Related papers: Linear stochastic equations in the critical case

Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to description of the long time time asymptotics of two critical cases of these equations, when the division…

We present a condition for a stochastic differential equation dX_{t}={\mu}(t,X_{t})dt+{\sigma}(t,X_{t})dB_{t} to have a unique functional solution of the form Z(t,B_{t}). The condition expresses a relation between {\mu} and {\sigma}. A…

The `archetypal' equation with rescaling is given by $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(\mathrm{d}a,\mathrm{d}b)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, with…

We establish the existence and nonexistence of entire solutions to a semilinear elliptic problem whose nonlinearity is the critical power multiplied by a function that takes the value 1 in an open bounded region and the value -1 in its…

We consider a non-homogeneous nonlinear stochastic difference equation X_{n+1} = X_n (1 + f(X_n)\xi_{n+1}) + S_n, and its important special case X_{n+1} = X_n (1 + \xi_{n+1}) + S_n, both with initial value X_0, non-random decaying free…

Several nonlinear stochastic differential equations have been proposed in connection with self-organized critical phenomena. Due to the threshold condition involved in its dynamic evolution an infinite number of nonlinearities arises in a…

This short survey article stems from recent progress on critical cases of stochastic evolution equations in variational formulation with additive, multiplicative or gradient noises. Typical examples appear as the limit cases of the…

We consider linear n-th order stochastic differential equations on [0,1], with linear boundary conditions supported by a finite subset of [0,1]. We study some features of the solution to these problems, and especially its conditional…

We study asymptotic behaviour of stochastic approximation procedures with three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function.…

For i.i.d. random vectors $(M_{1},Q_{1}),(M_{2},Q_{2}),\ldots$ such that $M>0$ a.s., $Q\geq 0$ a.s. and $\mathbb{P}(Q=0)<1$, the random difference equation $X_{n}=M_{n}X_{n-1}+Q_{n}$, $n=1,2,\ldots$, is studied in the critical case when the…

Let $d \ge 2$. In this paper, we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dS_{t}+b(s+t, X_{t})dt, \quad X_{0}=x, \] where $(s,x)\in \mathbb{R}_+ \times \mathbb{R}^{d}$ is the initial starting…

In this paper we show under weak assumptions that for $R\stackrel{d}{=}1+M_1+M_1M_2+\ldots$, where $P(M\in[0,1])=1$ and $M_i$ are independent copies of $M$, we have $\ln P(R>x)\sim C\, x\ln P(M>1-\frac1x)$ as $x\to\infty$. The constant $C$…

Given the matrix equation ${\bf A X} + {\bf X B} + f({\bf X }) {\bf C} ={\bf D}$ in the unknown $n\times m$ matrix ${\bf X }$, we analyze existence and uniqueness conditions, together with computational solution strategies for $f \,:…

In this paper we consider the existence of positive solutions for a singular elliptic problem involving an asymtotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison…

Consider the multidimensional SDE $\mathrm d X(t) = a(X(t))\mathrm d t + b(X(t))\mathrm d W(t).$ We study the asymptotic behavior of its solution $X(t)$ as $t \to \infty$, namely, we study sufficient conditions of transience of its solution…

Let $\{X_i\}$ be a sequence of independent identically distributed random variables with an intermediate regularly varying (IR) right tail $\bar{F}$. Let $(N, C_1, ..., C_N)$ be a nonnegative random vector independent of the $\{X_i\}$ with…

Models of self-organized criticality, which can be described as singular diffusions with or without (multiplicative) Wiener forcing term (as e.g. the Bak/Tang/Wiesenfeld- and Zhang-models), are analyzed. Existence and uniqueness of…

In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution,…

In this paper, we study the existence of nonnegative solutions for a class of multivalued $(p,N)$-Laplace problems having discontinuous nonlinearity with critical exponential growth in $\mathbb{R}^N$. To demonstrate the existence results,…

Approximations to sums of stationary and ergodic sequences by martingales are investigated. Necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained. It is first…