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Register a new userOn regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients
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In this paper we solve a selection problem for multidimensional SDE $d X^varepsilon(t)=a(X^varepsilon(t)) d t+varepsilon sigma(X^varepsilon(t)), d W(t)$, where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H$. It is assumed that $X^varepsilon(0)=x^0in H$, the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H$, and the limit ODE $d X(t)=a(X(t)), d t$ does not have a unique solution. We show that if the drift pushes the solution away of $H$, then the limit process with certain probabilities selects some extreme solutions to the limit ODE. If the drift attracts the solution to $H$, then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.
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We study the limit behavior of differential equations with non-Lipschitz coefficients that are perturbed by a small self-similar noise. It is proved that the limiting process is equal to the maximal solution or minimal solution with certain probabilities $p_+$ and $p_-=1-p_+$, respectively. We propose a space-time transformation that reduces the investigation of the original problem to the study of the exact growth rate of a solution to a certain SDE with self-similar noise. This problem is interesting in itself. Moreover, the probabilities $p_+$ and $p_-$ coincide with probabilities that the solution of the transformed equation converges to $+infty$ or $-infty$ as $ttoinfty,$ respectively.
We study ODEs with vector fields given by general Schwartz distributions, and we show that if we perturb such an equation by adding an infinitely regularizing path, then it has a unique solution and it induces an infinitely smooth flow of diffeomorphisms. We also introduce a criterion under which the sample paths of a Gaussian process are infinitely regularizing, and we present two processes which satisfy our criterion. The results are based on the path-wise space-time regularity properties of local times, and solutions are constructed using the approach of Catellier-Gubinelli based on non-linear Young integrals.
We consider the $[0,1]$-valued solution $(u_{t,x}:tgeq 0, xin mathbb R)$ to the one dimensional stochastic reaction diffusion equation with Wright-Fisher noise [ partial_t u = partial_x^2 u + f(u) + epsilon sqrt{u(1-u)} dot W. ] Here, $W$ is a space-time white noise, $epsilon > 0$ is the noise strength, and $f$ is a continuous function on $[0,1]$ satisfying $sup_{zin [0,1]}|f(z)|/ sqrt{z(1-z)} < infty.$ We assume the initial data satisfies $1 - u_{0,-x} = u_{0,x} = 0$ for $x$ large enough. Recently, it was proved in (Comm. Math. Phys. 384 (2021), no. 2) that the front of $u_t$ propagates with a finite deterministic speed $V_{f,epsilon}$, and under slightly stronger conditions on $f$, the asymptotic behavior of $V_{f,epsilon}$ was derived as the noise strength $epsilon$ approaches $infty$. In this paper we complement the above result by obtaining the asymptotic behavior of $V_{f,epsilon}$ as the noise strength $epsilon$ approaches $0$: for a given $pin [1/2,1)$, if $f(z)$ is non-negative and is comparable to $z^p$ for sufficiently small $z$, then $V_{f,epsilon}$ is comparable to $epsilon^{-2frac{1-p}{1+p}}$ for sufficiently small $epsilon$.
In this paper we study zero-noise limits of $alpha -$stable noise perturbed ODEs which are driven by an irregular vector field $A$ with asymptotics $% A(x)sim overline{a}(frac{x}{leftvert xrightvert })leftvert xrightvert ^{beta -1}x$ at zero, where $overline{a}>0$ is a continuous function and $beta in (0,1)$. The results established in this article can be considered a generalization of those in the seminal works of Bafico cite% {Ba} and Bafico, Baldi cite{BB} to the multi-dimensional case. Our approach for proving these results is inspired by techniques in cite% {PP_self_similar} and based on the analysis of an SDE for $tlongrightarrow infty $, which is obtained through a transformation of the perturbed ODE.
The problem on identification of a limit of an ordinary differential equation with discontinuous drift that perturbed by a zero-noise is considered in multidimensional case. This problem is a classical subject of stochastic analysis. However the multidimensional case was poorly investigated. We assume that the drift coefficient has a jump discontinuity along a hyperplane and is Lipschitz continuous in the upper and lower half-spaces. It appears that the behavior of the limit process depends on signs of the normal component of the drift at the upper and lower half-spaces in a neighborhood of the hyperplane, all cases are considered.
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