Fabrice Baudoin (University of Connecticut)
Generalized stochastic areas, Winding numbers, and hyperbolic Stiefel fibration
Abstract: We study the Brownian motion on the non-compact Grassmann manifold \( U(n−k,k)/U(n−k)U(k) \) and some of its functionals. The key point is to realize this Brownian motion as a matrix diffusion process, use matrix stochastic calculus and take advantage of the hyperbolic Stiefel fibration to study a functional that can be understood in that setting as a generalized stochastic area process. In particular, a connection to the generalized Maass Laplacian of the complex hyperbolic space is presented and applications to the study of Brownian windings in the Lie group \( U(n−k,k) \) are then given. This is a joint work with Nizar Demni and Jing Wang.
Generalized stochastic areas, Winding numbers, and hyperbolic Stiefel fibration
Abstract: We study the Brownian motion on the non-compact Grassmann manifold \( U(n−k,k)/U(n−k)U(k) \) and some of its functionals. The key point is to realize this Brownian motion as a matrix diffusion process, use matrix stochastic calculus and take advantage of the hyperbolic Stiefel fibration to study a functional that can be understood in that setting as a generalized stochastic area process. In particular, a connection to the generalized Maass Laplacian of the complex hyperbolic space is presented and applications to the study of Brownian windings in the Lie group \( U(n−k,k) \) are then given. This is a joint work with Nizar Demni and Jing Wang.
Marco Carfagnini (University of California-San Diego)
Spectral gaps and small deviations
Abstract: In this talk we will discuss the spectral gap and its connection to limit laws such as small deviations and Chung’s laws of the iterated logarithm. The main focus is on hypoelliptic diffusions such as the Kolmogorov diffusion and horizontal Brownian motions on Carnot groups. We will also discuss spectral properties and existence of spectral gaps on general Dirichlet metric measure spaces. This talk is based on joint works with Maria Gordina and Alexander Teplyaev.
Spectral gaps and small deviations
Abstract: In this talk we will discuss the spectral gap and its connection to limit laws such as small deviations and Chung’s laws of the iterated logarithm. The main focus is on hypoelliptic diffusions such as the Kolmogorov diffusion and horizontal Brownian motions on Carnot groups. We will also discuss spectral properties and existence of spectral gaps on general Dirichlet metric measure spaces. This talk is based on joint works with Maria Gordina and Alexander Teplyaev.
Prakash Chakraborty (Pennsylvania State University)
Reinforcement Learning using Rough Paths
Abstract: This study delves into an optimal control formulation for continuous time reinforcement learning in rough environments. To model uncertainty and facilitate effective exploration of the environment, our approach incorporates relaxed controls into the controller's actions. We rigorously define and explore essential properties of the pathwise version of the rough control problem. This research was conducted collaboratively with Harsha Honnappa and Samy Tindel from Purdue University.
Reinforcement Learning using Rough Paths
Abstract: This study delves into an optimal control formulation for continuous time reinforcement learning in rough environments. To model uncertainty and facilitate effective exploration of the environment, our approach incorporates relaxed controls into the controller's actions. We rigorously define and explore essential properties of the pathwise version of the rough control problem. This research was conducted collaboratively with Harsha Honnappa and Samy Tindel from Purdue University.
Le Chen (Auburn University)
Moment growth and intermittency for SPDEs in the sublinear-growth regime
Abstract: In this talk, we present a recent work joint work with Panqiu Xia from Auburn University (preprint available at arXiv:2306.06761). In this paper, we investigate stochastic heat equation with sub-linear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds for the solution. These moment bounds shed light on the smoothing intermittency effect under weak diffusion (i.e., sub-linear growth) previously observed by Zeldovich et al. The method we employ is highly robust and can be extended to a wide range of stochastic partial differential equations, including the one-dimensional stochastic wave equation.
