Stochastic Processes

Lévy Processes and Jump Diffusions

Investigation of stochastic processes with discontinuous sample paths and their applications in financial modeling and risk assessment.

RESEARCH GAP FRONTIERS

Fractional Brownian Motion Theory

Study of non-Markovian Gaussian processes with long-range dependence and self-similarity properties.

RESEARCH GAP FRONTIERS

Backward Stochastic Differential Equations

Analysis of BSDEs and their connections to partial differential equations, optimal control, and mathematical finance.

RESEARCH GAP FRONTIERS

Interacting Particle Systems

Study of collective behavior emerging from interactions between stochastic particles with applications to phase transitions and spin systems.

RESEARCH GAP FRONTIERS

Stochastic Volatility Models

Development and analysis of models where asset price volatility is itself a random process, fundamental in derivative pricing.

RESEARCH GAP FRONTIERS

Markov Chain Monte Carlo Methods

Development of computational algorithms using Markov chains for Bayesian inference and sampling from complex distributions.

RESEARCH GAP FRONTIERS

Gaussian Process Regression

Nonparametric Bayesian approach using stochastic processes as priors for machine learning and prediction tasks.

RESEARCH GAP FRONTIERS

Hawkes Point Processes

Analysis of self-exciting point processes where event arrivals cluster temporally with applications in seismology and finance.

RESEARCH GAP FRONTIERS

Branching Processes and Population Dynamics

Study of reproduction mechanisms in populations with applications to epidemiology, extinction probability, and genealogical structures.

Stochastic Partial Differential Equations

Theory and numerical methods for partial differential equations driven by noise, arising in fluid dynamics and quantum mechanics.

Measure-Valued Processes

Analysis of stochastic processes taking values in spaces of measures with applications to branching and particle systems.

Large Deviations Theory

Study of probability of rare events and asymptotic behavior of stochastic systems under extreme conditions.

Stochastic Filtering and State Estimation

Optimal estimation of unobservable system states from noisy observations with applications to signal processing and control.

Random Matrix Theory

Asymptotic properties of eigenvalues and eigenvectors of random matrices with applications to wireless communications and statistics.

Weak Convergence and Donsker Theorems

Functional limit theorems for sequences of stochastic processes converging to Brownian motion or other limiting processes.

Cox Processes and Doubly Stochastic Models

Point processes with random intensity functions modulated by an underlying stochastic process.

Martingale Theory and Optimal Stopping

Foundational theory of martingales and principles for determining optimal times to stop stochastic processes.

Rough Path Analysis

Framework for analyzing stochastic differential equations driven by highly irregular paths with limited regularity.

Mean Field Games

Study of Nash equilibria in games with infinitely many agents through stochastic processes and optimal control.

Stochastic Gradient Descent Convergence

Analysis of convergence properties of stochastic optimization algorithms widely used in machine learning.

Piecewise Deterministic Markov Processes

Study of processes with deterministic motion punctuated by random jumps with applications to queueing and switching systems.

Extreme Value Theory for Stochastic Processes

Asymptotic behavior of maximum and minimum values of stochastic processes relevant to risk management.

Stochastic Reaction-Diffusion Systems

Dynamics of chemical reactions in spatially extended systems with diffusion under stochastic perturbations.

Regenerative Processes and Renewal Theory

Study of processes that return to initial states with applications to queueing, reliability, and maintenance problems.

Stochastic Optimal Transport

Optimal transport problems solved through stochastic processes and martingale methods for probability measures.

Multidimensional Diffusion Processes

Analysis of Itô diffusions in multiple dimensions including properties of covariance structure and boundary behavior.

Time-Changed Processes and Subordination

Study of stochastic processes obtained by replacing deterministic time with random subordinators.

Jump-Diffusion Models in Finance

Modeling asset prices with both continuous diffusion and discrete jump components for option pricing.

Ergodicity and Mixing Properties

Long-run statistical behavior of stochastic processes and convergence to stationary distributions.

