Probability Theory

Stochastic Differential Equations and Numerical Methods

Development and analysis of numerical schemes for solving SDEs with applications to financial modeling and physical systems.

RESEARCH GAP FRONTIERS

Extreme Value Theory and Rare Events

Study of tail behavior of random variables and asymptotic distributions of extreme order statistics with applications to risk management.

RESEARCH GAP FRONTIERS

Markov Chain Monte Carlo Methods

Development of advanced sampling algorithms and convergence diagnostics for Bayesian inference and high-dimensional integration.

RESEARCH GAP FRONTIERS

Large Deviations Theory and Asymptotic Analysis

Analysis of exponential decay rates of probabilities of rare events and applications to queueing and statistical mechanics.

RESEARCH GAP FRONTIERS

Gaussian Processes and Functional Data Analysis

Theory and applications of infinite-dimensional Gaussian measures with focus on regression and uncertainty quantification.

RESEARCH GAP FRONTIERS

Branching Processes and Population Dynamics

Probabilistic models of population evolution with applications to extinction probability and critical phenomena in biological systems.

RESEARCH GAP FRONTIERS

Interacting Particle Systems and Mean Field Limits

Analysis of large systems of interacting particles and convergence to deterministic or stochastic mean field equations.

RESEARCH GAP FRONTIERS

Random Matrix Theory and Spectral Methods

Study of eigenvalue distributions and spectral properties of random matrices with applications to quantum chaos and machine learning.

Optimal Transport and Wasserstein Distances

Geometric measure theory approach to comparing probability distributions with applications to generative models and gradient flows.

Fractional Brownian Motion and Self-Similar Processes

Study of long-range dependence in stochastic processes and Hurst parameter estimation with applications to finance and turbulence.

Levy Processes and Infinite Divisibility

Theory of processes with independent and stationary increments including stable processes and subordination techniques.

Point Processes and Spatial Statistics

Statistical analysis of randomly distributed events in space and time with applications to ecology, seismology, and astronomy.

Percolation Theory and Phase Transitions

Study of connectivity thresholds in random graphs and lattice models with applications to network resilience and critical phenomena.

Regenerative Processes and Queueing Theory

Analysis of systems with renewal epochs and waiting times with applications to service systems and performance evaluation.

Weak Convergence and Functional Limit Theorems

Study of convergence in distribution for random variables and stochastic processes with applications to asymptotic theory.

Poisson Point Processes and Thinning Methods

Analysis of fundamental point processes and operations on them with applications to simulation and network modeling.

Coupling Methods and Probability Inequalities

Development of probabilistic techniques for comparing distributions and bounding tail probabilities in stochastic systems.

Deep Learning and Neural Network Optimization

Probabilistic analysis of gradient descent dynamics and generalization bounds in deep neural networks.

Stochastic Gradient Descent and Online Learning

Analysis of convergence rates and regret bounds for iterative algorithms in streaming and online settings.

Random Graphs and Network Evolution

Study of threshold phenomena and giant components in random graph models with applications to social networks and epidemiology.

Mixing Times and Rapid Mixing Markov Chains

Analysis of convergence rates to stationary distributions in discrete Markov chains with applications to sampling.

Moment Problems and Measure Identification

Study of when probability measures are uniquely determined by their moments with applications to inverse problems.

Survival Analysis and Competing Risks Models

Probabilistic modeling of time-to-event data with multiple failure modes in biomedical and reliability applications.

Bayesian Nonparametrics and Dirichlet Processes

Development of flexible prior distributions for infinite-dimensional parameters with applications to clustering and density estimation.

Stein Method and Distributional Approximations

Technique for proving limit theorems and quantifying approximation errors between probability distributions.

Empirical Process Theory and Statistical Learning

Study of convergence properties of empirical distributions and generalization in statistical learning theory.

Backward Stochastic Differential Equations

Theory and applications of BSDEs for pricing derivatives, optimal control, and partial differential equations.

Martingale Theory and Optional Stopping

Analysis of martingales and submartingales with applications to harmonic analysis and financial mathematics.

Stochastic Control and Dynamic Programming

Optimal decision-making in uncertain environments with applications to finance, operations research, and robotics.

Variance Reduction Techniques in Simulation

Methods for decreasing estimation variance in Monte Carlo simulations including importance sampling and antithetic variates.

Copula Methods and Dependence Modeling

Study of joint distributions through copulas with applications to risk aggregation and multivariate analysis.