Moment growth and intermittency for SPDEs in the sublinear-growth regime
Abstract: In this talk, we present a recent work joint work with Panqiu Xia from Auburn University (preprint available at arXiv:2306.06761). In this paper, we investigate stochastic heat equation with sub-linear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds for the solution. These moment bounds shed light on the smoothing intermittency effect under weak diffusion (i.e., sub-linear growth) previously observed by Zeldovich et al. The method we employ is highly robust and can be extended to a wide range of stochastic partial differential equations, including the one-dimensional stochastic wave equation.
Li Chen (Louisiana State University)
Dirichlet fractional Gaussian fields on the Sierpinski gasket
Abstract: We study Dirichlet fractional Gaussian fields on the Sierpinski gasket. Heuristically, such fields are defined as distributions \( X_s=(-\Delta)^{-s}W \), where \(W\) is a Gaussian white noise and \(\Delta\) is the Laplacian with Dirichlet boundary condition. The construction is based on heat kernel analysis and spectral expansion. We also discuss regularity properties and discrete graph approximations of those fields. This is joint work with Fabrice Baudoin.
Dirichlet fractional Gaussian fields on the Sierpinski gasket
Abstract: We study Dirichlet fractional Gaussian fields on the Sierpinski gasket. Heuristically, such fields are defined as distributions \( X_s=(-\Delta)^{-s}W \), where \(W\) is a Gaussian white noise and \(\Delta\) is the Laplacian with Dirichlet boundary condition. The construction is based on heat kernel analysis and spectral expansion. We also discuss regularity properties and discrete graph approximations of those fields. This is joint work with Fabrice Baudoin.
Alexander Dunlap (NYU Courant)
Stochastic heat equations and Cauchy distributions
Abstract: I will describe how an invariant measure with Cauchy-distributed marginals arises from a supercritical stochastic heat equations with an additional, independent additive noise. Joint work with Chiranjib Mukherjee.
Stochastic heat equations and Cauchy distributions
Abstract: I will describe how an invariant measure with Cauchy-distributed marginals arises from a supercritical stochastic heat equations with an additional, independent additive noise. Joint work with Chiranjib Mukherjee.
Yu Gu (University of Maryland)
Effective diffusivities in periodic KPZ
Abstract: There are two magical exponents in the 1+1 KPZ universality class, the 1/3 fluctuation exponent and the 2/3 wandering exponent. Despite remarkable progress during the past decade, our understanding of the two exponents is still rather limited beyond the so-called solvable models. In this talk, I will explain how we study the corresponding problem on a torus, a sort of infra-red cutoff regime, and how we derive the central limit theorem for the fluctuation and the wandering in this case. It turns out that the asymptotic behaviors of the effective diffusivities as the length of the torus goes to infinity are intimately related to the 1/3 and 2/3 exponents. The tools we use are a combination of Malliavin calculus and homogenization of diffusions in random environments. Based on joint works with Tomasz Komorowski and Alex Dunlap.
Effective diffusivities in periodic KPZ
Abstract: There are two magical exponents in the 1+1 KPZ universality class, the 1/3 fluctuation exponent and the 2/3 wandering exponent. Despite remarkable progress during the past decade, our understanding of the two exponents is still rather limited beyond the so-called solvable models. In this talk, I will explain how we study the corresponding problem on a torus, a sort of infra-red cutoff regime, and how we derive the central limit theorem for the fluctuation and the wandering in this case. It turns out that the asymptotic behaviors of the effective diffusivities as the length of the torus goes to infinity are intimately related to the 1/3 and 2/3 exponents. The tools we use are a combination of Malliavin calculus and homogenization of diffusions in random environments. Based on joint works with Tomasz Komorowski and Alex Dunlap.
Pierre Yves Gaudreau Lamarre (University of Chicago)
Moments of the Parabolic Anderson Model with Asymptotically Singular Noise
Abstract: The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.
One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.
In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.
This talk is based on a joint work with Promit Ghosal and Yuchen Liao.