Stochastic Differential Game Theory

Analysis of competitive and cooperative strategies in games governed by stochastic differential equations.

Path Space Analysis and Skohorod Topology

Functional analytic methods for studying convergence and properties of stochastic processes in path spaces.

Stochastic Control and Hamilton-Jacobi-Bellman

Optimal control of systems governed by stochastic processes through dynamic programming equations.

Continuous-Time Markov Chains

Theory of stochastic processes on discrete state spaces with continuous parameter for jump processes.

Reflected and Constrained Brownian Motion

Brownian motion with boundary conditions and constraints relevant to queueing and biological systems.

Sparse Grids and High-Dimensional Stochastic PDEs

Computational methods for solving stochastic PDEs in high dimensions using polynomial chaos and sparse approximations.

Stochastic Homogenization and Multiscale Processes

Analysis of effective behavior of heterogeneous stochastic systems across multiple temporal and spatial scales.

Stochastic Volatility Jump Models

Combination of stochastic volatility and jump components for modeling realistic financial asset price dynamics.

Information Filtration and Progressive Processes

Study of stochastic processes adapted to information structures with applications to optimal stopping and filtering.

Pathwise Integrals and Stratonovich Calculus

Alternative approaches to stochastic integration preserving certain classical calculus rules.

Multifractal Processes and Self-Affinity

Analysis of stochastic processes exhibiting fractal structure at multiple scales with varying regularity.

Coupling and Transport Methods

Probabilistic techniques of coupling processes on common probability spaces for convergence and stability analysis.

Stochastic Equations with Memory Effects

Study of non-Markovian stochastic processes where future evolution depends on entire past history.

Continuous-Space Particle Systems

Analysis of infinitely many interacting particles in continuous spaces with applications to statistical mechanics.

Stein Method and Normal Approximation

Probabilistic techniques for proving central limit theorems and approximations for dependent stochastic processes.

Occupation Time and Local Time Processes

Study of cumulative time spent at specific levels by stochastic processes with applications to boundary crossing.

Regime-Switching Models

Stochastic processes with parameters that change according to hidden Markov chains for modeling regime changes.

Stochastic Approximation Algorithms

Iterative algorithms using noisy observations for solving equations or optimization problems.

Inverse Problems and Stochastic Inversion

Probabilistic methods for recovering unknown parameters or functions from noisy indirect observations.

Free Probability and Random Matrices

Noncommutative probability theory for independent random matrices with limiting spectral distributions.

Stochastic Network Models and Epidemic Spread

Random graph evolution and disease propagation through populations modeled as stochastic processes.

Stochastic Delay Differential Equations

Analysis of differential equations with time-delayed stochastic terms, addressing stability, existence, and uniqueness of solutions in systems with memory.

Branching Random Walks and Genealogies

Study of spatial branching processes combining random walk dynamics with genealogical structure to model population expansion and evolution.

Stochastic Filtering for Nonlinear Systems

Development and analysis of particle filters and sequential Monte Carlo methods for optimal estimation in nonlinear dynamical systems.

Variational Methods in Stochastic Analysis

Application of calculus of variations to stochastic processes for deriving optimality conditions and studying variational representations.

Moderate Deviations Principles and Asymptotics

Investigation of moderate deviation rates between large and large deviation regimes for refined asymptotic behavior of stochastic systems.

Stochastic Combustion and Reaction Kinetics

Modeling and analysis of chemically reactive systems with random fluctuations using stochastic differential equations and chemical master equations.

Non-Gaussian Random Fields and Processes

Theory and applications of random fields beyond Gaussian assumptions, including stable processes and infinitely divisible distributions.

Spin Glass Models and Disorder Systems

Study of random Hamiltonian systems with disorder using stochastic methods to understand phase transitions and ground state properties.

Stochastic Modeling of Biological Sequences

Development of hidden Markov models and stochastic context-free grammars for analyzing DNA, RNA, and protein sequences.

