Journal of Theoretical Probability

Abstract The free positive multiplicative Brownian motion $$(h_t)_{t\ge 0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> is the large N limit in non…

Abstract We study a single-server priority queue with a finite number of classes, in which the arrivals follow a fractional Poisson process of index $$\alpha \in (0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> and…

Abstract In the voter model, vertices of a graph (interpreted as voters) adopt one out of two opinions (0 and 1) and update their opinions at random times by copying the opinion of a neighbor chosen uniformly at random. This process is dual to a system of coalescing random walks. The duality implies that the set of stationary measures of the voter model on a graph is linked to the dynamics of the…

Abstract We use the correlation matrix of the generating distribution to determine the mixing time for random walks on the torus $$(\mathbb {Z}/q\mathbb {Z})^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mi>q</mml:mi> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> …

Abstract We prove a central limit theorem (CLT) for the number of joint orbits of random tuples of commuting permutations. In the uniform sampling case this generalizes the classic CLT of Goncharov for the number of cycles of a single random permutation. We also consider the case where tuples are weighted by a factor other than one, per joint orbit. We view this as an analogue of the Ewens measur…

Abstract We define a fractional Itô stochastic integral with respect to a randomly scaled fractional Brownian motion via an S -transform approach. We investigate the properties of this stochastic integral, prove an Itô formula for functions of such stochastic integrals and apply this Itô formula to the investigation of related generalized time-fractional evolution equations. We show that the cons…

Abstract We introduce and study renewal processes defined by means of extensions of the standard relaxation equation through “stretched” non-local operators (of order $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and with parameter $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> ). In a fir…

Abstract We consider the stochastic partial differential equation (SPDE) $$\begin{aligned} \partial _t u = \tfrac{1}{2} \partial ^2_x u + b(u) + \sigma (u) \dot{W}, \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:m…

We consider a certain class of Riemannian submersions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mi>N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>M</mml:mi></mml:mrow> </mml:math> and study lifted geodesic random walks from the base manifold <i>M</i> to the total manifold <i>N</i>. Under appropriate conditions on the distribution of the s…