Moments of the Parabolic Anderson Model with Asymptotically Singular Noise
Abstract: The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.
One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.
In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.
This talk is based on a joint work with Promit Ghosal and Yuchen Liao.
Yier Lin (University of Chicago)
Multi-point Lyapunov Exponents of the Stochastic Heat Equation
Abstract: We study the Stochastic Heat Equation with multiplicative space-time white noise. Extensive research has already been conducted on the one-point Lyapunov exponents of this equation. In this talk, I will present how we can compute the multi-point Lyapunov exponents by leveraging a combination of integrability and probability. Additionally, we solve a non-trivial optimization problem as a byproduct.
Multi-point Lyapunov Exponents of the Stochastic Heat Equation
Abstract: We study the Stochastic Heat Equation with multiplicative space-time white noise. Extensive research has already been conducted on the one-point Lyapunov exponents of this equation. In this talk, I will present how we can compute the multi-point Lyapunov exponents by leveraging a combination of integrability and probability. Additionally, we solve a non-trivial optimization problem as a byproduct.
Phanuel Mariano (Union college)
Improved Upper Bounds for the Hot Spots Constant of Lipschitz Domains
In this talk we discuss the Hot Spots constant for bounded smooth domains that was recently introduced by S. Steinerberger as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We use probabilistic techniques to derive a general formula for a dimension-dependent upper bound that can be tailored to any specific class of bounded Lipschitz domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in \(\mathbb{R}^{d}\) for both small and asymptotically large \(d\) that significantly improve upon the existing results. Moreover, we prove new bounds for the Hot Spots constant for Lipschitz domains on Rimannian manifolds with non-negative Ricci curvature. This is joint work with Hugo Panzo and Jing Wang.
Improved Upper Bounds for the Hot Spots Constant of Lipschitz Domains
In this talk we discuss the Hot Spots constant for bounded smooth domains that was recently introduced by S. Steinerberger as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We use probabilistic techniques to derive a general formula for a dimension-dependent upper bound that can be tailored to any specific class of bounded Lipschitz domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in \(\mathbb{R}^{d}\) for both small and asymptotically large \(d\) that significantly improve upon the existing results. Moreover, we prove new bounds for the Hot Spots constant for Lipschitz domains on Rimannian manifolds with non-negative Ricci curvature. This is joint work with Hugo Panzo and Jing Wang.
Tai Melcher (University of Virginia)
Functional inequalities for degenerate diffusions and some applications
Abstract: We will discuss various functional inequalities for diffusions with degenerate noise, with a particular focus on Wang Harnack inequalities. The coefficients appearing in these inequalities typically feature distances inherent to the underlying geometry of the diffusion. In some degenerate cases of interest, there are known notions of distance that appear naturally; however, in other cases, there is perhaps no obvious candidate for a replacement. We'll discuss some new examples and revisit some old ones, introducing notions of distance appropriately adapted to the diffusion to allow proof of the inequalities. A feature of Wang Harnack inequalities is their dimension independence. As an application, we'll show how our inequalities may be used to prove smoothness results for some infinite-dimensional versions of these diffusions.
Functional inequalities for degenerate diffusions and some applications
Abstract: We will discuss various functional inequalities for diffusions with degenerate noise, with a particular focus on Wang Harnack inequalities. The coefficients appearing in these inequalities typically feature distances inherent to the underlying geometry of the diffusion. In some degenerate cases of interest, there are known notions of distance that appear naturally; however, in other cases, there is perhaps no obvious candidate for a replacement. We'll discuss some new examples and revisit some old ones, introducing notions of distance appropriately adapted to the diffusion to allow proof of the inequalities. A feature of Wang Harnack inequalities is their dimension independence. As an application, we'll show how our inequalities may be used to prove smoothness results for some infinite-dimensional versions of these diffusions.