Functional Data Analysis with Stochastic Methods

Statistical inference techniques for functional data treating observations as realizations of stochastic processes with applications in smoothing and classification.

Stochastic Algorithms for Machine Learning

Analysis of stochastic gradient methods, variance reduction techniques, and convergence guarantees for large-scale optimization in neural networks.

Cluster Expansion Methods in Statistical Mechanics

Rigorous perturbative analysis of random systems through cluster expansions to prove convergence and uniqueness in high-dimensional models.

Stochastic Poisson Equations and Resolvents

Solvability theory for stochastic Poisson equations and analysis of resolvents of infinitesimal generators of Markov processes.

Multilevel Monte Carlo Methods and Analysis

Development of hierarchical sampling strategies to dramatically reduce computational complexity in uncertainty quantification for stochastic systems.

Quantum Stochastic Processes and Open Systems

Extension of stochastic calculus to quantum systems using Fock space methods to model decoherence and dissipation in open quantum dynamics.

Stochastic Morphogenesis and Pattern Formation

Analysis of noise-driven pattern generation in biological systems through stochastic reaction-diffusion and Turing instability mechanisms.

Brane Dynamics in Random Environments

Study of elastic membranes and interfaces evolving in random potentials using pinning models and stochastic interface equations.

Stochastic Finite Element Methods

Numerical discretization and error analysis for stochastic PDEs combining finite element approximations with intrinsic randomness representations.

Catalytic Reactions and Stochastic Kinetics

Mathematical modeling of chemical catalysis using continuous-time Markov chains and stochastic rate equations with fast-slow dynamics.

Directed Polymers in Random Media

Investigation of polymer paths through random environments combining probability theory with replica methods and disorder effects.

Stochastic Demography and Population Genetics

Modeling of population dynamics under randomness including genetic drift, mutations, and selection using branching and birth-death processes.

Optimal Transport in Stochastic Settings

Extension of optimal transport theory to random measures and applications in distributional robustness of stochastic control problems.

Scattering and Resonances in Random Media

Rigorous analysis of wave propagation through disordered systems using stochastic perturbation theory and localization phenomena.

Stochastic Heat Kernel Estimates

Precise asymptotic analysis of fundamental solutions to stochastic parabolic equations and heat kernels on manifolds with random geometry.

Bayesian Nonparametrics and Stochastic Process Priors

Theory of Dirichlet processes, Gaussian process priors, and their generalizations for flexible Bayesian inference with uncertainty quantification.

Stochastic Bifurcation Theory and Critical Transitions

Analysis of qualitative changes in long-term dynamics of noisy systems near bifurcation points and early warning indicators for tipping points.

Nonlinear Filtering and Particle Approximations

Convergence analysis of particle systems approximating conditional distributions in nonlinear filtering problems with applications to data assimilation.

Stochastic Geometry and Poisson Point Processes

Theory of random spatial point patterns and geometric structures including random graphs, tessellations, and percolation phenomena.

Variational Inference for Stochastic Models

Optimization of variational bounds to approximate intractable posterior distributions in Bayesian inference for complex stochastic systems.

Stochastic Control of Epidemic Processes

Optimal intervention strategies for stochastic epidemic models minimizing disease spread through vaccination, quarantine, or treatment policies.

Anomalous Diffusion and Subdiffusive Processes

Analysis of non-Gaussian limiting behavior in processes with long-range dependence, including continuous-time random walks and fractional equations.

Stochastic Synchronization and Coupled Oscillators

Study of collective dynamics and phase locking phenomena in networks of noisy oscillators using averaging and bifurcation techniques.

Backward Stochastic Volterra Integral Equations

Theory and applications of backward equations with memory kernels extending classical backward SDEs to systems with functional dependencies.

Random Graph Evolution and Preferential Attachment

Analysis of growing random networks with preferential attachment mechanisms characterizing scale-free degree distributions and network properties.

