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Register a new userOn a Selection Problem for Small Noise Perturbation in Multidimensional Case
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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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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.
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.
In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff] pointed out that it would be natural for $pi_t$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y={Y_s,sin[0,t]}$. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function $f$, there exists a continuous map $theta^f_t$, defined on the space of continuous paths $C([0,t],mathbb{R}^d)$ endowed with the uniform convergence topology such that $pi_t(f)=theta^f_t(Y)$, almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43-56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125-139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160-167], Kushner [Stochastics 3 (1979) 75-83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260-278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505-528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403-424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process $Y$ is lifted to the process $mathbf{Y}$ that consists of $Y$ and its corresponding L{e}vy area process, and we show that there exists a continuous map $theta_t^f$, defined on a suitably chosen space of H{o}lder continuous paths such that $pi_t(f)=theta_t^f(mathbf{Y})$, almost surely.
Here we will use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games on $Z^d$, $dge 3$ when the interaction kernel is finite range, symmetric, and has covariance matrix $sigma^2I$. The games we consider have payoff matrices of the form ${bf 1} + wG$ where ${bf 1}$ is matrix of all 1s and $w$ is small and positive. Since our population size $N=infty$, we call our selection small rather than weak which usually means $w =O(1/N)$. The key to studying these games is the fact that when the dynamics are suitably rescaled in space and time they convergence to solutions of a reaction diffusion equation (RDE). Inspired by work of Ohtsuki and Nowak and Tarnita et al we show that the reaction term is the replicator equation for a modified game matrix and the modifications of the game matrix depend on the interaction kernel only through the values of two or three simple probabilities for an associated coalescing random walk. Two strategy games lead to an RDE with a cubic nonlinearity, so we can describe the phase diagram completely. Three strategy games lead to a pair of coupled RDE, but using an idea from our earlier work, we are able to show that if there is a repelling function for the replicator equation for the modified game, then there is coexistence in the spatial game when selection is small. This enables us to prove coexistence in the spatial model in a wide variety of examples where the replicator equation of the modified game has an attracting equilibrium with all components positive. Using this result we are able to analyze the behavior of four evolutionary games that have recently been used in cancer modeling.
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.
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