Andrea Nahmod (University of Massachusetts Amherst)
Invariant Gibbs measures for 2D NLS and 3D cubic NLW
Waves arise in quantum mechanics, fiber optics, ferromagnetism, atmospheric and water waves, and many other models. Such wave phenomena are never too smooth or too simple, the byproduct of nonlinear interactions. To accurately predict wave phenomena in nature, it is important to understand and describe the dynamical behavior of such models under certain noisy conditions or given an initial statistical ensemble and have a precise description of how the inherent randomness built in these models propagate.
In this talk, we will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. We will continue by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this
context and how it opened the door to unveil the random structures of nonlinear waves that live on high frequencies and fine scales. We will then discuss how these ideas broke the logjam in the study of the Gibbs measures associated to nonlinear Schrödinger equations in the context of equilibrium statistical mechanics and of the hyperbolic \(\Phi^4_3\) model in the context of constructive quantum field theory. We will end with some open challenges about the long-time propagation of randomness and out-of-equilibrium dynamics.
Invariant Gibbs measures for 2D NLS and 3D cubic NLW
Waves arise in quantum mechanics, fiber optics, ferromagnetism, atmospheric and water waves, and many other models. Such wave phenomena are never too smooth or too simple, the byproduct of nonlinear interactions. To accurately predict wave phenomena in nature, it is important to understand and describe the dynamical behavior of such models under certain noisy conditions or given an initial statistical ensemble and have a precise description of how the inherent randomness built in these models propagate.
In this talk, we will start by describing how classical tools from probability offer a robust framework to understand the dynamics of waves via appropriate ensembles on phase space rather than particular microscopic dynamical trajectories. We will continue by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this
context and how it opened the door to unveil the random structures of nonlinear waves that live on high frequencies and fine scales. We will then discuss how these ideas broke the logjam in the study of the Gibbs measures associated to nonlinear Schrödinger equations in the context of equilibrium statistical mechanics and of the hyperbolic \(\Phi^4_3\) model in the context of constructive quantum field theory. We will end with some open challenges about the long-time propagation of randomness and out-of-equilibrium dynamics.
Mickey Salins (Boston University)
Regularity of the law of solutions to the stochastic heat equation with non-Lipschitz reaction term on an unbounded spatial domain
Abstract: The law of the solution to the stochastic heat equation has a density with respect to Lebesgue measure when the multiplicative stochastic noise is sufficiently non-degenerate and the reaction term satisfies the so-called “half-Lipschitz” condition. The proof of the existence of a density uses Malliavin calculus techniques. Because the spatial domain is unbounded, the solutions to the stochastic heat equation is generally unbounded in space, complicating the analysis. Joint work with Samy Tindel.
Regularity of the law of solutions to the stochastic heat equation with non-Lipschitz reaction term on an unbounded spatial domain
Abstract: The law of the solution to the stochastic heat equation has a density with respect to Lebesgue measure when the multiplicative stochastic noise is sufficiently non-degenerate and the reaction term satisfies the so-called “half-Lipschitz” condition. The proof of the existence of a density uses Malliavin calculus techniques. Because the spatial domain is unbounded, the solutions to the stochastic heat equation is generally unbounded in space, complicating the analysis. Joint work with Samy Tindel.
Hao Shen (University of Wisconsin)
Introduction to stochastic quantization
Abstract: Stochastic quantization is a bridge between quantum field theory on one hand and stochastic analysis on the other hand. Through this connection, it is possible to put certain functional integrals in quantum field theory on rigorous footing, and prove various properties of them; I will discuss some recent progress and ongoing efforts. These studies also foster the developments of stochastic analysis and stochastic PDE theories. In the first lecture I will explain these ideas using the Phi4 model. In the second lecture I will focus on the Yang-Mills model in 2 and 3 dimensions, discuss the construction of singular orbit space, local stochastic dynamics, singular holonomies etc. On lattice, stochastic quantization can be also used to prove interesting properties such as mass gap at strong coupling.