Quantum Probability and Von Neumann Algebras

Extension of classical probability theory to noncommutative operator algebras with connections to quantum mechanics.

Time Series Analysis and Autoregressive Models

Statistical modeling and forecasting of temporal data including ARIMA, GARCH, and spectral methods.

Robust Statistics and Outlier Detection

Development of statistical methods resistant to deviations from model assumptions and contaminated data.

Information Theory and Entropy Methods

Study of information measures and their probabilistic interpretations with applications to statistics and machine learning.

Random Geometry and Shape Analysis

Probabilistic study of random geometric objects including random polytopes, tessellations, and shape spaces.

Semigroups and Generator Theory

Analysis of operator semigroups and their generators with applications to PDEs and diffusion processes.

Gibbs Measures and Statistical Mechanics

Probabilistic models in statistical physics including phase transitions and ferromagnetic systems.

Graphical Models and Bayesian Networks

Probabilistic inference in directed and undirected graphical models with applications to causal reasoning.

Multiscale Analysis and Homogenization

Study of effective behavior of systems with multiple scales in random and periodic environments.

Change Point Detection and Anomaly Discovery

Statistical methods for identifying abrupt changes in time series and detecting unusual patterns in data streams.

Concentration Inequalities and High-Dimensional Geometry

Study of probability concentration phenomena in high dimensions with applications to machine learning.

Cox Processes and Doubly Stochastic Intensity

Generalization of point processes with random intensity functions for modeling overdispersion and latent variability.

Renewal Theory and Regeneration

Analysis of recurring events and renewal epochs with asymptotic results applicable to reliability and inventory models.

Hypothesis Testing and Statistical Inference

Theory of optimal tests and confidence sets with applications to multiple testing and sequential analysis.

Spatial Point Patterns and Ripley Functions

Analysis of clustering and regularity in spatial point distributions using distance-based summary statistics.

Stochastic Volatility and Jump Models

Financial modeling incorporating time-varying volatility and jump discontinuities in asset prices.

Importance Sampling and Rare Event Simulation

Advanced Monte Carlo techniques for efficient estimation of small probabilities in complex systems.

Wiener Chaos and Malliavin Calculus

Functional calculus on Wiener space with applications to stochastic analysis and sensitivity analysis.

Hidden Markov Models and Filtering

State estimation in partially observed Markov processes including Kalman filtering and particle filters.

Free Probability Theory and Random Operators

Studies noncommutative probability spaces and asymptotic freeness phenomena in operator algebras with applications to random matrix models.

Optimal Stopping and Sequential Decision Making

Analyzes optimal timing strategies for stopping random processes under uncertainty using martingale techniques and dynamic programming.

Spin Glasses and Disordered Systems

Investigates probabilistic models of disordered magnetic systems exhibiting complex phase transitions and replica symmetry breaking.

Stochastic Partial Differential Equations

Develops solution theory and regularity results for partial differential equations driven by infinite-dimensional noise sources.

Self-Normalized Limit Theorems and Stable Processes

Establishes asymptotic distributional results for normalized sums without moment conditions using stable law theory.

Coalescent Theory and Genealogical Structures

Analyzes random genealogical trees describing ancestry relationships in population genetics and evolutionary biology.

Nonparametric Bayesian Methods and Stick-Breaking Processes

Develops inference procedures using random probability measures including stick-breaking and Pitman-Yor processes.

Anderson Localization and Random Schrodinger Operators

Studies spectral properties and eigenfunctions of random quantum Hamiltonians exhibiting localization phenomena.

Branching Random Walks and Genealogies

Analyzes spatial distribution and genealogical structure of branching processes with random displacement mechanisms.

Harmonic Analysis on Groups and Random Walks

Studies spectral properties and convergence rates of random walks on algebraic structures using representation theory.

Dynamical Systems and Chaos Theory

Investigates long-term probabilistic behavior and invariant measures of chaotic deterministic and random systems.

Probabilistic Combinatorics and Random Structures

Applies probability methods to prove existence and analyze properties of combinatorial objects via probabilistic arguments.

Ergodic Theory and Mixing Properties

Studies long-run statistical behavior of invariant transformations and decay of correlations in dynamical systems.

Boolean Models and Random Covers

Analyzes probabilistic properties of random unions of geometric objects with applications to coverage and percolation.

Stochastic Network Models and Epidemic Spreading

Models disease transmission and information diffusion on random networks using continuous-time branching processes.

