The aim of this paper is to establish new pointwise regularity results for solutions degenerate second-order partial differential equations with a Kolmogorov-type operator the form $$\begin{aligned} {\mathscr {L}}:=\sum _{i,j=1}^m \partial ^2_{x_i x_j } +\sum _{i,j=1}^N b_{ij}x_j\partial _{x_i}-\partial _t, \end{aligned}$$ where $$(x,t) \in {{\mathbb {R}}}^{N+1}$$ , $$1 \le m N$$ and matrix $$B...

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We deal with a class of semilinear parabolic PDEs on the space continuous functions that arise, for example, as Kolmogorov equations associated to infinite-dimensional lifting path-dependent SDEs. investigate existence smooth solutions through their representation via forward–backward stochastic systems, which we provide necessary regularity theory. Because lack smoothing properties operators a...

We introduce and study some backward Kolmogorov equations associated to filtering problems. In the stochastic framework, SDEs for measure-valued processes arise naturally (Zakai Kushner–Stratonovich equation). The have been intensively studies, assuming that admit a density then by exploiting calculus in Hilbert spaces. Our approach differs from this since we do not assume existence of work dir...

Abstract. We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly no...

The main aim of the paper is to incorporate the nonlinear kinetic term into non-Markovian transport equations described by a continuous time random walk (CTRW) with nonexponential waiting time distributions. We consider three different CTRW models with reactions. We derive nonlinear Master equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and w...