Introduction to stochastic quantization
Abstract: Stochastic quantization is a bridge between quantum field theory on one hand and stochastic analysis on the other hand. Through this connection, it is possible to put certain functional integrals in quantum field theory on rigorous footing, and prove various properties of them; I will discuss some recent progress and ongoing efforts. These studies also foster the developments of stochastic analysis and stochastic PDE theories. In the first lecture I will explain these ideas using the Phi4 model. In the second lecture I will focus on the Yang-Mills model in 2 and 3 dimensions, discuss the construction of singular orbit space, local stochastic dynamics, singular holonomies etc. On lattice, stochastic quantization can be also used to prove interesting properties such as mass gap at strong coupling.
Li-Cheng Tsai (University of Utah)
High moments of the SHE and spacetime limit shapes of the KPZ equation
Abstract: Consider the n-point, fixed-time large deviations of the Kardar--Parisi--Zhang (KPZ) equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a range of scaling regimes that allows time to be short, unit-order, and long. I will present a result (joint with Yier Lin) on the n-point Large Deviation Principle (LDP) and the corresponding spacetime limit shape. The proof is based on another work (of myself) on the multipoint moments of the Stochastic Heat Equation (SHE). I will explain how to analyze the moments via a system of attractive Brownian particles and how to use the moments to obtain the LDP and spacetime limit shape.
High moments of the SHE and spacetime limit shapes of the KPZ equation
Abstract: Consider the n-point, fixed-time large deviations of the Kardar--Parisi--Zhang (KPZ) equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a range of scaling regimes that allows time to be short, unit-order, and long. I will present a result (joint with Yier Lin) on the n-point Large Deviation Principle (LDP) and the corresponding spacetime limit shape. The proof is based on another work (of myself) on the multipoint moments of the Stochastic Heat Equation (SHE). I will explain how to analyze the moments via a system of attractive Brownian particles and how to use the moments to obtain the LDP and spacetime limit shape.
Xuan Wu (University of Chicago)
From the KPZ equation to the directed landscape
This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.
From the KPZ equation to the directed landscape
This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.
Abstracts of contributing talks
Hongyi Chen (University of Illinois at Chicago)
Multiplicative Stochastic Heat Equation on Compact Riemannian Manifolds of Nonpositive Curvature
We establish well-posedness and exponential upper bounds of the solution of the multiplicative stochastic heat equation on compact manifolds with measure-valued initial condition. Unlike most previous work on this equation, Fourier analysis is not employed. Existence of the second moment of the solution in large time for all compact Riemannian manifolds will be shown. The large time existence of higher moments on all compact manifolds is proven, confirming a previous result by Tindel and Viens with more restrictive initial conditions. In small time, the difficulty induced by nonuniqueness of geodesics will be presented, along with how to overcome it on manifolds of nonpositive curvature.
Multiplicative Stochastic Heat Equation on Compact Riemannian Manifolds of Nonpositive Curvature
We establish well-posedness and exponential upper bounds of the solution of the multiplicative stochastic heat equation on compact manifolds with measure-valued initial condition. Unlike most previous work on this equation, Fourier analysis is not employed. Existence of the second moment of the solution in large time for all compact Riemannian manifolds will be shown. The large time existence of higher moments on all compact manifolds is proven, confirming a previous result by Tindel and Viens with more restrictive initial conditions. In small time, the difficulty induced by nonuniqueness of geodesics will be presented, along with how to overcome it on manifolds of nonpositive curvature.
Shalin Parekh (Maryland - College Park)
A nonlinear Strassen law for singular SPDEs
We derive very general conditions under which one has a functional law of the iterated logarithm for some process derived from a Gaussian measure and some finite number of its higher chaoses. We focus in particular on the example of obtaining a nontrivial compact limit set in the example of the KPZ equation.
A nonlinear Strassen law for singular SPDEs
We derive very general conditions under which one has a functional law of the iterated logarithm for some process derived from a Gaussian measure and some finite number of its higher chaoses. We focus in particular on the example of obtaining a nontrivial compact limit set in the example of the KPZ equation.
