The ODE method for some self-interacting diffusions on \(\mathbb R^d\)
Self-interacting processes, also known as reinforced processes or Brownian polymers [\textit{R. T. Durrett} and \textit{L. C. G. Rogers}, Probab. Theory Relat. Fields 92, No.~3, 337--349 (1992; Zbl 0767.60080)] are solutions of SDEs of the type \[ dX_t = dB_t + dt\int_0^t f(X_t-X_s)\,ds, \] where \(B_t\) is a standard \(d\)-dimensional Brownian motion. Depending on \(f\), the solution process is self-repelling or self-attracting. In the present paper, the following generalization of this SDE is considered: \[ \begin{aligned} dX_t &= dB_t - \left(\left.\nabla V(X_t)+\nabla_x \int_{\mathbb{R}^d} W(x,y)\,\mu_t(dy)\right|_{x=X_t}\right)\,dt,\\ d\mu_t &= \big(\delta_{X_t}-\mu_t\big)\,\frac{dt}{r+t}, \end{aligned} \] with \(X_0 = x\) and \(\mu_0=\mu\). Here, \(V:\mathbb{R}^d\to[1,\infty)\), \(W:\mathbb{R}^d\times\mathbb{R}^d\to[0,\infty)\) are the confinement, resp., interaction potentials and \(\mu\) is a bounded Borel measure on \(\mathbb{R}^d\). The author studies the long-time behaviour of the solution. Typically, it is assumed that \(V,W\) are twice continuously differentiable, \(\nabla V\) is Lipschitz, \(W(x,y)\) is controlled by \(V(x)+V(y)\), \(\nabla^2_{xx}W(x,y)\) and \(\nabla_{x}W(x,y)\) are controlled by \(V(x)+W(x,y)\), \((|\nabla V(x)|^2 + 2\nabla V(x)\cdot\nabla_xW(x,y))/(V(x)+W(x,y))\) tends (uniformly in \(y\)) to \(+\infty\) as \(|x|\to\infty\). Finally, the following ``curvature'' assumptions are needed: \[ x\cdot \nabla_x(V(x)+W(x,y)) \geq a|x|^{2\delta}-\alpha,\qquad \xi\cdot\nabla_{xx}^2(V(x)+W(x,y)\xi \geq M|\xi|^2. \] Under these assumptions, the following results are shown: The set of accumulation points of \((\mu_t)_{t\geq 0}\) is weakly compact, invariant by the flow generated by \(\dot\mu = \Pi(\mu)-\mu\) and admits no other subattractor than itself. \(\Pi(\mu)\) is given by \(\Pi(\mu)=\exp\left(-2V -2\int W(x,y)\,\mu(dy)\right)/Z(\mu)\) with a suitable normalization constant \(Z(\mu)\). If \(W\) is symmetric, then the limit set is a connected subset of the set of fixed points of the map \(\mu\mapsto\Pi(\mu)\). If \(V=V(\rho)\) is rotationally invariant and \(W(x,y)=x\cdot R_\theta y\) where \(R_\theta\) is the rotation by \(\theta\) degrees, then conditions are given such that either \(\mu_t\) converges weakly to (an explicitly given, deterministic) probability measure \(\gamma\) or, depending on \(\theta=\pi\) or \(\theta\neq\pi\), \(\mu_t\) converges weakly to a random measure \(\mu_\infty\) or circles around, respectively.
scientific article; zbMATH DE number 5795077
scientific article