On random perturbations of dynamical systems and diffusions with a Brownian potential in dimension one
Let \(\beta \) be a one-dimensional Wiener process, \(W:\mathbb R_{+} \to \mathbb R\) a continuous function. A stochastic differential equation \[ dX_{t} = -\frac 12\nabla W(X_{t}) dt + k d\beta_{t},\quad X_0 = x_0,\tag{1} \] with reflection at 0 is studied. Results both on the long time behaviour of solution to (1) and on their asymptotics as \(k\to 0+\) are available in the literature provided the potential \(W\) of the drift is sufficiently smooth. In the paper under review, two important particular cases of non-smooth potentials \(W\) are considered. For \(x_{0}\in \mathbb R_{+}\), the solution to (1) is defined by \[ X_{t} = x_{0} + S^{-1} (\beta (T^{-1}(t))), \] \(S(x)=\int ^{x}_{0}\exp (k^{-1}W(z+ x_{0})) dz\), \(T(t)=k^{-1}\int ^{t}_{0} \exp ( 2k^{-1}W(S^{-1}(\beta (x))+x_{0})) dx\). First, (1) is treated as a small random perturbation of a dynamical system \(dy_{t} = -\frac 12\nabla V(y_{t}) dt\), \(y_{0}=x_{0}\), \(V\) being a smooth function such that \(\nabla V(x)<0\) for all \(x\in \mathbb R_{+}\) and \(W\equiv W^{k} = V + k^{a}B\). It is shown that if \(a\in ]0,2 [\), \(B\) is a \(b\)-Hölder continuous real function on \(\mathbb R_{+}\) and \(a+2b>2\), then \(X_{1}\) converges in probability to \(y_{1}\) as \(k\to 0+\). An example is given such that \(a=1\), \(B\) is \(b\)-Hölder continuous for any \(b<\frac 12\), nevertheless, the above convergence does not take place. Second, the case of \(W\) that is a sum of a deterministic part and a Brownian motion is dealt with. Let \(x_{0}=0\), \(k=1\), \(W= -V+B\), where \(V(x) = x^\alpha \) and \(B\) is a Wiener process independent of \(\beta \). Denote by \(Q\) the law of \(B\). It is proven that the process \(X_{t}\) has the same long time behaviour as the solution \(Y_{t}\) of a stochastic differential equation \(d Y_{t} = \frac {\alpha}2(Y_{t})^{\alpha -1} dt +d \beta_{t}\), \(Y_{0} = 0\), with reflection at 0 if \(\alpha \in ]1,2 [\). Namely, in this case \(t^{1/(\alpha -2)} X_{t}\) converges in probability to \((\alpha (2-\alpha)/2)^{1/(2-\alpha)}\) \(Q\)-almost surely as \(t\) goes to infinity. On the other hand, the behaviour of \(X_{t}\) and \(Y_{t}\) as \(t\to +\infty \) is shown to be different if \(\alpha \in ]\frac 12,1 [\).
stochastic differential equations with reflection
random perturbations of dynamical systems
scientific article; zbMATH DE number 1364500
scientific article