Peter Rudzis (University of North Carolina - Chapel Hill)
Fluctuations of the inhomogeneous Atlas model
The Atlas model is a prototypical example of a rank-based diffusion, a system of Brownian particles whose interaction potential is a function of the order of the particles on the real line. In this model, infinitely many Brownian particles are started from some initial distribution such that a lowest particle exists; the lowest particle always gets an upward drift of \(\gamma \geq 0\). The particle gap distribution of the Atlas model, of interest in its own right, can be understood as a reflected Brownian motion in the positive orthant \(\mathbb{R}_+^\infty\). The centered Atlas model has a rich ergodic structure, with uncountably many (known) equilibrium regimes, which can be obtained through an explicit transformation of a family of Poisson random measures on the real line; the corresponding gap distributions have an explicit product form. In this talk, we will review some known results on the equilibrium regimes of the Atlas model, and we will present new results on the fluctuations of the Atlas model, started from an inhomogeneous ergodic distribution. After appropriately scaling space, the limiting fluctuations are described by a random field which is the solution of an SPDE.
This work is joint with Amarjit Budhiraja and Sayan Banerjee, and extends fluctuation results obtained by A. Dembo and L-C. Tsai.
Fluctuations of the inhomogeneous Atlas model
The Atlas model is a prototypical example of a rank-based diffusion, a system of Brownian particles whose interaction potential is a function of the order of the particles on the real line. In this model, infinitely many Brownian particles are started from some initial distribution such that a lowest particle exists; the lowest particle always gets an upward drift of \(\gamma \geq 0\). The particle gap distribution of the Atlas model, of interest in its own right, can be understood as a reflected Brownian motion in the positive orthant \(\mathbb{R}_+^\infty\). The centered Atlas model has a rich ergodic structure, with uncountably many (known) equilibrium regimes, which can be obtained through an explicit transformation of a family of Poisson random measures on the real line; the corresponding gap distributions have an explicit product form. In this talk, we will review some known results on the equilibrium regimes of the Atlas model, and we will present new results on the fluctuations of the Atlas model, started from an inhomogeneous ergodic distribution. After appropriately scaling space, the limiting fluctuations are described by a random field which is the solution of an SPDE.
This work is joint with Amarjit Budhiraja and Sayan Banerjee, and extends fluctuation results obtained by A. Dembo and L-C. Tsai.
Kihoon Seong (Cornell University)
Phase transition of singular Gibbs measures for three-dimensional Schrödinger-wave system
We study the phase transition phenomenon of the singular Gibbs measure associated with the Schr¨odinger-wave systems, initiated by Lebowitz, Rose, and Speer (1988). In the three-dimensional case, this problem turns out to be critical, exhibiting a phase transition according to the size of the coupling constant. In the weakly coupling region, the Gibbs measure can be constructed as a probability measure, which is singular with respect to the Gaussian free field. On the other hand, in the strong coupling case, the Gibbs measure can not be normalized as a probability measure. In particular, the finite-dimensional truncated Gibbs measures have no weak limit, even up to a subsequence. The singularity of the Gibbs measure makes an additional difficulty in proving the non-convergence in the strong coupling case
Phase transition of singular Gibbs measures for three-dimensional Schrödinger-wave system
We study the phase transition phenomenon of the singular Gibbs measure associated with the Schr¨odinger-wave systems, initiated by Lebowitz, Rose, and Speer (1988). In the three-dimensional case, this problem turns out to be critical, exhibiting a phase transition according to the size of the coupling constant. In the weakly coupling region, the Gibbs measure can be constructed as a probability measure, which is singular with respect to the Gaussian free field. On the other hand, in the strong coupling case, the Gibbs measure can not be normalized as a probability measure. In particular, the finite-dimensional truncated Gibbs measures have no weak limit, even up to a subsequence. The singularity of the Gibbs measure makes an additional difficulty in proving the non-convergence in the strong coupling case
Liet Vo (University of Illinois at Chicago)
Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise.