Tensor Methods and High-Dimensional Statistical Inference

Develops probabilistic techniques for analyzing multilinear data structures in high-dimensional statistical problems.

Randomized Algorithms and Probabilistic Analysis

Studies computational complexity and correctness guarantees of algorithms using probabilistic methods and randomization.

Stochastic Dominance and Order Statistics

Analyzes partial orderings of distributions and properties of ranked samples in probabilistic frameworks.

Random Field Models and Kriging Interpolation

Develops spatial prediction methods using Gaussian and non-Gaussian random field models with covariance structure.

Voter Models and Interacting Systems

Studies competing particle systems and consensus dynamics in spatial populations under random interaction.

Pathwise Calculus and Rough Paths

Develops differential equation theory for irregular paths using algebraic structures beyond classical Ito calculus.

Extreme Dependence and Copula Tail Behavior

Analyzes joint extreme value behavior and tail dependence structures in multivariate distribution models.

Urn Models and Polya Processes

Studies sampling with replacement and reinforced random processes arising from urn schemes and exchangeability.

Stochastic Resonance and Noise-Induced Phenomena

Investigates beneficial effects of noise on signal detection and synchronization in nonlinear systems.

Conformal Field Theory and Scaling Limits

Studies universal scaling behavior and conformal invariance of lattice models at critical points.

Cramer Large Deviations and Rate Functions

Establishes exponential decay rates for sample means deviating from expected values via contraction principle.

Stochastic Homogenization and Effective Coefficients

Derives effective macroscopic equations for differential equations with rapidly oscillating random coefficients.

Probabilistic Recursion and Iterated Random Functions

Analyzes fixed points and distributional limits of compositions of random functions and recursive equations.

Bootstrap Methods and Resampling Theory

Develops data-dependent resampling techniques for statistical inference and confidence interval construction.

Harmonic Functions and Potential Theory

Studies harmonic functions and capacity theory related to random walks and Brownian motion excursions.

Moderate Deviations and Log-Log Asymptotics

Establishes intermediate-rate asymptotic probabilities between large deviations and central limit regimes.

Stochastic Acceleration and Particle Trapping

Analyzes unbounded growth of particle velocities and trapping phenomena in random forcing environments.

Moment Closure Methods and Approximate Inference

Develops truncation techniques for infinite moment equations to approximate solutions of stochastic systems.

Extremes of Dependent Sequences and Clusters

Studies clustering and dependence structure of extreme events in non-independent data sequences.

Phase Separation and Ising Model Dynamics

Investigates temporal evolution of disordered phases and interface dynamics in statistical mechanical systems.

Sobolev Spaces and Regularity Theory

Develops functional analytic foundations for analyzing existence and smoothness of solutions to stochastic equations.

Quantum Markov Chains and Open Systems

Studies decoherence and master equations for quantum systems using noncommutative probability frameworks.

Renewal-Reward Processes and Regeneration

Analyzes cost-benefit analysis and long-term performance metrics using renewal structure and regenerative blocks.

Random Polytopes and Convex Geometry

Studies volume and surface area distribution of convex hulls of random point sets in Euclidean space.

Variational Methods and Euler-Lagrange Equations

Develops calculus of variations for stochastic functionals and optimality conditions via variational principles.

Interacting Brownian Motions and Collision Dynamics

Analyzes systems of interacting diffusions with collisions, repulsion, and excluded volume interactions.

Random Surfaces and Interface Growth Models

Studies scaling properties and height fluctuations of random surfaces arising from growth processes.

Probabilistic Finite Element Methods

Develops discretization schemes incorporating uncertainty quantification for differential equation approximations.

First Passage Time Distributions and Exit Problems

Analyzes hitting time laws and exit probabilities for Markov processes from domains and barriers.

Ultracontractivity and Log-Sobolev Inequalities

Establishes smoothing properties and mixing bounds using functional inequalities for Markov semigroups.

Boolean Percolation and Continuum Models

Studies connectivity and clustering phenomena in random geometric graphs and continuum percolation.

Functional Data Analysis and Infinite Dimensions

Develops statistical methods for analyzing random functions and functional time series data.

Stochastic Averaging and Slow-Fast Dynamics

Derives effective dynamics for systems with multiple timescales using averaging and homogenization theory.

Palm Theory and Point Process Conditioning

Studies conditional distributions and reduced point processes revealing local properties of random measures.

Machine Learning Generalization and Sample Complexity

Analyzes probability bounds for learning algorithm error using empirical process and PAC-learning theory.

