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I am an Associate Professor in the Department of Mathematics at Michigan State University (Go Green, Go White!). My office is located at C334 Wells Hall , and my email is "rschwab" at the usual math email server, "math.msu.edu".

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I am teaching MTH 320 during Fall 2019. All course documents can be found by students at MSU's D2L portal.

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I study Partial Differential Equations , and frequently I can't help but be drawn to questions involving an overlap between PDE and Probability (although I am not necessarily an expert in Probability). It is a very natural connection to have in the fields of elliptic, parabolic, and integro-differential equations, e.g. random homogenization , diffusion processes and large deviations of their long time behavior, jump processes , interacting particle systems, random walks in a random environment, stochastic optimal control and games -- just to name a few-- are all amenable to study via (usually nonlinear!) PDE methods.

Lately much of my recent and ongoing work is focused in the realm of systems which exhibit some sort of long range and or nonlocal behavior. This leads to PDE with nonlocal terms , typically of an integro-differential type, which in their simplest forms are similar to (or actually are) fractional powers of the Laplacian .

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(13) Min-max formulas for nonlocal elliptic operators on Euclidean Space (Joint with Nestor Guillen) -- Nonlinear Analysis, 2019 pdf , journal page , or arXiv:1812.09642 [math.AP]

(12) Some free boundary problems recast as nonlocal parabolic equations (Joint with Hector Chang Lara and Nestor Guillen) -- Nonlinear Analysis, 2019 pdf , journal page , or arXiv:1807.02714 [math.AP]

(11) Estimates for Dirichlet-to-Neumann maps as integro-differential operators (Joint with Jun Kitagawa and Nestor Guillen) -- Potential Analysis, 2019 pdf , journal page , or arXiv:1710.03152 [math.AP]

(10) Min-max formulas for nonlocal elliptic operators (Joint with Nestor Guillen) -- (To Appear in) Calculus of Variations and PDE pdf or arXiv:1606.08417 [math.AP]

(9) Neumann Homogenization via Integro-Differential Operators, Part 2: Singular Gradient Dependence (Joint with Nestor Guillen)-- SIAM Journal of Mathematical Analysis, 2018 pdf , journal page , or arXiv:1512.06027 [math.AP]

(8) Regularity for parabolic integro-differential equations with very irregular kernels (Joint with Luis Silvestre) -- Analysis and PDE, 2016 pdf , journal page , or arXiv:1412.3790 [math.AP]

(7) Neumann Homogenization via Integro-Differential Operators (Joint with Nestor Guillen)-- Discrete and Continuous Dynamical Systems, Series A, 2016 pdf , journal page or arXiv:1403.1980 [math.AP]

(6) Hölder Regularity For Integro-Differential Equations With Nonlinear Directional Dependence (Joint with Moritz Kassmann and Marcus Rang)-- Indiana University Mathematics Journal, 2014; pdf , journal page , or arXiv:1306.0082 [math.AP]

(5) Regularity results for nonlocal parabolic equations (Joint with Moritz Kassmann)-- Rivista di Matematica della Universita di Parma, 2014; pdf , journal page , or arXiv:1305.5418 [math.AP]

(4) Stochastic Homogenization for Some Nonlinear Integro-Differential Equations-- Communications in Partial Differential Equations, 2012; pdf , journal page , or arXiv:1101.6052v1 [math.AP]

(3) Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations, (Joint with Nestor Guillen) -- Archive for Rational Mechanics and Analysis, 2012; pdf , journal page , or arXiv:1101.0279 [math.AP]

(2) Periodic Homogenization For Nonlinear Integro-Differential Equations, -- SIAM Journal of Mathematical Analysis, 2010; pdf or journal page

(1) Stochastic Homogenization of Hamilton-Jacobi Equations in Stationary Ergodic Spatio-Temporal Media -- Indiana University Mathematics Journal, 2009; pdf or journal page