In this talk, I will present a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new H\"older continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Finally, we consider the stochastic Allen-Cahn equation and the stochastic Stokes equations as numerical examples to verify the sharpness of the proposed numerical scheme.
Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise.
In this talk, I will present a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new H\"older continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Finally, we consider the stochastic Allen-Cahn equation and the stochastic Stokes equations as numerical examples to verify the sharpness of the proposed numerical scheme.
Guang Yang (Purdue University)
A probabilistic proof of the fundamental gap conjecture on spheres
The fundamental gap conjecture asserts that the difference between the first two Dirichlet eigenvalues (the fundamental gap) of the Laplacian on a bounded convex domain in R^n is bounded below by the fundamental gap on an interval of the same diameter. This conjecture was proved by B. Andrews and J. Clutterbuck. More recently, the same conjecture was verified for spheres by S. Seto, L. Wang and G. Wei using pure geometric method that relies on the two-point maximal principle. We present a probabilistic proof using diffusion coupling on Riemannian manifolds, which is more flexible than the geometric proof and is suitable for studies on more general manifolds. This talk is based on joint work with G. Cho and G. Wei.
A probabilistic proof of the fundamental gap conjecture on spheres
The fundamental gap conjecture asserts that the difference between the first two Dirichlet eigenvalues (the fundamental gap) of the Laplacian on a bounded convex domain in R^n is bounded below by the fundamental gap on an interval of the same diameter. This conjecture was proved by B. Andrews and J. Clutterbuck. More recently, the same conjecture was verified for spheres by S. Seto, L. Wang and G. Wei using pure geometric method that relies on the two-point maximal principle. We present a probabilistic proof using diffusion coupling on Riemannian manifolds, which is more flexible than the geometric proof and is suitable for studies on more general manifolds. This talk is based on joint work with G. Cho and G. Wei.
Pavlos Zoubouloglou (University of North Carolina - Chapel Hill)
Large Deviations for Empirical Measures of Self-Interacting Markov Chains
Let \(\Delta^o\) be a finite set and, for each probability measure \(m\) on \(\Delta^o\), let \(G(m)\) be a transition kernel on \(\Delta^o\). Consider the sequence \(\{X_n\}\) of \(\Delta^o\)-valued random variables such that, and given \(X_0,\ldots,X_n\), the conditional distribution of \(X_{n+1}\) is \(G(L^{n+1})(X_n,\cdot)\), where \(L^{n+1}=\frac{1}{n+1}\sum_{i=0}^{n}\delta_{X_i}\). Under conditions on \(G\) we establish a large deviation principle for the sequence \(\{L^n\}\). As one application of this result we obtain large deviation asymptotics for the Aldous-Flannery-Palacios (1988) approximation scheme for quasi-stationary distributions of finite state Markov chains. The conditions on $G$ cover other models as well, including certain models with edge or vertex reinforcement.
Large Deviations for Empirical Measures of Self-Interacting Markov Chains
Let \(\Delta^o\) be a finite set and, for each probability measure \(m\) on \(\Delta^o\), let \(G(m)\) be a transition kernel on \(\Delta^o\). Consider the sequence \(\{X_n\}\) of \(\Delta^o\)-valued random variables such that, and given \(X_0,\ldots,X_n\), the conditional distribution of \(X_{n+1}\) is \(G(L^{n+1})(X_n,\cdot)\), where \(L^{n+1}=\frac{1}{n+1}\sum_{i=0}^{n}\delta_{X_i}\). Under conditions on \(G\) we establish a large deviation principle for the sequence \(\{L^n\}\). As one application of this result we obtain large deviation asymptotics for the Aldous-Flannery-Palacios (1988) approximation scheme for quasi-stationary distributions of finite state Markov chains. The conditions on $G$ cover other models as well, including certain models with edge or vertex reinforcement.