Boolean Network Dynamics and Noise Robustness

Studies stochastic behavior and stability of discrete dynamical systems with random inputs and Boolean logic transitions.

Stochastic Partial Differential Equations and Regularity

Investigates existence, uniqueness, and regularity of solutions to SPDEs with multiplicative and additive noise.

Exchangeability and de Finetti Theorems

Analyzes permutation-invariant probability structures and their representation through mixing distributions.

Pathwise Integration and Rough Paths Theory

Develops integration theory for irregular paths using algebraic and analytic structures beyond classical Ito integration.

Additive Processes and Subordination

Explores processes with independent increments and their composition with stopping times and other random subordinators.

Recurrence and Transience of Random Walks

Characterizes return properties of random walks on different graph structures and metric spaces.

Self-Normalized Limit Theorems and Statistics

Studies asymptotic behavior of ratios of sums and self-normalized statistics without moment conditions.

Brownian Motion Excursions and Local Time

Analyzes paths of Brownian motion between zero crossings and properties of accumulated time at levels.

Stochastic Homogenization and Effective Coefficients

Determines effective behavior of PDEs with rapidly oscillating random coefficients through multiscale analysis.

Palm Theory and Conditional Distributions

Develops theory of conditioning point processes to contain a point at origin or specific location.

Coupling and Monotonicity Methods

Uses coordinated representations of random variables to establish stochastic ordering and dependence properties.

Free Probability and Operator Algebras

Studies probability theory in the context of non-commutative operators without commutativity assumptions.

Conformal Invariance and Critical Phenomena

Investigates scale-invariant random structures and their conformal symmetries at phase transitions.

Schramm-Loewner Evolution and SLE Curves

Studies random conformal curves defined by growing slits in the complex plane with Loewner dynamics.

Multiplicative Chaos and Log-Correlated Fields

Analyzes singular random measures obtained by exponentiating correlated Gaussian fields with logarithmic covariance.

Branching Random Walks and Genealogies

Studies particles that move randomly and reproduce, analyzing spatial distribution and ancestral structure.

Universality and Limit Shape Phenomena

Examines convergence of random growth models and interface processes to universal deterministic shapes.

Reflected Boundary Problems and Sticky Boundaries

Studies behavior of diffusions at boundaries with various interactions including reflection and sticky points.

Occupation Time Density and Harmonic Analysis

Analyzes distribution of cumulative time random processes spend in given sets using spectral methods.

Maximum Likelihood and M-Estimation Asymptotics

Studies consistency and limiting distributions of extremum estimators under weak regularity conditions.

Empirical Likelihood and Bootstrap Methods

Develops distribution-free inference techniques using empirical data distributions and resampling strategies.

Urn Models and Reinforcement Processes

Studies self-reinforcing processes where outcomes feed back to influence future probabilities.

Interacting Branching Diffusions and Competition

Analyzes systems of particles that branch and move while competing for resources or space.

Voter Models and Opinion Dynamics

Investigates stochastic processes modeling consensus formation and coexistence in heterogeneous populations.

Coalescing Processes and Genealogical Trees

Studies merging of random paths and dual representations relating genealogies to spatial structure.

Fluctuation Dissipation and Response Theory

Connects response of systems to perturbations with fluctuations of unperturbed equilibrium dynamics.

Limit Theorems for Non-Stationary Sequences

Extends convergence results to sequences with time-varying distributions and dependent increments.

Spectral Gap and Mixing Rate Bounds

Estimates convergence speed to stationarity using spectral properties of transition operators.

Contraction Properties and Lyapunov Exponents

Studies exponential rates of convergence in Markov processes and stability of random dynamical systems.

Age-Dependent Branching and Bellman Equations

Analyzes branching processes where reproduction depends on individual age and cumulative history.

Kinetic Theory and Boltzmann Equations

Derives macroscopic kinetic equations from stochastic particle collisions and interactions.

Nonequilibrium Statistical Mechanics and Driven Systems

Studies probability distributions and phase behavior of systems far from thermodynamic equilibrium.

Fluctuation Theorems and Jarzynski Equality

Investigates symmetry properties of work and fluctuation probabilities in driven stochastic systems.

Ergodicity Breaking and Glassy Dynamics

Studies non-ergodic behavior and slow relaxation in disordered systems with random energy landscapes.

Directed Polymers in Random Media

Analyzes paths of polymers wandering through random obstacle potentials with extremal and scaling properties.