Stochastic Riemannian Geometry and Curvature Effects

Investigation of diffusion processes on curved manifolds with emphasis on how curvature influences heat kernels, geodesics, and exit times.

Entropic Methods in Stochastic Analysis

Application of information-theoretic concepts and entropy methods to prove functional inequalities and concentration bounds for stochastic processes.

Stochastic Resource Allocation and Scheduling

Optimal dynamic control of allocating resources to queues and jobs under uncertainty using Markov decision processes and throughput optimization.

Scaling Limits of Discrete Stochastic Systems

Rigorous passage from discrete Markov chains and particle systems to continuous-time continuous-space stochastic processes through scaling limits.

Stochastic Thermodynamics and Entropy Production

Foundation of non-equilibrium statistical mechanics using stochastic dynamics to derive entropy production, fluctuation theorems, and dissipation.

Stochastic Delay and Neutral Type Equations

Analysis of stochastic functional differential equations with delays in both drift and diffusion for systems with hereditary dynamics.

Risk-Sensitive Stochastic Control Problems

Optimal control under exponential risk measures and distortion functions modeling decision-maker risk aversion in stochastic environments.

Poisson Approximation and Stein Bounds

Quantitative bounds for approximating rare event probabilities using Stein method with applications to limit theorems for sparse processes.

Stochastic Wave Equations and Acoustic Scattering

Well-posedness and regularity theory for stochastic hyperbolic PDEs with applications to random wave propagation and noisy acoustic fields.

Multitype and Branching with Immigration Processes

Asymptotic behavior and extinction criteria for branching processes with multiple types and random immigration modulating population growth.

Stochastic Numerical Methods for SDEs

Convergence analysis of discretization schemes including Euler, Milstein, and higher-order methods for approximating solutions of SDEs.

Hidden Markov Models with Parameter Learning

Statistical inference algorithms including EM and gradient methods for identifying hidden state parameters from partially observed sequence data.

Stochastic Homology and Persistence Processes

Topological data analysis techniques studying how features persist in random complexes and evolving point clouds using persistent homology.

Small Noise Asymptotics and Exit Problems

Analysis of metastability, mean exit times, and transition rates in systems with small stochastic perturbations near stable equilibria.

Stochastic Portfolio Optimization Under Constraints

Dynamic asset allocation accounting for transaction costs, market impact, and regulatory constraints using backward SDEs and duality methods.

Tau-Leaping and Approximation Schemes in Chemotaxis

Multiscale stochastic simulation methods for biological systems with chemical reactions coupled to directional motion through random environments.

Functional Data Analysis via Stochastic Processes

Development of statistical methods for analyzing infinite-dimensional functional data arising from continuous stochastic processes and their applications.

Stochastic Bifurcation and Critical Phenomena

Study of phase transitions and bifurcations in stochastic dynamical systems near critical points using renormalization group techniques.

Stochastic Geometry and Random Graphs

Examination of geometric properties and percolation phenomena in random point processes and network structures.

Singular Perturbation of Stochastic Systems

Analysis of multiscale stochastic systems with separated timescales using averaging and homogenization principles.

Stochastic Integration with Respect to General Semimartingales

Rigorous development of integration theory for semimartingales without assuming continuity or standard assumptions.

Volatility Clustering and Long-Memory Processes

Analysis of persistence and clustering phenomena in stochastic volatility using fractionally integrated and long-range dependent models.

Nonlinear Stochastic Filtering and Particle Methods

Development and convergence analysis of particle filters and sequential Monte Carlo methods for nonlinear state estimation.

Stochastic Resonance and Noise-Induced Phenomena

Study of constructive effects of noise in amplifying weak signals and enhancing detection in nonlinear stochastic systems.

Generalized Autoregressive Conditional Heteroscedasticity

Theoretical and applied analysis of GARCH models and their extensions for modeling time-varying volatility in financial time series.

