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The purpose is to study the Cauchy problem for non-linear in time and space pseudo-differential equations. These include the fractional in time versions of HJB equations governing the controlled scaled CTRW. As a preliminary step which is of independent interest we analyse the corresponding linear equation proving its well-posedness and smoothing properties.

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Recent studies have established that the circadian clock influences onset, progression and therapeutic outcomes in a number of diseases including cancer and heart diseases. Therefore, there is a need for tools to measure the functional state of the molecular circadian clock and its downstream targets in patients. Moreover, the clock is a multi-dimensional stochastic oscillator and there are few tools for analysing it as a system. In this paper we consider the methodology behind TimeTeller, a machine learning tool that analyses the clock as a system and aims to estimate circadian clock function from a single transcriptome by modelling the multi-dimensional state of the clock. We demonstrate its potential for clock systems assessment by applying it to mouse, baboon and human microarray and RNA-seq data and show how to visualise and quantify the global structure of the clock, quantitatively stratify individual transcriptomic samples by clock dysfunction and globally compare clocks across individuals, conditions and tissues thus highlighting its potential relevance for advancing circadian medicine.

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