Kesten's bound for subexponential densities on the real line and its multi-dimensional analogues

We study the tail asymptotic of sub-exponential probability densities on the real line. Namely, we show that the n-fold convolution of a sub-exponential probability density on the real line is asymptotically equivalent to this density times n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular sub-exponential functions and use it to find an analogue of Kesten's bound for functions on R^d. The results are applied for the study of the fundamental solution to a nonlocal heat-equation

An Ikehara-type theorem for functions convergent to zero

We prove an analogue of the Ikehara theorem for positive non-increasing functions convergent to zero, generalising the results postulated in Diekmann, Kaper (1978, Nonlinear Anal. 2(6), 721--737) and Carr, Chmaj (2004, Proc. AMS 132(8), 2433--2439)

Front propagation in the non-local Fisher-KPP equation

Tkachov P. Front propagation in the non-local Fisher-KPP equation. Bielefeld: Universität Bielefeld; 2017

Doubly nonlocal Fisher–KPP equation: Speeds and uniqueness of traveling waves

We study traveling waves for a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions. We describe relations between speeds and asymptotic of profiles of traveling waves, and prove the uniqueness of the profiles up to shifts

Doubly nonlocal Fisher–KPP equation: front propagation

We study propagation over R^d of the solution to a doubly nonlocal reaction-diffusion equation of the Fisher-KPP-type with anisotropic kernels. We present both necessary and sufficient conditions which ensure linear in time propagation of the solution in a direction. For kernels with a finite exponential moment over R^d we prove front propagation in all directions for a general class of initial conditions decaying in all directions faster than any exponential function (that includes, for the first time in the literature about the considered type of equations, compactly supported initial conditions)

The hair-trigger effect for a class of nonlocal nonlinear equations

We prove the hair-trigger effect for a class of nonlocal nonlinear evolution equations on R^d which have only two constant stationary solutions, 0 and \theta>0. The effect consists in that the solution with an initial condition non identical to zero converges (when time goes to \infty) to \theta locally uniformly in R^d. We find also sufficient conditions for existence, uniqueness and comparison principle in the considered equations

Spatial growth processes with long range dispersion: microscopics, mesoscopics, and discrepancy in spread rate

We consider the speed of propagation of a {continuous-time continuous-space} branching random walk with the additional restriction that the birth rate at any spatial point cannot exceed $1$. The dispersion kernel is taken to have density that decays polynomially as $|x|^{- 2\alpha}$, $x \to \infty$. We show that if $\alpha > 2$, then the system spreads at a linear speed, {while for $\alpha \in (\frac 12 ,2]$ the spread is faster than linear}. We also consider the mesoscopic equation corresponding to the microscopic stochastic system. We show that in contrast to the microscopic process, the solution to the mesoscopic equation spreads exponentially fast for every $\alpha > \frac 12$.Comment: v2 update: A new result is added covering the case $alpha < 2$ for the microscopic model. Further remarks and heuristic comments are added, including connections to other models. Many minor changes are mad