Tempered Stable Processes and Subordinators

Investigation of heavy-tailed Lévy processes with exponential tempering and their applications to financial modeling.

Stochastic Master Equation and Coarse Graining

Development of reduced-order models from high-dimensional stochastic systems through projection and information-theoretic methods.

Entrance and Exit Boundary Behavior of Diffusions

Characterization of boundary classification and scale functions for one-dimensional diffusion processes near singular points.

Stochastic Mortality Modeling and Longevity Risk

Analysis of time-dependent mortality hazard rates using age-structured stochastic models for actuarial applications.

Poisson Functionals and Determinantal Point Processes

Investigation of repulsive point processes with determinantal kernels and their statistical properties and inference methods.

Stochastic Lipschitz Regression and Monotonicity Constraints

Development of nonparametric regression methods with shape constraints under random noise using optimal transport ideas.

Local Limit Theorems for Dependent Random Variables

Derivation of density asymptotics for sums of dependent random variables arising from stochastic processes.

Weak Solutions to Nonlinear Stochastic PDEs

Existence and uniqueness of solutions to nonlinear SPDEs using weak convergence and monotonicity methods.

Regeneration and Coupling for Random Walks

Application of regeneration theory and coupling constructions to study exit times and return distributions of multidimensional walks.

Stochastic Representations via Feynman-Kac Formulas

Probabilistic interpretation of solutions to parabolic PDEs and their relation to path integrals and expectation values.

Transition Density Asymptotics for Jump Processes

Detailed analysis of heat kernel estimates and short-time asymptotics for processes with jumps and discontinuities.

Stable Limit Theorems for Dependent Sequences

Convergence to stable distributions under weak dependence conditions using characteristic functions and regular variation theory.

Stochastic Epidemic Models with Intervention Dynamics

Modeling disease spread through branching processes and reaction-diffusion systems with control policies and vaccination strategies.

Recurrence Classification of Markov Chains on Groups

Analysis of transience and recurrence for random walks on infinite groups using spectral and harmonic analysis.

Approximation of Stochastic Integrals via Quadrature

Convergence analysis of discretization schemes for computing stochastic integrals using multilevel and variance reduction methods.

Semi-Parametric Inference for Lévy Processes

Development of estimators for Lévy measure components and jump characteristics from high-frequency observations.

Stochastic Dynamics of Self-Organized Criticality

Analysis of power-law distributions and avalanche dynamics in random systems driven to critical states.

Hypoelliptic Diffusions and Sub-Riemannian Geometry

Study of diffusions on manifolds with degenerate diffusion coefficients using horizontal lifts and Hörmander conditions.

Time-Inhomogeneous Stochastic Differential Equations

Theory and applications of SDEs with time-dependent coefficients including existence, uniqueness, and stability properties.

Sequential Testing and Optimal Stopping Boundaries

Analysis of optimal stopping times in sequential decision problems using free boundary methods and smooth pasting conditions.

Stochastic Convolution and Functional Integration

Treatment of convolution of deterministic functions with stochastic processes and functional analytic methods for SPDEs.

Stochastic Calculus for Processes with Infinite Activity

Development of integration theory and Itô formulas for processes with infinite variation paths and jump activities.

Clustering and Community Detection in Random Graphs

Statistical methods for identifying community structure in random network models using spectral and variational approaches.

Stochastic Resonance in Neuron Models

Study of noise-enhanced signal detection in spiking neurons and neural networks via stochastic dynamical analysis.

Mixing Times for Markov Processes on Networks

Analysis of convergence rates to stationary distributions for random walks on complex networks and graphs.

Likelihood Ratio Methods for Rare Event Simulation

Development of importance sampling schemes for estimating probabilities of rare events in stochastic systems.

Stochastic Volatility with Microstructure Noise

Analysis of high-frequency financial data incorporating market microstructure effects and noise using hidden state models.

Functional Inequalities for Markov Processes

Development and application of Sobolev, Poincaré, and log-Sobolev inequalities to control mixing and concentration.

