-changed processes include the so-called Fontes-Isopi-Newman (FIN) diffusion, the introduction of which was motivated by the study of the localization and aging properties of physical spin systems. This FIN diffusion is known to be the scaling limit of the one-dimensional Bouchaud trap model.
We will provide a general framework for studying such time changed processes and their discrete approximations in the case when the underlying stochastic process is strongly recurrent, in the sense that it can be described by a resistance form, as introduced by J. Kigami. In particular, this includes the case of Brownian motion on tree-like spaces and low-dimensional self-similar fractals. If time permits, we also discuss heat kernel estimates for the relevant time-changed processes.
This is a joint work with D. Croydon (Warwick) and B.M. Hambly (Oxford).
A Solvable Homopolymer Model(Michael Cranston)
With the reference measure induced by Brownian motion on starting at 0 we wish to define, for , the Gibbs probability measure with density with respect to heuristically given by
where . We achieve this using a one parameter family of self-adjoint extensions of the Laplacian on This Gibbs probability measure provides a simple continuum model for a homopolymer with an attractive potential at the origin. In this talk we give a comprehensive study of the behavior of paths with respect to these Gibbs measures. In particular, there is a phase transition in the behavior of these paths from diffusive behavior for to positive recurrent behavior for . The critical value is determined by means of the spectral properties of the operator , the self-adjoint extension of the Laplacian on corresponding to the parameter value . This corresponds to a transition from a diffusive or stretched out phase to a globular phase for the polymer. We consider various quantities associated to the paths and examine their behavior near the critical point. We also draw comparisons to a similar model on previously studied by the authors. The program can also be carried out for stable processes in low dimensions. This talk is based on joint work with L. Koralov, S. Molchanov, N. Squartini and B. Vainberg.
On the regularity of stochastic partial differential equations on non-smooth domains(PETRU A. CIOICA-LICHT)
Although there exists an (almost) fully-fledged regularity theory for (semi-)linearsecond order stochastic partial differential equations (SPDEs, for short) on smoothdomains, very little is known about the regularity of these equations on domainswith non-smooth boundaries that have singularities, such as corners or edges. Inthis talk we present an approach for analysing the behaviour of the solution toSPDEs in the direct vicinity of boundary singularities. We focus on a simplifiedsetting that is still general enough to reveal some of the main issues: We consider thestochastic heat equation with additive noise and zero Dirichlet boundary conditionon a planar angular domain. In this way we deal with a basic but typical exampleof a second order parabolic SPDE that is forced to vanish on the boundary of adomain which is smooth except at one point, the vertex. In order to capture thesingular behaviour of the solution and its derivatives at the vertex, we use weightedSobolev spaces, where the weights are appropriate powers of the distance to thevertex. Our approach relies on suitable Green function estimates, which we use toprove a fundamental weighted Lp-estimate for the stochastic convolution associatedto the corresponding equation.(∗) This is joint work with Kyeong-Hun Kim (Korea University, Korea), Kijung Lee(Ajou University, Korea), and Felix Lindner (TU Kaiserslautern, Germany).
Rescaling Ward identities and random normal matrix ensembles near the edge of the spectrum(Nam-Gyu Kang)
I present existence and universality of scaling limits for the eigenvalues of a random normal matrix near the boundary of the spectrum when the external potential is radially symmetric. Our approach uses rescaling Ward identities which lead to a non-linear equation for the one-point function. This is a joint work with Y. Ameur and N. Makarov.
Conservativeness criteria for generalized Dirichlet forms(Gerald Trutnau)
We present conservativeness criteria for non-sectorial perturbations
of strongly local symmetric Dirichlet forms on a locally compact separable metric space.
This is joint work with M. Gim (Seoul National University).
An application of large deviation principle for the Markov processes
(Hyun Jae Yoo)
In this talk we discuss an application of large deviation principle for the Markov processes. Markov processes will be defined from diagonally dominant matrices. The large deviation principle for those processes can say the invertibility of the diagonally dominant matrices, which is a classical result of Taussky.
Poisson kernel estimates for subordinate Brownian motions in Lipschitz domain(Jaehoon Kang)
In this talk, we discuss Poisson kernel estimates for subordinate Brownian motions in bounded Lipschitz domains. To derive Poisson kernel estimates, we use Green function estimates in bounded Lipschitz domains, which are proved in P. Kim and A. Mimica’s paper. We also use lower bound of exit time of bounded cone to cover geometric stable process, whose Laplace ex- ponent is slowly varying function at infinity.
On heat equation with general random noise as the initial condition
(Jinoh Kim)
In this talk we discuss the decreasing speed of the total heat and total flux when the heat diffusion starts with a random noise field. The mathematical model is in the form
where event is randomly chosen from a probability space and is a random field on the space domain Even with a very wild initial condition, the control at the boundary and the averaging effect caused by diffusion make the total heat and flux inside the space domain decrease gradually. We are interested in the decreasing speeds. This question is inspired by D. Khoshnevisan’s work that considered a white noise as Our random noise models of include less white, white, and whiter noises in one frame.
