In this paper we introduce two novel convolutions for the fractional Fourier transforms (FRFT), and prove natural algebraic properties of the corresponding multiplications such as commutativity, associativity and distributivity, which may be useful in signal processing and other types of applications. We analyze a consequent comparison with other known convolutions, and establish a necessary an...

In this paper we study the solvability of backward doubly stochastic differential equations (BDSDEs for short) with polynomial growth coefficients and their connections with SPDEs. The corresponding SPDE is in a very general form, which may depend on the derivative of the solution. We use Wiener-Sobolev compactness arguments to derive a strongly convergent subsequence of approximating SPDEs. Fo...

A general approach is developed to solvability theorems involving a broad class of functions, here called H-convex functions and inf-H-convex functions. The concept of Minkowski duality is exploited to provide dual characterizations for certain innnite inequality systems. The results not only cover the recently developed solvability results involving DSL functions, concave functions and diieren...

The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular (ordinary and generalized) equations over torsion free groups, we ...

Following a recently considered generalization of linear equations to unordered data vectors, we perform a further generalization to ordered data vectors. These generalized equations naturally appear in the analysis of vector addition systems (or Petri nets) extended with ordered data. We show that nonnegative-integer solvability of linear equations is computationally equivalent (up to an expon...

We study the Lp Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed p > 2, we show that a reverse Hölder condition with exponent p is sufficient for the solvability of the Dirichlet problem with boundary data in LpN (∂Ω, R d). Then we obtain a much simpler condition which implies the reverse Hölder condition. Finally, we establish the solvability of the Lp Dirichlet prob...

These lecture notes are based on Ivan Corwin’s summer 2014 MSRI summer school on SPDEs, as well as Jeffrey Kuan’s TA sessions accompanying these lectures. Please email [email protected] if you have questions or find mistakes. 1. Mild solution to the stochastic heat equation The stochastic heat equation (SHE) with multiplicative noise looks (in differential form) like { ∂tz = 1 2∂xxz + zξ z(...