Stochastic Dominance and Order Statistics

Comparison of random variables and stochastic processes using dominance orderings with applications to decision theory.

Invariant Distributions for Stochastic Flows

Characterization of stationary measures and invariant foliations for random dynamical systems and stochastic flows.

Stochastic Equations Driven by Fractional Noise

Analysis of differential equations driven by fractional Gaussian processes and their regularity and stability properties.

Urn Models and Pólya Processes

Investigation of urn models with reinforcement and their continuous-time generalizations via Pólya processes.

Stochastic Representation of Harmonic Functions

Probabilistic proofs of harmonic function properties using martingales and optional stopping for diffusion processes.

Coupling Techniques for Convergence Analysis

Use of optimal couplings and coupling-from-the-past to establish convergence rates and invariance principles.

Stochastic Partial Differential Equations on Manifolds

Theory and numerical analysis of SPDEs on curved spaces using geometric analysis and differential forms.

Extremes of Dependent Stationary Sequences

Asymptotic distributions of maxima for sequences with temporal dependence using cluster analysis and point process methods.

Stochastic Scheduling and Queuing Networks

Optimal control of multi-class queueing systems under uncertainty using dynamic programming and fluid limits.

Functional Limit Theorems for Dependent Data

Convergence theory for functional central limit theorems applied to weakly dependent stochastic sequences.

Numerical Methods for SPDEs

Development and analysis of discretization schemes for stochastic partial differential equations in infinite dimensions.

Affine Processes and Markov Structures

Study of affine jump-diffusions and their mathematical properties in multidimensional settings.

Stochastic Heat Equations with Singular Noise

Analysis of heat equations driven by space-time white noise and regularization of singular solutions.

Optimal Stopping in Dynamic Markets

Theory of optimal stopping times under market frictions and incomplete information in finance.

Subordinated Processes and Tempered Dynamics

Investigation of time-subordinated stochastic processes and tempered stable distributions for heavy-tailed phenomena.

Variational Inference for Latent Processes

Approximate Bayesian inference methods for complex latent stochastic process models.

Stochastic Burgers Equation and Shocks

Analysis of the stochastic Burgers equation with focus on shock formation and solution structure.

Particle Filtering and Sequential Monte Carlo

Development of advanced particle filtering algorithms for nonlinear state-space models with applications.

Heston Model Extensions and Calibration

Extensions of the Heston stochastic volatility model with jumps and refined calibration techniques.

Excursion Theory and Path Decomposition

Analysis of excursions of Markov processes away from sets and Itô excursion theory.

Moderate Deviations Principles

Study of moderate deviations for stochastic processes filling the gap between CLT and large deviations.

Stochastic Allen-Cahn Equations

Analysis of stochastic reaction-diffusion equations with bistable potentials and front propagation.

Ruin Probabilities in Risk Theory

Study of survival and ruin metrics for insurance risk models driven by Lévy processes.

Wasserstein Distance and Optimal Coupling

Applications of optimal transport theory to measure distances between stochastic processes.

Stochastic Navier-Stokes Equations

Well-posedness and regularity of stochastic incompressible fluid equations with multiplicative noise.

Persistence and Fluctuation of Gaussian Fields

Analysis of level set crossings and persistence probabilities for Gaussian random fields.

Onsager-Machlup Theory and Action Functionals

Study of action functionals characterizing most probable paths of diffusion processes.

Stochastic Schrödinger Equations

Analysis of quantum stochastic systems with nonlinear noise coupling and conservation laws.

Couplings and Synchronization of Processes

Maximal couplings and synchronization phenomena in stochastic processes with applications.

Multi-Scale Analysis and Averaging Techniques

Perturbation and averaging theory for stochastic systems with multiple time scales.

Branching Random Walks and Genealogy

Study of branching random walks and genealogical structures with applications to population genetics.

