A theorem on continuous dependence of solutions to stochastic evolution equations on coefficients is established, covering the classical averaging procedure for stochastic parabolic equations with rapidly oscillating both the drift and the diffusion term.

In this paper, we consider weak solutions to stochastic inclusions driven by a semimartingale and a martingale problem formulated for such inclusions. Using this we analyze compactness of the set of solutions. The paper extends some earlier results known for stochastic differential inclusions driven by a diffusion process.

Existence of a weak solution to the $n$-dimensional system of stochastic differential equations driven by a fractional Brownian motion with the Hurst parameter $H\in (0,1)\setminus \left\{\frac{1}{2}\right\}$ is shown for a time-dependent but state-independent diffusion and a drift that may by split into a regular part and a singular one which, however, satisfies the hypotheses of the Girsanov Theorem. In particular, a stochastic nonlinear oscillator driven by a fractional noise is considered.

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: $$\frac{\mathrm{d}}{\mathrm{d}t}u+\mathrm{i}\left(-\Delta +V\left(x\right)\right)u=\nu \left(\Delta u-{\gamma }_R{\left|u\right|}^{2p}u-\mathrm{i}{\gamma }_I{\left|u\right|}^{2q}u\right)+\sqrt{\nu }\eta (t,x).(*)$$ The force $\eta$ is white in time and smooth in $x$; the potential $V\left(x\right)$ is typical. We are concerned with the limiting, as $\nu \to 0$, behaviour of solutions on long time-intervals $0\le t\le {\nu }^{-1}T$, and with behaviour of these solutions under the double limit $t\to \infty$ and $\nu \to 0$. We show that these two limiting behaviours may be described in terms of solutions for thesystem of effective equations for(...

If the space $𝒬$ of quadratic forms in ${ℝ}^n$ is splitted in a direct sum ${𝒬}_1\oplus ...\oplus {𝒬}_k$ and if $X$ and $Y$ are independent random variables of ${ℝ}^n$, assume that there exist a real number $a$ such that $E\left(X\right|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E\left(q\left(X\right)\right|X+Y)=b_{i}q(X+Y)$ for any $q$ in ${𝒬}_i.$ We prove that this happens only when $k=2$, when ${ℝ}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

Wiener integral for the coordinate process is defined under the σ-finite measure unifying Brownian penalisations, which has been introduced by [Najnudel et al., C. R. Math. Acad. Sci. Paris345 (2007) 459–466] and [Najnudel et al., MSJ Memoirs19. Mathematical Society of Japan, Tokyo (2009)]. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study [K. Yano, J. Funct. Anal.258 (2010) 3492–3516] of Cameron-Martin formula for the...

Wiener integral for the coordinate process is defined under the σ-finite measure unifying Brownian penalisations, which has been introduced by [Najnudel et al., C. R. Math. Acad. Sci. Paris 345 (2007) 459–466] and [Najnudel et al., MSJ Memoirs 19. Mathematical Society of Japan, Tokyo (2009)]. Its decomposition before and after last exit time from 0 is studied. This study prepares for the author's recent study [K. Yano, J. Funct. Anal. 258 (2010) 3492–3516] of Cameron-Martin formula for the σ-finite...