Gaussian Free Fields and Harmonic Functions

Studies properties of Gaussian fields defined implicitly through Laplace operators and potential theory.

Loop Soup and Gaussian Multiplicative Chaos

Analyzes random collections of non-intersecting loops and measures constructed from exponential fields.

Metric Measure Spaces and Ricci Curvature

Develops probability theory on non-smooth spaces using synthetic geometry and curvature bounds.

Gradient Flows and Entropy Methods

Studies evolution equations as gradient flows of free energy functionals with convexity applications.

Regenerative Block Bootstrap and Dependent Sampling

Develops resampling methods for dependent data using regeneration structure and blocking strategies.

Moderate Deviations and Refined Asymptotic Analysis

Studies probabilities of rare events at intermediate scales between local limit and large deviations.

Spectral Methods for Matrix-Valued Processes

Analyzes eigenvalue distributions and spectral density of random matrix-valued time series.

Stable Laws and Tail Behavior

Investigates distributions with power-law tails and their role in limit theorems without finite moments.

Self-Normalized Sums and Recent Progress

Studies asymptotic distributions of normalized sums with denominator depending on sample itself.

Particle Filters and Sequential Monte Carlo

Develops approximation algorithms for computing filter distributions in hidden Markov state spaces.

Operator Scaling and Functional Limit Laws

Studies convergence of multi-parameter processes under matrix-valued normalization and scaling.

Branching in Varying Environments

Analyzes branching processes where reproduction rates change over time or depend on external conditions.

Queueing Networks and Fluid Limits

Studies heavy-traffic behavior and fluid approximations of large-scale interconnected queue systems.

Branching Measure-Valued Processes

Investigates branching processes where population state is described by random measures on space.

Multitype Branching and Nonlinear Dynamics

Studies branching processes with multiple types where reproduction rates depend on population composition.

Stochastic Partial Differential Equations and Noise

Investigates well-posedness, regularity, and long-time behavior of SPDEs with multiplicative and additive noise perturbations.

Palm Distributions and Conditional Probability

Analyzes conditional distributions of point processes and their applications to stochastic geometry and spatial point patterns.

Metastability and Quasi-Stationary Distributions

Examines long-time behavior of transient stochastic processes and rare exit events before absorption.

Subordination and Operator-Stable Processes

Studies time-changed stochastic processes and their spectral properties through subordination functions and operator-stable generators.

Graphon Convergence and Limit Graphons

Develops theory of graphon limits for dense networks and their application to random graph sampling and exchangeability.

Voter Models and Coalescing Processes

Analyzes stochastic models of opinion dynamics and genealogical merging with phase transitions and critical behavior.

Stein Discrepancy and Goodness of Fit

Develops kernel Stein discrepancy metrics for assessing distributional approximation quality in computational inference.

Reflected Brownian Motion and Boundaries

Studies Brownian motion constrained to domains with reflecting or sticky boundaries and their excursion theory.

Tensor Models and Random Structures

Investigates random tensor models exhibiting phase transitions and connections to quantum gravity and statistical mechanics.

Branching Brownian Motion and Extremes

Studies maximum displacement and genealogies in branching Brownian motion systems with applications to spatial competition.

Rough Path Theory and Pathwise Analysis

Develops analysis of highly irregular paths and their integration theory for stochastic modeling without explicit noise assumptions.

Dyson Brownian Motion and Eigenvalue Dynamics

Analyzes the dynamics of random matrix eigenvalues under Brownian perturbations with applications to universality and spacing statistics.

Symmetry Properties and Exchangeability

Explores de Finetti theorems and representations of exchangeable random sequences with applications to Bayesian modeling.

Hawkes Processes and Self-Exciting Events

Studies self-exciting point processes with clustering behavior applied to seismology, neuroscience, and financial markets.

Ultrametric Spaces and p-adic Probability

Develops probability theory on non-Archimedean ultrametric spaces with applications to hierarchical and tree-like structures.

Excursion Decomposition and Local Times

Studies excursions of Markov processes and local time behavior using the Tanaka formula and occupation density measures.

Boolean Networks and Random Automata

Analyzes the dynamics and attractors of random Boolean networks modeling gene regulatory systems and biological complexity.

Regularity Structures and Singular SPDEs

Develops Haag-Kastler regularity theory for distributional solutions to highly singular stochastic PDEs.

Ballot Problems and Cyclic Lemmas

Applies combinatorial probabilistic methods to path counting and reflection principle problems in random walks.