Spectral Properties of Markov Semigroups

Analysis of eigenvalues and spectral gaps of Markov transition semigroups and mixing times.

Stochastic Optimal Control in Singular Regimes

Optimal control problems with singular controls and finite fuel constraints.

Volterra Processes and Stochastic Convolution

Study of Volterra equations with stochastic kernels and convolutional noise.

Stochastic Viscous Conservation Laws

Well-posedness of stochastic conservation laws with diffusive regularization.

Harmonic Analysis of Stochastic Processes

Fourier and spectral methods for analyzing properties of random processes.

Quantum Stochastic Calculus

Development of stochastic calculus in quantum probability spaces and operator algebras.

Stochastic Compartmental Models in Epidemiology

Markovian stochastic models for disease transmission with parameter inference methods.

Riemannian Manifold-Valued Brownian Motion

Analysis of diffusion processes on Riemannian manifolds with applications to geometric statistics.

Consensus Dynamics and Multi-Agent Systems

Study of stochastic consensus algorithms for distributed multi-agent networks.

Free Boundary Problems and Stefan Equations

Stochastic versions of Stefan problems with moving interfaces and phase transitions.

Gibbsian and Grand Canonical Ensembles

Statistical mechanics of infinite-volume stochastic systems and Gibbs measures.

Convergence Rates in Markov Chains

Quantitative analysis of convergence rates and burn-in periods for Markov chains.

Stochastic Gradient Descent with Momentum

Convergence analysis of accelerated stochastic gradient methods with variance reduction.

Branching Diffusions and Superprocesses

Study of superprocesses as limits of branching particle diffusions.

Contraction Properties and Lyapunov Exponents

Analysis of exponential stability via contraction metrics and random Lyapunov exponents.

Stochastic Demography and Life Tables

Stochastic modeling of population demographics with parameter uncertainty.

Perturbation Theory for Random Matrices

Eigenvalue perturbation bounds for random matrix ensembles with applications.

Tightness Criteria and Relative Compactness

Prokhorov tightness and relative compactness theorems for functional limit theory.

Stochastic Geometric Networks and Percolation

Analysis of percolation and connectivity in random geometric graph models.

Signal Detection in Stochastic Noise

Optimal detection and estimation theory for signals corrupted by colored noise.

Persistence of Intermittency in Turbulence

Stochastic models for turbulent cascade and intermittent fluctuations.

Stochastic Porous Media Equations

Analysis of stochastic degenerate parabolic equations arising in porous media flow.

Stochastic Heat Equations with Multiplicative Noise

Investigation of well-posedness, regularity, and long-time behavior of heat equations driven by multiplicative noise using renormalization techniques and paraproduct theory.

Generator Expansion and Infinitesimal Analysis

Taylor expansions and infinitesimal generators of Markov semigroups.

Pathwise Uniqueness and Strong Solutions

Analysis of conditions for pathwise uniqueness in stochastic differential equations with irregular coefficients using Tanaka''s formula and Yamada-Watanabe theory.

Entropic Regularization in Stochastic Control

Study of entropy-regularized optimal control problems and their connections to stochastic variational inference and controlled diffusion processes.

Stochastic Population Genetics and Neutral Theory

Markovian models for neutral genetic drift and fixation probabilities.

Non-Local Operators and Stable Processes

Research on fractional Laplacians, stable Lévy processes, and their analytical properties through harmonic measure and capacity theory.

Degenerate Diffusions and Hypoellipticity

Analysis of degenerate diffusion operators and Hörmander hypoellipticity conditions.

Fluctuation Exponents in Random Interface Growth

Study of scaling limits and KPZ universality class in models of stochastic interface evolution using regularity structures and paracontrolled distributions.

Sparsity and Compressed Sensing in Filtering

Development of computationally efficient particle filters and sequential Monte Carlo methods using sparsity constraints and compressed sensing principles.

Deep Learning Approximation of Processes

Neural network methods for approximating solutions to stochastic differential equations.