Multiplicative Chaos and Multifractal Analysis

Studies logarithmically correlated Gaussian fields and their exponential transformations exhibiting phase transitions.

Conformal Invariance and SLE Theory

Develops Schramm-Loewner evolution for critical phenomena exhibiting conformal invariance in two-dimensional systems.

Stochastic Resonance and Noise Benefits

Investigates constructive roles of noise in signal detection and dynamical systems enhancement through stochastic resonance mechanisms.

Ancestral and Genealogical Processes

Studies backward-in-time genealogical structures and coalescent processes in population genetics and evolutionary models.

Isoperimetric Inequalities and Probability

Applies geometric isoperimetric principles to derive concentration bounds and measure extension theorems.

Thinning and Superposition of Point Processes

Analyzes operations on point processes including random deletion, merging, and marked point process decompositions.

Chaos Expansion and Multiple Stochastic Integrals

Develops orthogonal expansions of random functionals using iterated stochastic integrals with applications to approximation theory.

Functional Central Limit Theorems and Tightness

Establishes convergence of stochastic process paths in infinite-dimensional spaces using tightness criteria and relative compactness.

Entropic Regularization and Optimal Transport

Uses entropy regularization to approximate optimal transport problems with computational efficiency improvements and curvature analysis.

Subordinators and Increasing Levy Processes

Studies monotone Levy processes as random time changes with applications to first passage times and inverse processes.

Branching Processes Conditioned on Survival

Analyzes the conditional distribution and asymptotic behavior of branching processes given non-extinction events.

Scattering and Inverse Problems

Develops probabilistic methods for solving inverse problems in wave propagation and imaging through random medium propagation.

Polya Urns and Reinforcement Learning

Studies urn models with reinforcement mechanisms exhibiting rich limiting behavior with applications to adaptive sampling.

Convergence Rates and Berry-Esseen Bounds

Quantifies the speed of convergence to limiting distributions through explicit error bounds for normal and other approximations.

Nonstandard Analysis and Hyperfinite Probability

Applies nonstandard analysis to probability theory providing infinitesimal approaches to limits and stochastic calculus.

Aldous-Brouwer Diffusion and Limits

Studies limiting behavior of random trees and forests through diffusion approximations and continuum-random-tree convergence.

Stable Processes and Alpha-Stable Distributions

Investigates stable Levy processes with heavy tails and their subordination properties in anomalous transport.

Localization and Anderson Transitions

Studies spectral phase transitions in random Schrodinger operators modeling electron transport in disordered materials.

Randomized Algorithms and Probabilistic Methods

Applies probabilistic techniques to algorithm design and analysis with efficiency guarantees for computational problems.

Cramer Large Deviations and Rate Functions

Extends large deviations theory to empirical measures with applications to statistical inference and hypothesis testing.

Spectral Methods in Stochastic Analysis

Uses spectral theory of generators and semigroups to study eigenvalues and eigenfunctions of stochastic processes.

Self-Organized Criticality and Avalanches

Analyzes stochastic dynamical systems exhibiting power-law behavior and avalanche distributions without tuning.

Skorohod Embedding and Martingale Representations

Studies embeddings of random variables into Brownian motion paths with applications to martingale approximation.

Stochastic Homogenization and Effective Behavior

Derives effective equations for processes in heterogeneous random media through multiscale averaging techniques.

Schramm-Loewner Evolution and Interfaces

Studies random curves and interfaces at criticality using Loewner chains and conformal maps in scaling limits.

Survival Probabilities and Green Functions

Analyzes transient behavior of Markov processes through Green functions and hitting probability calculations.

Nonlocal Operators and Fractional Calculus

Develops probabilistic approaches to nonlocal and fractional differential operators with applications to anomalous diffusion.

Mixing and Ergodic Theory of Markov Chains

Studies asymptotic properties and convergence to stationarity using spectral gaps and coupling constructions.

Rough Path Theory and Regularity Structures

This research area develops the theory of rough paths and regularity structures to solve stochastic differential equations driven by highly irregular signals, with applications to singular stochastic PDEs and pathwise analysis of stochastic systems.

Barrier Crossing and Escape Times

Investigates first passage and exit times for diffusion processes crossing random and deterministic barriers.

Probabilistic Methods in Combinatorics and Extremal Problems

This field applies probabilistic techniques and random graph models to solve extremal combinatorial problems, existence proofs, and threshold phenomena through the use of the probabilistic method and concentration bounds.